THERMODYNAMICS APPLIED TO EARTH MATERIALS

James Connolly, NW/E/78.1, 2-7804 [email protected]

These notes and other materials for this course are available at: www.perplex.ethz.ch/thermo_course

SOME PRELIMINARIES

MAPLE TECHNICAL ISSUES

Installing Maple - Expand the Mac (OSX), LINUX or PC (Windows) zip archive in www.perplex.ethz.ch\thermo_course\maple. Follow the instructions in the Maple installation text file! N.B. if you use Windows, then I recommend you install the 32 bit (x86) version regardless of whether you have 64 bit machine, the “Classic” Maple (cwmaple.exe) is only available in the x86 version. I have not checked myself, but I have been told that the “Classic” Maple is not available in the LINUX/OSX versions.

Display settings – if you wish to keep the format of your scripts similar to those on the course web-page and you are not using “Classic” Maple, then: under Tools->Options->Display set Input display to Maple Notation and Typsetting level to Maple Standard, these minimize complications caused by invisible formatting characters; and under Tools->Options->Interface set Default format for new worksheets to Worksheet.

Maple tutorial – this course does not require much skill with Maple and I will attempt to provide you with the essential syntax. However, if you want a better understanding of its syntax and capabilities, then the tutorial present in www.perplex.ethz.ch\thermo_course\maple may be useful.

Classic vs Java scripts – the Maple scripts prepared for this course were prepared in the Classic Maple version (file type *.mws). These scripts can be used in the Java version of Maple, but if you do so please save them as Java version files (file type *.mw), Classic version files that have been manipulated in the Java version become illegible if saved as Classic version files.

Running Classic Maple on Windows – The Classic version can be accessed from Start-> All programs -> Maple 18 -> Classic Worksheet Maple 18, or simply create a direct shortcut to the Classic version, which is usually at C:\Program Files\Maple 18\bin.win\cwmaple.exe.

A word to the wise – Save your frequently; do not rely on the auto-backup function, especially not if you are using the Java version of Maple.

PROBLEM SETS

The problem sets may be solved as a group effort. If you solve a problem set as a group, then please submit one copy of the solved problems for the entire group. In the first half of the semester, the problem sets are due within two weeks of their assignment.

If you do a problem with Maple, then I am happy to accept an e-mail copy of your Maple script rather than a hand-written answer. However, be sure that the script can be executed sequentially, to verify this, type “restart;” or press the Maple menu restart button and then step through the script. Just as in a handwritten problem set, maple scripts should be documented by explanatory comments, preferably inserted as text (the “T” menu button). In particular, it is essential that the units be given for any dimensional results. Some firewalls do not allow Maple scripts as attachments; therefore, it is best to enclose the Maple script in a zip file.

If necessary, then I will offer a tutorial for the more complicated problem sets, probably on Thursday’s, 14:15- 16:15. The penalty for solving problems in the tutorial is that the note for a perfect solution is 5. If you attend a tutorial, then please indicate this on your problem set.

THE FIRST LECTURE AND THE CORRESPONDING PROBLEM SET

To illustrate the purpose and goals of this course, in the first lecture I will begin at the end by considering how diagrams are constructed from thermodynamic models. There is no script for this lecture, but a short review of the basic concepts is in p 12-16 of www.perplex.ethz.ch\thermo_course\chapter_0\potenza.pdf. The problem set for this lecture consists of the four problems outlined in the maple scripts at ...\thermo_course\chapter_0.

1: THERMODYNAMIC PROCESSES AND follows that the integral of a along a LAWS closed path, i.e., a path that returns the system to its “Thermodynamics is the science of the impossible. initial state must be zero. Thus an alternative It enables you to tell with certainty what cannot statement of the first law is happen. Thermodynamics is noncommittal about ∫ d0U = . 0.2 the things that are possible. Thermodynamics is at But to define a closed path you must first define its best when nothing can happen, a condition state. Early thermodynamicists understood that a called equilibrium. The concept of equilibrium has system could only do work by changing its been fruitfully extended to reversible processes. or mass, therefore the properties V, M1, …, Mk are Here everything is impossible except one very state functions that define mechanical and chemical specific process and even this process is on the state. Furthermore, (T) was assumed verge of being impossible.” to be a state function although the relationship – An anonymous, slightly inaccurate, wit. between temperature and was not entirely clear. is a theory developed to predict and understand the consequences of If a system loses heat, it can be considered to have processes. In principle it can be used to predict any done positive thermal work, accordingly Eq 0.1 can process, but it is mainly used to understand the be written more compactly processes of heat and mass transfer and isostatic k +2 dUW+= d0. dilation (i.e., a change of volume (V) in an ∑ i i=1 isotropic stress field). This restriction is implicit in Early thermodynamicists were concerned primarily the remainder of these notes. In this context, energy with the construction of steam engines, which is referred to as the (U) to convert heat to mechanical work; thus they had the emphasize that it is only the energy accessible prejudiced view that mechanical work was useful through the processes of interest. For a body of and that thermal work was useless except to the matter, i.e., a system, composed of k independently extent that it could be converted to useful work. variable kinds of mass (M1, …, Mk), there are thus While this attitude is no longer appropriate, it is k+2 independent processes. Intuitively one expects sometimes useful to isolate heat as a special kind of that any of these processes will change the energy work in order to understand the perspective of the of a system on which they operate. The first law of early thermodynamicists who were responsible for thermodynamics is a formal statement of this thermodynamic theory. intuition. Specifically, the first law states that energy of a system may only change if the system A MECHANICAL ANALOGY does work on its environment. Moreover, the first Although mechanics is more complex than law states that energy (U) is conservative, i.e., k +2 chemical thermodynamics in that it involves kinetic

ddUQ−+∑ d Wi = 0 0.1 energy and vectors, it is familiar via elementary i=2 physics. Therefore it is helpful to introduce a where Q is the heat gained by the system and Wi is mechanical analogy to Eq 0.1, specifically the the work done by the system in the ith process, potential energy U of a stationary ball as a function where d is used to indicate an inexact (path of its horizontal position x along a frictionless 1- dependent) . The sign convention in Eq differential dimensional surface with height H(x) in a gravita- 0.1 is by no means universal; in particular, in many tional field (Fig 1.1). The potential energy of the textbooks W represents work done on the system in ball is proportional to the height H of the surface, which case the work and heat differentials have the i.e., U ∝ H(x) and by the first law (Eq 0.1) same sign. ddUW+= 0 0.3

the ball can do work if it lowers its potential energy Properties that are not path dependent are said to be by moving. In mechanics this work is state functions and the importance of the first law ddW= fx 0.4 is that it establishes energy as a state function. It where f is the force resisting the movement dx, thus

1

= − ddU fx 0.5 U In this particular case, the work differential happens to be exact because there is only one process (path) f=-∂ U/∂θx= possible.

Eqs 0.3 and 0.4 can be considered, although it is not conventional, to define force as the negative spatial gradient in potential energy dU f ≡− . 0.6 x dx Fig 1.1 Energy of a stationary ball along a 1-d The ball will have no tendency to move if this surface, the force that would induce the ball to gradient is zero, thus the gradient can be viewed as roll is the negative of the gradient in the the potential for the only process (displacement of energy along the surface. A spatial gradient in energy is usually referred to as a potential. The the ball) possible in the system in the absence of potential is a measure of the amount of work external influence. For this reason spatial gradients that can be extracted from the ball by a displacement (dx ) in the along the surface. If in energy are often referred to as potentials for the system consists of only one ball it can be in work equilibrium only if the potential of the ball is dU zero. However, if the system of interest θ≡ consists of two balls, it will be in mechanical dx equilibrium when the potential of each ball is thus forces may be regarded as the negative of equal in magnitude (but opposite in sign). potentials for work and 0.5 may also be written = θ . 0.7 ddUx process that changes a property Ψ, is that the The topographic highs and lows of the surface (Fig potential 1.1) are the only points at which a stationary ball is ∂U θ≡ 0.8 in mechanical equilibrium, i.e., in force balance. ∂Ψ We may further characterize these equilibria as for the process is uniform throughout the system. being stable or unstable if they correspond to maxima or minima, it being apparent that an Integrating Eq 0.5 at constant force equilibrium at a maximum is unstable with respect Ux= θ 0.9 to small perturbations. and the differential of Eq 0.9 is then dU=θ+θ dd xx . 0.10 For the case of a single ball we have the trivial Comparison of 0.5 and 0.10, leads to the conclusion result that the condition for equilibrium, i.e., a state that the exactness of 0.5 requires where no processes are possible is that θ = 0. At xd0θ= , 0.11 this point the example is not especially relevant to which is the mathematical statement that changing thermodynamics because no processes are possible the potential of a system does not affect the systems within the system. Therefore, consider now a energy if the system does no work. system consisting of two balls that are constrained to remain in contact. In this system, we have the THERMODYNAMIC WORK possibility of an internal process in that one ball can Just as in the foregoing mechanical example where push the other. In this case, the condition for the work differential can be written variously equilibrium is not that the balls are at the same dU dW= − d x = −θ dd x = fx. height, but rather that the potentials of the balls dx must be equal in magnitude and opposite in sign. The work differentials in Eq 0.1 can be written The necessity for the opposing signs is that in kk++22∂U k+2 dW =− d=Ψ − θΨ d 0.12 mechanics forces have direction and this is the only ∑∑i  i ∑ ii ii=22= ∂Ψ i=2 i Q,Ψ ≠ real distinction to thermodynamics where the forces ji or potentials are scalar. Thus in thermodynamics a where the work differentials are now inexact requirement for equilibrium with respect to a because more than one process is possible and the thermodynamic potential θi is defined as the partial

2

derivative of energy with respect to property Ψi for PV = nRT. 0.16 an adiabatic system (dQ = 0) at constant Ψ ji≠ . In Eq Since the system is pure and at constant mass, its 0.13 the potentials are differential coefficients P(bar) defined, at least in the adiabatic case, entirely in terms of state functions, therefore it follows that the potentials are also state functions. Using explicit nR=1 J/K notation, the differential coefficient on volume is 4 known from mechanics to be , i.e., the isochoric (W =0) “force” that resists dilational work, thus 4 cooling (Q4<0 ) dWV = PdV 0.14 and pressure can be defined thermodynamically as the negative of the potential for volume adiabatic (Q3 =0)

∂U compression (W3 <0) isochoric (W2 =0) P ≡− . heating (Q >0 ) ∂ 2 V QM, ik M 0 adiabatic (Q =0) In contrast, chemical work is entirely a thermody- 1 decompression (W1 >0) namic concept and defined in terms of a potential 1 10 for chemical work V(J/bar) ∂U µ≡ i  Fig 1.2 Illustration of a closed path for a fixed mass of ∂M i QV,, Mji≠ an initially at V0=1 J/bar and T0=1 K (P0=1 such that bar). The path consists of 4 segments: adiabatic dW= µ dM . decompression to V1=10 J/bar, isochoric (constant Mii volume) heating back to T , adiabatic compression Making use of these definitions, Eq 0.1 is 0 back to V , and isochoric cooling back to T . The k 0 0 mechanical work done along any segment is the area ddU= Q − PV d +µ∑ ii d M 0.15 i=1 under the path from its initial to its final volume The peculiar mixture of forces and potentials coordinate, thus the gas does no work in the isochoric reflects the historical fact the dilational work term segments and the amount of work done in the adiabatic segments is not equal. This partitioning had been defined in mechanics in terms of pressure, demonstrates that work and heat are path dependent whereas chemical work was not defined prior to the functions, however the first law requires that these work of Gibbs on chemical thermodynamics. functions sum to zero for closed path as the total energy of the gas must be conserved. EXACT DIFFERENTIALS After reading thermodynamic textbooks one could energy is a function of two inexact functions, heat easily be forgiven for concluding that the true and dilational work dU= dd QW − . statement of the 1st law is that energy is an exact V first order homogeneous differential, rather than Now let us devise a set of paths involving work and that these mathematical qualities are a consequence heat that ultimately leave the gas in its original of energy conservation. I suspect this confusion, of state. First, we thermally insulate the system and which I was a victim, results from emphasis on allow it to expand. During this part of the cycle the mathematical rather than physical concepts, so to system can neither gain nor lose heat (i.e., it is adiabatic, dQ ≡ 0) and the pressure of the gas at allay this confusion here let us consider illustrations of the inexactness (i.e., path dependence) of heat any point can be shown to be 5/3 and work and the mathematical requirement for P= PVV00( ) , 0.17 exactness (we leave an explanation of “first order where the zero subscripts denote initial conditions. homogeneity” for later). From Eqs 0.14 and 0.17, the change in energy and work done in this segment is

V1 For the first illustration (Fig 1.2), consider a system 5/3 ∆=−=−U W PVVd V 0.18 composed of a constant amount of a pure monatom- 1 1∫ 00( ) V0 ic ideal gas for which

3

which is negative of the area under the path in a V- Applying this approximation successively for the P diagram (Fig 1.2). From Eq 0.16 the gas must ∂f ∂f ∂f ′ f2′ = f0 + δy + δx cool during this work to f1 = f0 + δy ∂ ∂y ∂x 0,δy y 0,0 0,0 PV11 T = ∂f ∂f 1 f = f + δx + δy nR 2 0 ∂ ∂ x 0,0 y δx,0 as a consequence of its expansion. Next, we remove the insulation and heat the gas at constant volume back to its original temperature T0. As the gas does not change its volume it does no mechanical work, y but its pressure increases to f0 ∂f f = f + δx 1 0 ∂ nRT0 x x 0,0 P2 = . V1 = ′ We now adiabatically compress the gas back to its if f2 f2 then ∂f ∂f ∂f ∂f original volume and restore the gas to its original f + δx + δy = f + δy + δx 0 δx δy 0 ∂y ∂x state by isochoric (constant volume) cooling. Thus, 0,0 δx,0 0,0 0,δy ∂f ∂f ∂f ∂f we have a closed path for which energy conserva- − − ∂y ∂y ∂x ∂x tion requires δx,0 0,0 = 0,δy 0,0 δx δy ∆= = UU∫ d0  ∂f ∂f ∂f ∂f    − −  ∂ ∂ 2 2  y δx,0 y 0,0 ∂x 0,δy ∂x 0,0  ∂ f ∂ f yet because the V-P paths during compression and lim = = = δx,δy→0  δx δy  ∂y∂x ∂x∂y   decompression are different it is evident that     ∆=Wd0 W = WW + ≠, ∫ V 12 Fig 1.3 Proof that the cross derivatives of two variable i.e., that work is not a state function in that the net function f(x,y) must be equal if the differential of the amount of work done is dependent on the amount function is exact. If the differential is exact, then the of heating done during the isochoric segments of value of the function must be independent of path, the path. therefore consider two alternative paths involving finite increments of x and y. The value of the function The path independence of exact differentials after each increment is approximated by a first order attracted much attention in science and mathemat- Taylor series. If the value of the function is the same ics during the 1700’s because the existence of such at path endpoints, then variation in the x-derivative perfection was construed as evidence for the hand with a finite y increment must equal the change in the of god. To illustrate the mathematical requirement y-derivative with a finite x increment; in the limit of for exactness, consider the total differential of infinitesimal increments this equality reduces to f(x,y) Euler’s Criterion. ∂∂ff df= dd xy + . 0.19 two possible paths, the result can only be independ- ∂∂xy ent of path if If this differential is exact, its integral must be the ∂∂ff∂∂ff same regardless of path, so let us consider the two − − ∂∂yyδ ∂∂xxδ alternate paths defined by small, but finite, x,0 0,0 = 0,y 0,0 . δδxy increments δx and δy (Fig 1.2). Because the The Taylor series approximation is exact in the increments are small, we approximate f by a first infinitesimal limit (δx, δy → 0) in which case we order Taylor series as ∂f obtain the result =+δ 22 ffδx,0 0,0 x ∂∂ff ∂x 0,0 = ∂∂yx ∂∂ xy or that the cross derivatives of an exact function must ∂f ffδ =+δ y be independent of the order of differentiation. This 0,y 0,0 ∂y 0,0 result is known Euler’s criterion (the Euler where the subscripts indicate the x-y coordinates at formerly on the 10 CHF note) for exactness. which the function, or its derivatives are evaluated. Ordinarily in thermodynamics the partial deriva-

4

tives that are differential coefficients of a differen- The equalities in 0.21 and 0.22 apply for any real tial equation are given special names, e.g., process and the equalities correspond to the ∂∂=µUM or ∂UV ∂=− P as in Eq 0.15. theoretical limit that the processes of are non- Adopting this shorthand Eq 0.19 is dissipative, in thermodynamics this type of df= px dd + qy behavior is said to be reversible. and Euler’s criterion yields ∂∂p2 fq ∂ Unlike the macroscopic properties energy, mass = = . 0.20 ∂∂∂∂yxyx and volume, is unique to thermodynamics if f is exact. and therefore non-intuitive. Entropy is best understood as a macroscopic measure of disorder. From a thermodynamic perspective, the important In the specific case of heat transfer this disorder feature of an is that its integrals may simply be an increase in atomic vibrational can be evaluated given only knowledge of initial energy. From the first law, dissipation cannot affect and final states, i.e., the energy of a system, rather it is a lowering of the xyff, quality of the energy. This reduction in quality is = − ≡∆ ∫ dfffi f f manifest by an increase in disorder. In this regard, xy, ii the second law is pessimistic in that it implies that = + −+ ( pf x f q f y f) ( px ii qy ii) any real process increases the net disorder of the Where, if necessary, the differential coefficients universe. A simple illustration is the heating of a may be evaluated as explicit functions of x and y, system at constant mass and volume, if the heating e.g., is reversible ∂f ∆=Q T S − TS p = . f,, rev f rev i i i ∂x xyii, where the subscripts i and f designate the initial and From the foregoing considerations, it is apparent final states; if instead the same amount of heat is that it would be desirable to decompose the inexact added irreversibly heat differential in Eq 0.15 into the product of an ∆

dU≤ TS dd − PV +µ∑ ii d M 0.22 Another illustration of dissipation is provided by i=1 the mechanical analogy mentioned earlier (Fig 1.1).

5

Previously we discounted the fact that the "work" temperature is uniform, it is defined by a reversible the ball does is translation (a process that is not variation in entropy at constant mass and volume, usually considered in thermodynamics), i.e., the i.e., ∂U balls potential energy (U) is converted into kinetic T ≡ . energy in which case the total energy of the ball is ∂S EUv= + 2 2 . 0.23 As the same argument can be made for any of the Taking 0.23 as truth together with the first law, potentials in 0.22, it follows that for any reversible implies that if we place a ball above a minimum, process the potentials must be uniform throughout then the ball will oscillate back and forth across the the system. Thus, for a system to be heated minimum forever (a perpetual motion machine). reversibly it is necessary that temperature remain The second law says that this, and therefore 0.23 uniform during heating. Since rates of heat cannot be true. Rather Eq 0.23 but must be conduction, mass transport and dilation are finite, extended to this requirement leads to the conclusion that E=++ U v2 2 TS 0.24 reversible processes must be infinitely slow and manifest by infinitesimal potential gradients. In this and that every motion of the ball must increase its regard, the form of 0.22 is deceptive as the entropy (temperature), an effect usually attributed coefficients cannot be defined for an irreversible to friction but which may also arise as a conse- process. For this reason, a restricted statement of quence of deformation. Consequently the potentials the first and second laws known as the Gibbs for potential (height) and kinetic (velocity) differential processes will dissipate until the ball settles to the k minimum. Of course, this process the ball dU= TS dd − PV +µ∑ ii d M 0.25 raising the potential for thermal work. i=1 is preferred for reversible processes. Although the designation of energy derivatives as potentials follows Gibbs, it is nonetheless uncon- The Gibbs differential defines the fundamental ventional in thermodynamic literature. More variables of thermodynamics as entropy, volume commonly U itself, and free energy functions and mass, i.e., U(S, V, M1, …, Mk). Knowledge of derived from U that will be introduced later, are these variables must provide a complete definition of any ; in other words, any referred to as thermodynamic potential functions. The logic of this usage is that the functions measure attribute of a system that cannot be defined from the total energy that can be extracted from a these variables is not a thermodynamic attribute of system. For example, in the previous mechanical the system. All other thermodynamic properties are analogy (Fig 1.1) U is a direct measure of the derived, and specifically ∂U height of the ball, and the possible variation in the T ≡ ∂S height along the surface is therefore a direct ∂U measure of the potential of the ball to do work. In −≡P 0.26 ∂V contrast, potentials as defined here (0.8) are a ∂U measure of the potential for individual processes µ≡i ∂M i within a system. Thus, the thermodynamic potentials that we are

accustomed to thinking of as independent variables The thermodynamic potentials {T, −P, µ1, …, µk are dependent parameters in the context of the } corresponding to the differential coefficients of Gibbs differential, i.e., T(S, V, M1, …, Mk), etc. As 0.22 can only be defined for a system in a non- the Gibbs differential is simply a summation of arbitrary way if they are uniform in all parts of the reversible work differentials, it is analogous to the system. For example, imagine a two part system mechanical system discussed earlier. This analogy with distinct . What is the temperature can be made more succinct by introducing the of the entire system? There are numerous ways we general notation may compute an average temperature, and thus k +2 there is no unique answer. In contrast, if the ddU =∑ θΨii 0.27 i=1

6

where defined. Caratheodory’s statement of the 0th law, ∂U which was designated as such by Fowler (1956), θ≡i . 0.28 ∂Ψi was that if two bodies are in thermal equilibrium Eq 0.28 defines a special relationship between each with a third body, then they are in thermal equilib- fundamental variable and its potentials, properties rium with each other. The 0th law is necessary for related in this manner are said to be conjugate, e.g., legalistic arguments about temperature, but of little

T is conjugate to S, θi is conjugate Ψi. Just as in practical importance because it is usually implicitly mechanical systems, where force balance defines assumed. the conditions of a mechanical equilibrium, the uniformity of thermodynamic potentials define The third law, or Nernst’s Law, is of greater conditions of thermodynamic equilibrium, a state practical importance because it establishes an in which no reversible processes may take place in absolute scale for entropy. A weak form of this law a system without external influence. Equilibrium is the statement that the entropy of a system at zero processes are thus processes that occur entirely in Kelvin is zero. This form is “weak” because it is response to external influence since any system strictly correct only for a perfect crystal, but the described by 0.25 must be in internal equilibrium statement is adequate for present purposes. A more (Fig 1.2). robust statement is that no real processes are possible at absolute zero. Ergo, from the second As in the case of the mechanical system (0.9), 0.25 law if no real processes are possible, the entropy may be integrated holding the potentials constant to change of any process must be zero at absolute obtain zero. k U= TS − PV +µ M 0.29 ∑ ii LTERNATIVE TATEMENTS i=1 A S or http://en.wikiquote.org/wiki/Thermodynamics k +2 U = θΨ 0.30 ∑ ii EADING ATERIAL i=1 R M and as the total differential of 0.30 is I do not recommend any particular text for this kk++22 course because I have only been able to convince ddU =∑∑ θi Ψ+ i Ψ ii d θ 0.31 ii=11= myself that I understand anything of thermodynam- comparison of 0.31 and 0.27, requires ics by looking at many different texts. In this k +2 regard, texts can be broadly categorized as theoreti- ∑Ψiid0 θ= 0.32 i=1 cal books that explain principles but presume or in non-general form practical applications are trivial; or as "how to" k books that explain the "trivia" but neglect or restrict STd− VP d + M d0 µ= 0.33 ∑ ii theory in a way that hinders a general understand- i=1 Eq 0.33 is known as the Gibbs-Duhem relation ing. A few examples of each genre are: and has the profound implication that only k+1 Basic principles potentials of an equilibrium system are capable of independent variation. Thus the variation of one Thermodynamics. N. A. Gokcen, R. G. Reddy thermodynamic potential during a reversible Thermodynamics and an introduction to thermosta- process can always be expressed as a function of tistics. H. B. Callen the remaining potentials. Thermodynamics : an advanced treatment for chemists and physicists. E. A. Guggenheim SECOND THOUGHTS: THE 0TH AND 3RD LAWS Twenty lectures on thermodynamics. H.A. Buch- It is often difficult to distinguish what is assumed in dahl thermodynamics from what is axiomatic. The 0th How to law is an example of this, as it was only realized by Caratheodory (1909) after the second law was Applied mineralogical thermodynamics: selected postulated that temperature was not formally topics. N. D. Chatterjee

7

2 12 The elements of physical chemistry. Peter Atkins ddf3 = PVV + d P is exact, then how does V (an excellent P-Chem text) depend on P? c) Write the total differential (Eq 23 The Principles of chemical equilibrium: with 1.19) of f4 = yx + y, is it exact? d) f(x,y) is applications in chemistry and chemical engi- single-valued (i.e., formally a function of x and y), neering. Kenneth George Denbigh is its differential exact? Note: it is easiest to solve Thermodynamics of natural systems. Greg Ander- a-c analytically; it is possible, but exceedingly son (this book is geologically oriented, but best painful, to solve the problems as outlined in Fig for hydrothermal geochemistry) 1.3; and it is nearly impossible to solve the Geochemical thermodynamics. Darrell Kirk problems in Maple (i.e., do not waste your time Nordstrom, James L. Munoz (easy reader text) trying).

PROBLEMS 1.4) A constant mass system is composed of an In this course, if you do a problem with Maple, then ideal gas (P = MRT/V) and is in thermal equilibri- I am happy to accept an e-mail copy of your Maple um with an infinite heat reservoir at temperature T. script rather than a hand-written answer. Because If the system does work by expanding, then heat some firewalls do not allow Maple scripts, it is best must flow into the system (in response to infinites- to enclose scripts in a zip file or send a pdf of the imal temperature gradients) to maintain thermal script. equilibrium with the reservoir. For this process we may define the differential of a function known as 2 1.1) For the function f() x= ax ++ bx c use Maple the A, such that dA ≡ dU – (preferably the classic version) to a) solve for the dQ = −PdV + µdM, which represents the maximum roots of the function with the solve command, b) amount of work that can be extracted from the integrate and differentiate the function using the int system. Evaluate the change in the Helmholtz and diff commands, and c) plot the function, its energy of the system if it expands isothermally definite integral from zero to x, and its derivative in from V0 to 10 V0 by integrating −PdV + µdM. Is this a single plot with the plot and subs commands for result identical to ∆A the obtained by evaluating the a=1, b=2, c=3. difference in the integrated form of the Helmoltz energy (A = −PV + µM) between the initial and 1.2) Evaluate the amount of work done by the ideal final state of system (the Gibbs-Duhem relation, Eq gas during the closed cycle illustrated by Fig 1.2. 0.33, may be helpful in this regard)? Use the First Law to compute the heat gained by the gas. Note: to evaluate a definite integral in NOTE: a dimensional numerical result is not an Maple use the command “int(1/x,x=x0..x1);”. In answer without units. You should always indicate some cases Maple will tell you that it cannot appropriate units for answers to problems in this evaluate the integral because it is unable to course. You can make Maple work with units, but determine if 0 is between x0 and x1, in such cases unless you are fanatical I do not recommend this. modify the command using the assuming qualifier as in: “int(1/x,x=x0..x1) assuming x0>0, x1>0;”. Beware: variable ranges are indicated in Maple by two values separated by two periods, e.g., x0..x1, if you use actual rather than symbolic numbers for such a range and inadvertently separate the numbers by an ellipsis (“…”) Maple interprets the third dot as a decimal point, e.g., Maple interprets the range 1…10 to be from 1 to 0.1.

1.3) a) Use Euler's criterion (Eq 0.20) to determine which of the differentials df1 = PV dd − VPand

df2 = PV dd + VPare exact. b) If

8

2: THERMODYNAMIC VARIABLES AND HETEROGENEOUS SYSTEMS Thermodynamic Variables

The variables of the Gibbs differential {S, V, M1, Intensive

…, Mk} characterize all thermodynamic processes Potentials/ Specific/ Extensive of interest. As such they are the determinative −forces Molar/ parameters of any real system, i.e., the parameters Compositional that we, as masters of the universe, control. Any Thermal S T s one of these properties may be used as a measure of Mechanical V −P v the size of a system. It follows that the properties Chemical Mi µi m

{U, S, V, M1,…, Mk} must vary linearly with the General Ψi θi ψi size or extent, such properties are said to be extensive. Thus, if the size of the system is changed A process that changes only the size of a system, by a factor q, then changes all the systems extensive attributes by a

qU= U( qS, qV , qM1 ,, qM k ) . 2.1 common factor. Any other process must change the relative proportions of the extensive attributes; such A function with this character is a first order a process changes the state of the system. Thus homogeneous function in all its variables. Euler’s state is defined by the relative proportions of theorem on homogeneous functions states that if f is extensive properties and is independent of extent. a homogeneous function of order o in x = {x1, …, An attribute of matter that is independent of size is xn} then fo=xx ⋅∇ f( ) . 2.2 said to be intensive. As it is only possible to form k+1 independent ratios from the k+2 independent Thus for U(S, V, M1,…, Mk) and extensive properties of a system, e.g., ∂∂UUk ∂ U U=++ SV M V SV ∑ i = 2.5 ∂∂SV= ∂ M i 1 i 2.3 M MS k it is evident that a thermodynamic system can have =TS − PV +µ∑ ii M i=1 only k+1 independent intensive properties. a result previously obtained by integration with the unstated assumption that U, S, V and M are Thermodynamic potentials, the ratio of two extensive properties. That U and M must be differentials of extensive properties, are, of course, extensive follows from the fact that they are also intensive. conservative properties by the first law and conservation of mass (an implicit law in thermody- UGLY REALITY: COMPOSITIONAL, SPECIFIC AND namics due to Lomonosov, 1747). The second law MOLARVARIABLES defines entropy as a conservative property for In practice it is often desired to separate variations reversible processes. Thus it must be conservative in extent from variations in state, where the latter in the absence of any processes. Oddly, volume is are usually the variations subject of interest, e.g., not conservative. Therefore that volume is exten- phase diagrams show phase relations only as a sive follows only from the unstated assumption that function of state. This separation is accomplished matter is continuous within any system. In mechan- by defining specific variables ics this assumption is referred to as the continuity Ψ U ψ≡i , u = 2.6 constraint. i αα where α is an arbitrary linear combination of Since all extensive properties are proportional to extensive properties chosen to define the extent or the size of a system, size is defined by any exten- amount of matter (e.g., total mass or volume). In sive property. However, for practical purposes it is other words, α is an arbitrarily defined extensive preferable to choose properties that are conserva- property. As 2.6 is a linear transformation, 0.29 tive, i.e., for which yields the integral form of the specific energy k d0Ψ= 2.4 u=−+ Ts Pv µ m 2.7 ∑i=1 ii for all processes that are possible in the system such that other than a change in size.

9

U( SV,, M) = α u 2.8 directly proportional to the mass of the constituent The desired separation of state and extent is then entity (mi). Any equation written in terms of mass obtained from the differential of 2.8 may be written equivalently in terms of moles via M dU=α+α dd uu 2.9 N = i i Nm as α is solely a function of extent and u solely a Ai where roman face font is used to denote constants. function of state. The ugliness alluded to in the Thus, in terms of moles 0.29 is section title arises because only k+1 of the k+2 k specific variables on the right hand side of 2.7 are U= TS − PV +µ∑ ii N . 2.12 independent; but unfortunately without prior i=1 knowledge of the definition of α, the explicit Molar variables are specific variables formed by dependence of the specific energy can only be defining amount as the total number of moles k written in general form as u (ψψ11... k + ) , i.e., by α≡ N e 2.13 ∑ iNi i=1 making use of a relation such as 2.5 to ob- where e is a unit quantity of Νi, introduced so that Ni tain ψkk++2 =f ( ψψ 11... ) . This form is not only ugly the molar variables remain dimensionally consistent because of its vagueness, but also for its asymmetry with the chemical potentials of 2.12. The resulting with respect to the extensive properties. molar variables n1, …, nk are the mole fractions of For the sake of clarity equations for specific the different kinds of mass and subject to the properties are written here in terms of k+2 varia- constraint bles, with the implicit understanding that only k+1 k n e1= , of these variables are independent, e.g., u( svm,, ) . ∑ iNi i=1

which implies To illustrate the difficulties arising from the k −1 ddn= − e ne 2.14 definition of specific variables, three explicit k nki∑ in i formulations for u (ψψ... ) are outlined below. 11k + As 2.10 requires that for a variation in state at constant extent (dα=0) Gibbs’ Form k dddu= Ts − Pv +µ∑ ii d n 2.15 Not surprisingly, the most logical form is due to i=1 Gibbs, who defined amount substitution of 2.14 into 2.15 yields α≡M k u( svn, ,11 ,..., nk − ) , i.e., Noting that differential of a specific property ψi is k −1 e dΨ dα dddu= Ts − Pv + µ−nk µ dn. 2.16 dψ =i −Ψ , 2.10 ∑i ki ii2 i=1 e αα ni then the variation in the specific internal energy of Thus ∂u a system due to a change in state at constant = T amount is ∂s k −1 ∂u dddu= Ts − Pv +µ d m 2.11 = −P 2.17 ∑ ii ∂ i=1 v ∂u en in only k+1 differentials. The disadvantage of this =µ−i µ ∂neik form is that in graphical analysis (e.g., phase ink diagrams), the specific variables of a state in which from which it is apparent, that unlike Gibbs’ form Mk = 0 are infinite. (2.11), 2.16 does not maintain the relationships between specific variables and potentials present in Molar Specific Variables the Gibbs differential. The virtue of molar variables For historical reasons, in applied chemical thermo- is that it is always possible to choose mass variables dynamics an indirect measure of mass, known as a so that mole, is used in place of mass. A mole of any 01≤≤mi consists of Avogadro's number (NA) of the constituent entities of that substance and is thus

10

is true for all possible chemical compositions. Consequently, the chemical states of the system can S be represented within k−1 dimensional simplex (a polygon with k vertices in k−1 dimensions), the triangular (k=3) and tetrahedral (k=4) representa- T tions of which are familiar to petrologists. s General Compositional Variables µ -P The definition of molar variables treats mass differently than volume or entropy, however there is no fundamental reason for such a distinction. M V Indeed for certain types of systems it is desirable to v define specific variables that define the complete Fig 2.1 Ternary composition space for an physicochemical composition of a system. To this composed of one kind of mass. end, amount is defined k +2 α≡ Ψ e 2.18 ∑ i Ψi i M m = so that it is symmetric with respect to all the SV++ M extensive properties. All the specific variables are and from 2.19 then analogous to mass fractions, as used in v=−−1 sm chemistry, in that For this system, all states can be plotted within a

01≤ψi ≤ ternary simplex (Fig 2.1). If the system has a and composition such that m = 1 the system is infinitely k +2 dense (v = 0) and infinitely cold (s = 0). By varying ψ=e 1. 2.19 ∑ i ψi i entropy or volume from this state we can achieve As in the case of molar variables, from 2.19, the all possible thermal and mechanical states (i.e., dependent differential in can be expressed as states of lower pressure or higher temperature). k +1

dψ=−+ψ e de ψ ψ 2.20 ki2 ki+2 ∑ HETEROGENEOUS SYSTEMS i so that the total differential of the specific energy A system is heterogeneous if it consists of matter becomes in two or more states (e.g., vapor and liquid). k +1 Although a heterogeneous system may be com- eψ ddu = θ−k θ ψ. 2.21 ∑i ki posed of many parts (e.g., liquid droplets dispersed i=1 e ψi in a vapor) thermodynamics is only concerned with The virtue of this form is that all possible states of a those parts of the system that differ in state. All the system can be represented within k + 1 dimensional parts of a system that are in the same state are simplex, the vertices of which represent limiting grouped together and considered to comprise a states. Although specific properties defined in this phase of the system. A heterogeneous system is in was may seem exotic, their physical significance is internal equilibrium if there is no process possible not difficult to understand. For example, for a among the existing phases of the system (Fig 1.1). system containing only one kind of mass, dropping Since potentials are the thermodynamic forces that the unit factors for transparency, 2.6 in combination enable a process, it follows that a system is in with 2.18 yields internal equilibrium if each potential has the same U u = value in all p phases of the system, i.e., SV++ M θ=1 =θp ik =12 + S ii s = 1 p ++ TT= = SV M V 1 = = p v = PP SV++ M 1 p µ=ii =µik =1

11

11 1 Thus, in contrast to specific properties, in equilibri- ψ1 ψθk + 21 u um, systems potentials do not distinguish phase   = . 2.28 states and are only characteristic of a system as a pp p ψ1 ψθkk++ 22u whole. The extensive attributes of the system are While the system of equations need not be of full Ψsys =pp Ψj = αψjj =αsys ψ sys i∑∑jj=11 i= ii rank (i.e., p = k+2), it is evident that it cannot have pp Usys = Uj = α=αjj uusys sys a solution if p > k+2. It follows that in an equilibri- ∑∑jj=11= pp um system Ssys = Sj = α=αjj sssys sys 2.22 ∑∑jj=11= p ≤ k+2, 2.29 pp Vsys = Vj = α=αjj vvsys sys a deduction known as the phase rule. Phase ∑∑jj=11= pp equilibria are often characterized by the number of Msys = Mj = α=αjj mmsys sys i∑∑jj=11 i= ii potentials that can be varied without causing a where αj is the absolute amount of the phase. Given change of phase, i.e., the variance of an equilibri- that um. In a system consisting of one phase there are p α=αj sys thus k+1 independent variables of state. Since the ∑ j=1 of each phase adds one constraint the left-most equality in 2.22 can be rearranged as p (i.e., 2.26), it follows that the variance of a system x jjψ=ψsys ∑ j=1 ii or equilibrium is p xujj= usys fk=+−2 p, 2.30 ∑ j=1 p which is the most common expression of the phase xsjj= ssys 2.23 ∑ j=1 rule. p xvjj= vsys ∑ j=1 PHASE PROPORTIONS: THE LEVER RULE p xmjj= msys The constraints of 2.23 are simply a statement that ∑ j=1 ii the parts of the system must sum to the whole. If where xj is the relative or fractional amount of the the equilibrium compositions of a system and its jth phase constituent phases are known, these statements x jj=ααsys 2.24 furnish k+2 constraints on the p ≤ k+2 relative subject to the constraint amounts xx1,, p of the phases. Thus it is always p x j = 1 . 2.25 ∑ j=1 possible to determine the relative amounts of the phases, by making use of any subset of p of the k+2 THE PHASE RULE constraints, e.g., if p ≤ k+2 then 1 p sys Ordinarily the phase equilibrium problem is to find ss 1 s x the relative amounts and specific properties of the vv1  p vsys  = 2.31 phases in an equilibrium system. However, if the p  1 p x sys specific properties of the phases are known in mmkk m k advance, then the equations of state for each phase can be solved for xx1,, p . The geometric provide p equations of the form k expression of this logic is referred to as the lever jjj j u= Ts − Pv +µ∑ ii m 2.26 rule in the analysis of phase diagrams. i=1 Viewed from this perspective the equations PROBLEMS comprise a system of p equations in k+2 unknown 2.1) a) Given the following properties for molar potentials, i.e. in matrix form quantities of the aluminosilicate polymorphs T sv11 m 1 u1 k −P andalusite kyanite sillimanite   =  2.27 u(J) -2330280 -2334497 -2327357 pp p  p sv mk u s(J/K) 251.065 241.994 253.916 µk v(J/bar) 5.19585 4.45214 5.01131 or n(mol) 1 1 1

12

use Eq 2.27 compute the P-T-µ condition at which or=[blue,red], symbol=cross, axes=boxed, thick- all 3 polymorphs coexist. b) The 3 polymorphs ness=2, labels=["X-axis","Y-axis"]);” coexist in an isolated system that contains 1 mol of d) To overlay two, or more, plots assign them

Al2SiO5, and has an entropy of 246 J/K and a “handles” and use the “display” command, e.g., volume of 4.7 J/bar, use Eq 2.31to compute the “a_plot := plot(y,x); b_plot := plot(z,x); dis- molar and volume fractions of the polymorphs play({a_plot,b_plot});” [hint: the easiest way to get the volume proportions is to express the specific properties of the system and its phases per unit volume, rather than as per mole Al2SiO5 as in the above table]. c) Eq 2.31 can be used for either absolute or relative proportions. The relative proportions sum to unity, use this constraint to formulate the previous problem as a system of two equations in two unknowns. d) Construct an s-v phase diagram of the triple point, shade (or indicate) the physically accessible range of entropy and volume for this system [hint: if you wish to make the plot in Maple see “Plotting Points in Maple” below].

SOLVING MATRIX PROBLEMS IN MAPLE The Maple commands listed below may be helpful (a template for these commands is in ../chapter_2/problems_chapter_2_setup.zip): a) Load the linear algebra package by typing: “with(linalg);”. b) Create an n x n-dimensional matrix A by typing:

“A := matrix (n, n, [a11, a12,…, a1m,…, an1…, anm]);” where the elements aij may be numeric or symbolic. c) Create the transpose of A by typing: “At := transpose(A);”. c) Create an n-dimensional b vector by typing: “b

:= vector (n, [b1,…, bn]);”. d) Solve for the n-dimensional x vector by typing: “linsolve (a, x);” e) Concatenate 3 vectors a, b, c by typing: “d := concat (a,b,c);”

POINT PLOTS IN MAPLE To plot an array of points in Maple the following commands may be helpful: a) To create an array of pairs of x-y points: “pts :=

[[x1,y1],...,[xn,yn]];”. b) To make the default plot (points connected by lines): “plot(pts);” c) To make a fancier plot in which the individual points are marked and connected by lines and the axes are labeled and bound the entire range of points: “plot ([pts,pts], style=[line,point], col-

13

3: STABILITY AND SPONTANEOUS dSU,V,M < 0 3.2 PROCESSES where the subscripts are added to emphasize that Thermodynamics admits a special class of process- the variation is constrained to occur at constant U, es known as spontaneous processes that occur V and M. without any external influence. In an equilibrium system no internal processes are possible between An odd feature, resulting from the historical the phases of the system, therefore the only possible development of thermodynamics, is that 3.2 is spontaneous process is the formation of a new rarely used as a stability criteria, rather 3.1 is phase. If the state of a system is such that no substituted into 0.25 to obtain k ddU<− T S PdV + µ d M . 3.3 spontaneous processes are possible, then the system ∑i=1 ii is in a stable equilibrium Condition 3.3 can be viewed as a general criterion state. for possible processes, i.e., it must be true for any real process. It follows, that if a system is in stable equilibrium, then a variation, again by definition impossible, to any other real physical state must satisfy k ddU>− T S PdV + µ d M , 3.4 ∑i=1 ii which is therefore a general stability criterion. ∆UVM = ∆∆ = = 0 Taken literally, 3.4 implies Gibb’s second, and oft cited, second criterion for the stability of an isolated ∆ S > 0 system

dUS,V,M > 0. 3.5 Fig 3.1 A spontaneous process in an isolated system, i.e., a bomb exploding to form a gas The problem (Fig 3.2) with this criterion is that (or plasma). Experience suggests that the there is no spontaneous process in an isolated reverse process will not occur. From the first law the energy of the system must be system at constant entropy or, more importantly, at unchanged by the process, but by the second variable energy. The problem with employing the law the specific entropy must increase after the criterion 3.5 is somewhat metaphysical, if we have process. The second law is simply the formal statement of the experience that the if the an isolated system and there is another state of the bomb has exploded spontaneously, the gas will system such that dSU,V,M > 0, then it is apparent that never condense back into its bomb-like form, i.e., that all spontaneous processes are the current state is metastable, and that the system unidirectional (i.e., irreversible) and proceed in would be more stable in this alternative state. the direction that increases the net entropy of the universe. However if we have an isolated system and there is another state of the system such that dUS,V,M < 0, then it is indeed true that the system cannot be In the event of a spontaneous process, entropy can stable, but the alternative state is not a possible no longer be equated with heat (Q), because the state of the system since the energy of an isolated spontaneous process may also create entropy. system must remain constant (i.e., the first law of Explicitly the second law states that for any real thermodynamics). process system dQ dS > . 3.1 T It follows that a spontaneous process may occur in an isolated system (i.e., dQ = dU = dV = dM = 0) if there is a state of greater entropy possible for the system (Fig 3.1). Thus a system is in a state of stable equilibrium, i.e., no internal or spontaneous processes may occur, if its entropy is a maximum. For any variation from a stable equilibrium, which is by definition impossible,

14

U constant MV, Condition 3.5 has had the unfortunate consequence that people often think of spontaneous processes as why is ∂U /∂≥S 0? processes that minimize the energy of a system, dUSMV > 0 max S accessible states states which is, of course, nonsense. The same logic used

1 to derive 3.5, can be used to derive any number of U dS < 0 UMV equally valid, and misleading, criteria, e.g.: max S at UU = 1 "real" process stability crit. inaccessible states S dU0 const. U,S,M 0<−PdV or dV<0 dV>0 const. U,S,V 0<µdM or dM<0 dM>0 dVSMU > 0 max S accessible states states

1 V dSUMV < 0 max S at VV = 1 inaccessible states S

M constant UV,

dMSUV > 0 max S accessible states states

1 M dSUMV < 0 max S at MM = 1 inaccessible states S

Fig 3.2 S- U, SV,-- SM diagrams illustrating the extremal character of entropy for a stable system. Accessible states correspond to physically real states, inaccessible states are states that cannot be achieved by any real process. If a system is on the boundary between the these regions, then the system has the maximum entropy consistent with its energy and the minimum energy consistent with its entropy, i.e., the system is in a stable equilibrium state. This argument generalizes to

any extensive property Ψi as: for a stable

system, any Ψi property will be at a minimum

if θθi /T is < 0 and at a maximum if i /T is > 0.

15

4: OPEN SYSTEMS: RULES OF THE GAME In general isolated systems, with a few minor exceptions such as the universe, are not of interest, −P V dM =0 so it would be useful to be able to apply our stability criteria to systems that are able to interact with their environment in some limited way. The S problem is that in thermodynamics we are only allowed (or, more accurately, we only want to) T observe the system. This means that we can only derive a stability criterion for a specified system if Fig 4.1 Schematic isobaric-isothermal closed all the interactions between the system and its chemical system. If a (spontaneous) process occurs within the system, the pressure and environment are reversible. If this were not the temperature of the system are buffered by the case, it would always be possible for a spontaneous transfer of volume and entropy from (infinite) reservoirs at the temperature and pressure of process to alter the entropy of the system in some the system. way that we cannot predict without looking at the environment as well. dG = d U− T d Sexternal ++ d S internal PV d ( ) 4.3 =−+dddU TS PV ISOTHERMAL-ISOBARIC CHEMICALLY CLOSED Integrating 4.3 yields SYSTEMS G = U − TS + PV 4.4 Given these rules consider a system that can and combining 4.4 and 0.29 exchange entropy and volume with its surround- k GM= µ 4.5 ings, but is closed with respect to mass (dM = 0, Fig ∑i=1 ii 4.1). From the Gibbs differential (0.25), if the The total differential of G is obtained by taking the exchange of volume and entropy with the environ- differential of 4.4 and combining this with the ment is reversible we have: Gibbs differential (0.25) k dU = TdSexternal −P dV 4.1 dG=−+ d U d( TS ) d( PV ) +µ d M ∑i=1 ii and from Eqs 3.1 and 2.16: =−−+dddddU TS ST PV + VP 4.6 external internal dU − TdS +P dV ≤ TdS 4.2 k = µ−+dM ST dd VP where dSexternal is the entropy derived by reversible ∑i=1 ii heat exchange with the environment, and dSinternal is From which we deduce the independent variables the positive S production due to a spontaneous for G, the well-known Gibbs function, are {M1...Mk, internal process. Technically, the assumption that −P, T}. Since 4.6 is an exact differential heat exchange and dilation are reversible requires ∂G  = µi 4.7 that the system is perfectly transparent to heat and ∂M i PT,, Mji≠ has no viscous resistance to deformation. A system ∂G with these attributes is diathermal (infinite thermal = −S 4.8 ∂ conductivity) and inviscid (lacking viscosity). No T PM, such system exists (with the possible exception of ∂G = V 4.9 superfluid Helium), but the model is a reasonable  ∂P TM, approximation for systems where the time scale for the variables held constant during differentiation chemical equilibration is long compared to the are indicated by subscripts to emphasize that they time-scales for changes in pressure and tempera- are not the same as when the Gibbs differential is ture. The occurrence of high-pressure-temperature differentiated. G may either increase or decrease minerals on the earth's surface demonstrates that with changes in the environmental P-T conditions, such a model is often appropriate for geological but if the P and T of our system are kept constant, processes. then for any real process, from 4.2 and 4.6 internal dGPT,, M =−< TS d0 4.10 Rearrangement of 4.1 yields the differential of a function G, such that

16

So if an isobaric-isothermal chemically closed −P g(− PT ,) −P h(− Ps ,) system is in stable equilibrium, then for any A A S A S + A + = = K A variation in the state of the system K A > A=K+S d0GPT,, M 4.11 S S K = A change in Gibbs "energy" as described by 4.10 S K K K + does not arise through a change in the energy of the S system, but rather in an internal change in the entropy of the system. However, such a change T s does lower the potentials for useful work in the v avT(,) v uvs(,) system, and thus it would be more difficult to A A S A S A + + + + A K A K

extract energy from a system with lower G. It may A + be worth noting that the condition K S = +

S K S + S ∆=GPT,, M 0 4.12 A

K K is a stability criterion, but rather a criterion + NOT + K S K S for equilibrium, specifically that of a reaction. T s The Gibbs energy is often referred to as a free energy because it is a measure of the fraction of the Fig 4.2 Phase diagrams for the aluminosilicate (k =1) system (A-andalusite, K-kyanite, S- internal energy of a system that is available to do sillimanite) as a function of the variables of the work if the work occurs at constant pressure and Gibbs (g, c =1), (u, c =2), Helmholtz (a, c =2) and Internal (u, c= 3) specific free temperature. energy functions. Phase fields of p > c degenerate to singularities (heavy solid lines) at which the state of the system cannot be Heterogeneous Closed Systems determined from the state function for the phase diagram variables. These singularities In an isobaric-isothermal heterogeneous closed are referred to as reactions. Constant pressure- chemical system consisting of p phases, the specific temperature-mass reactions (A=K, A=S, K=S) conserve mass and Gibbs energy, constant gibbs energy of each phase provides an equation of pressure-entropy-mass reactions conserve entropy, mass and enthalpy (A=K+S), constant the form temperature-volume-mass reactions conserve k jj volume, mass and Helmholtz energy. Only the gm=∑ µii 4.13 internal energy (ck = +2) is capable of resolving i=1 all the phase relations of a thermodynamic and must satisfy system. For this reason the internal energy provides the only general equation of state for 11 1 mm11 k µ g heterogeneous systems. = . 4.14 mmpppµ g 1  kk such a variation is not arbitrary and in fact the Taken at face value, 4.14 implies that the maximum probability of observing such an equilibrium is zero number of phases in a closed chemical system is p for a perfectly diathermal and inviscid system, as ≤ k at an arbitrarily chosen pressure temperature would be required if the variations in pressure and condition. Goldschmidt (1911) combined this temperature is to be truly reversible, which of observation with the observation that the many course they are never. The reason for the vanishing minerals, particularly in skarns, have such restricted probability is that in a pressure-temperature compositions that they can be regarded as stoichi- coordinate frame the dimension of the stability field ometric compounds (i.e., that the specific masses of p > k phases is equal to the variance (2.30) of the are constants) to postulate that in closed chemical equilibrium. Thus, univariant and invariant systems p = k. This restricted statement of the phase equilibria are geometrically degenerate 1- and 0- rule (2.29) is known as Goldschmidt's mineralogi- dimensional fields in the 2-dimensional P-T field. cal phase rule. As such they occupy zero area of the 2-d field and as the probability of encountering a given field is In principle, it is possible to vary the pressure and proportional to its area (e.g., see the P-T diagram of temperature of an isochemical system to obtain a Fig 4.2). condition at which p > k phases coexist. However

17

The Stefan Problem The exact differential of Ω is obtained from the The geometric degeneracy of univariant and differential of 4.17 combined with the Gibbs invariant phase fields as a function has profound differential (0.25) k +2 dΩ= dU − d( Ψθ ) consequences, known as the Stefan problem, for ∑ic= +1 ii + , 4.19 modeling heat and deformation in real systems. =ck θ Ψ−2 Ψ θ ∑∑= idd i = + ii These arise because although g is an equation of i11ic state for a single phase, it is not an equation of state from which we deduce the independent variables for an equilibrium system consisting of p > k phases for Ω are Ψ1...Ψc, θc+1...θk+2. Since 4.19 is exact (Fig 4.2). This failing can be understood by ∂Ω  =θ=i ic1 4.20 ∂Ψi observing that for such an assemblage, 4.14 must Ψθji≠+, c12 ... θ k + be true for any permutation of k of the p > k phases. ∂Ω  =−Ψi ic = +12 k + 4.21 Therefore, in an invariant equilibrium, the chemical ∂θ i Ψ Ψθ potentials cannot depend on the proportions of the 1...c , ji≠ phases and the specific Gibbs energy of the system The function Ω measures the maximum amount of k sys sys energy that can be extracted from a system if the gm=∑ µii i=1 extraction is constrained to occur at constant must remain constant during heating or compres- θc+1...θk+2. As such it may be viewed as an extremal sion, which is constrained to occur at constant work function or a general free energy function. pressure and temperature. However, if g does not vary as a function of the proportions of the phases it The free energy function Ω may either increase or cannot define the state of the system; ergo it is not decrease with changes in the environmental an equation of state. Likewise, for a univariant variables θc+1...θk+2, but if these variables are kept equilibrium, g must remain constant for isobaric constant, then from 4.15, for any real process d0Ω<Ψ Ψθ θ 4.22 heating or isothermal compression and therefore 1...cc ,++ 12 ... k cannot resolve either process. In contrast, the So for a system subject to the constraints dΨ1 =...= natural variables of the specific internal energy dΨc = dθc+1 = … = dθk+2 = 0 at stable equilibrium, resolve equilibria of any variance. then for any variation in the state of the system d0Ω> 4.23 Ψ1 Ψθcc, ++ 12 θ k GENERALIZATION OF OPEN SYSTEMS which is a general criterion for stability. The foregoing derivation of a criterion for the stability of an isobaric-isothermal-isochemical The foregoing may be stated more succinctly: If a system can be made general for a system that is system is closed with respect to the properties closed with respect to the c-extensive properties {Ψ1...Ψc}, then we may derive a function

Ψ1...Ψc, i.e., dΨ1 =...= dΨc = 0 (Fig 4.3). From the Ω(Ψ1...Ψc, θc+1...θk+2) + Gibbs differential (0.25) and Gibbs' minimum Ω= −k 2 θΨ U ∑ = + ii energy criterion (3.5), if the exchange of ic1 4.24 =c θΨ Ψc+1... Ψk+2 with the environment is reversible ∑i=1 ii k +2 dU − θ d0 Ψ≤ 4.15 and ∑ic= +1 ii ck+2 ddΩ= θ Ψ − Ψ d θ 4.25 From which we can define the differential of a ∑∑i=11i i ic= + ii function Ω, such that such that for any variation from a stable equilibri- k +2 ddΩ=U − θ d Ψ 4.16 um at constant θc+1...θk+2 ∑ic= +1 ii d0Ω>Ψ Ψθ θ . 4.26 Integrating 4.16 at constant yields 1cc, ++ 12 k k +2 Ω≡U − θΨ 4.17 The transformation of U to Ω is known as a ∑ic= +1 ii Legendre transformation, such transformations and combining 4.17 and the integrated form of the are commutative, i.e., it is possible to change any Gibbs differential (1.6) c potential with its extensive conjugate (as defined by Ω= θΨ . 4.18 ∑i=1 ii 0.28), or any extensity with its intensive conjugate.

18

Thus in thermodynamics it is important to distin- θ c+3 guish the variables that are held constant during partial differentiation. Throughout the remainder of θc+2 θc+4 this text, partial differential notation is made Ψc+3 compact by dropping the subscripts on partial Ψc+2 Ψ c+4 derivatives and differential of thermodynamic functions, if the differential or partial derivative is θ c+1 Ψc+1 dΨΨ1 =...=dc =0 taken with respect to a natural variable of the function holding all other natural variables constant. Thus, it is to be understood that ∂∂=∂∂ UV( UV)SM, Ψk+2 or

∂Ω ∂Ψii =( ∂Ω ∂Ψ ) Ψθjji,12≠+, c ,..., θ k + θk+2 and ddΩ= Ω . Fig 4.3 A schematic system closed with Ψ1... Ψθcc ,++ 12 ... θ k respect to {ΨΨ ,..., } and open with respect to 1 c Although the choice of variables used to express a {Ψck+1 ,..., Ψ+2 } at constant { θθck+1,..., +2}. If a (spontaneous) process occurs within the given free energy function is in principle arbitrary,

system, the values of {θθck+1 ,..., +2 } within the thermodynamic stability criteria (4.23) are only Ψ system are buffered by the transfer of {c+1 ,..., valid for spontaneous processes that occur at Ψ } from (infinite) reservoirs into the system. k+2 constant values of the natural variables for the free energy function, i.e., it is not possible to formulate a stability criterion in terms of differentials such as Natural Variables and Partial Differential dUT,V,M or dGS,P,M. Notation Common Free Energy Functions The Gibbs differential and its Legendre transforms define state functions each of which has a specific There are several well-known thermodynamic set of independent variables. These variables are functions that were derived by Gibbs from the referred to natural variables of the function, e.g., Gibbs differential (0.25) for closed chemical the natural variables of the Gibbs function are −P, systems by Legendre transformations. The most important of these is the Gibbs energy for isobaric- T, M1, …, Mk. However, there is no fundamental reason that the resulting function cannot be isothermal chemically closed systems that we have expressed in terms of any arbitrary choice of derived previously (4.1-4.11). The additional independent state functions. For example, the functions are: the enthalpy function H(S, −P, M) specific internal energy, can be expressed legiti- for adiabatic-isobaric systems k H≡ U −− PV = TS + µ M 4.28 mately as u(T,v), i.e., ( ) ∑i=1 ii du=∂∂ u T dT − Pdv ; 4.27 k ( )v dH=++ TS dd VPµ d M 4.29 ∑i=1 ii where the partial derivative is a quantity that is and the Helmholtz energy A(T, V, M) for diather- often measured experimentally known as the mal isochoric systems isochoric k A≡− U TS =−+ PV µ M 4.30 ≡∂ ∂ ∑i=1 ii cv ( uT)v . k dA=−−+ ST dd PV µ d M 4.31 Comparing 4.27 to the form of specific energy ∑ i=1 ii derived from the Gibbs differential For adiabatic systems, the optimal stability criterion du= Tds − Pdv is always the maximum entropy criterion dS > 0 4.32 at constant volume yields ΩΨ,,1, , Ψcc−+ 1 θ 1, , θ k + 2 cv whereas for diathermal systems the optimal ddsTvv= T stability criterion (4.26) provided by minimizing from which it follows the appropriate free energy function. ∂∂ ≠∂∂ ( uv)TS( uv) .

19

∂∂ = ∂∂ KORZHINSKII'S PHASE RULE ( xy)zz1 ( yx) 4.39 We have established that for a system closed with equation 4.38 is rearranged to obtain Euler’s chain respect to Ψ1...Ψc at constant θc+1...θk+2 we rule for differentiation Ω ∂∂ ∂∂ ∂∂ =− minimize the function to determine the stable ( xz)yx( zy) ( yx) z1. 4.40 phase assemblages. In such a system consisting, the Given f(x,y) and z(x,y), the partial derivatives of f at specific free energy of each phase provides an constant z are obtained by differentiation of 4.35 as equation of the form (∂∂fx) =∂∂( fx) +∂∂( fy) ( ∂∂ yx) . 4.41 c z y xz ωjj = θψ ∑ ii These derivatives can be expressed entirely as i=1 derivatives at constant x and y, by substituting 4.38 and must satisfy 111 into 4.41 to obtain ψ ψθ ω 11c (∂∂fx) =∂∂( fx) −∂∂( fy) ( ∂∂ yz) ( ∂∂ zx) =  4.33 z y xxy ppp 4.42 ψ1  ψθcc ω Finally, it may be useful to recollect that or in matrix notation ∂∂ ∂∂ =∂∂ ( f x)zz( gx) ( f g) z 4.43 ψθpc× c= ω c 4.34 therefore from which we deduce that in a system that is (∂∂sP) ( ∂∂ vP) =∂∂( sv) closed with respect to c extensive properties the TT T maximum number of phases is p ≤ c. Various but ∂sP ∂ ∂ vP ∂ ≠∂ sv ∂ . authors have made similar observations and ( )TS( ) ( ) T suggested restricted statements of the phase rule. This result reemphasizes the importance of keeping Notably, Korzhinskii (1957) states (among other track of the variables that are held constant in variations) that for an isobaric-isothermal system thermodynamic partial derivatives. open to q kinds of matter (designated mobile components) the phase rule should be p ≤ k–q. To illustrate the utility of 4.42, suppose we know Such statements are subject to qualifications and the isochemical specific Gibbs energy as a function obscure the fundamental meaning of the phase rule of pressure and temperature, g(P,T) and we require ∂∂ as originally formulated by Gibbs. ( gP)v . Since we can only differentiate g(P,T) with respect to P and T , we cannot obtain PARTIAL DIFFERENTIAL RELATIONS ∂∂ ( gP)v directly, however given that v(P,T), 4.42 In a variety of problems it is necessary or, at least, allows us to write useful to relate thermodynamic function to (∂∂gP) =∂∂( gP) −∂∂( gT) ( ∂∂ Tv) ( ∂∂ vP) properties that are not measured as a function of the v T P PT where the derivatives of g on the right hand side are natural variables for the function. To this end, the entirely involve only P and T. following rules are useful. For an exact differential

df=∂∂( f x) dx +∂∂( f y) dy 4.35 yx Applying 4.36 to the Gibbs Differential at constant subsequent differentiation is independent of the mass yields order of differentiation, i.e. Euler’s criterion, ∂ ∂ =−∂ ∂ ( TV)SV( PS) 4.44 ∂∂( fx ∂) ∂ y =∂∂( ( fy ∂) ∂ x) 4.36 ( yx)x y the first of four relations known as the Maxwell Given a third variable z that is a function of x and y, relations. The remaining three as in e.g., P = g(V,T), the total differential of z is are obtained in the same manner from dA (4.31), =∂∂ +∂∂ dH (4.29), and dG (4.6). dz( z x)yx dx( z y) dy . 4.37 Therefore at constant z (dz=0), 4.37 requires PROBLEMS (∂∂z x) dxzz =−∂∂( z y) dy yx 4.1) Eq 4.44 is the first Maxwell relation, derive the or three additional Maxwell relations from dA (4.31), ∂∂ =−∂∂ ∂∂ 4.38 ( xy)z( zy) xy( zx) dH (4.29), and dG (4.6). and making use of the identity

20

4.2) Use the Legendre transform (4.24-4.26) to pressure-temperature derivatives given the above obtain the thermodynamic stability criterion for the relations. To express cP in terms of g observe that chemical system Fe-O at constant pressure, as h≡ u −−( Pv) and g≡ u − Ts −−( Pv) then temperature and of oxygen. How h= g + Ts (and s=−∂ gT ∂ ) . This is problem is many phases would you expect to be stable in this most easily done in Maple. See the paragraph system at an arbitrary P-T-µO2 condition? List 4 “Using Derivatives in Maple” for additional hints. minerals that might occur in this system. 4.5) The danger in using derived expressions from 4.3) Compute specific molar a-v diagrams for the the literature, such as Eq 4.48, is that they may have aluminosilicate system at 500 and 1000 K (using typographical errors or may assume secondary the data for problem 2.1 and the definition of A, definitions that are different from what you expect. 4.30). What phases are the possible stable phase To avoid this danger, derive sound velocity as a assemblages in each diagram? Use 4.33 to calculate function of N (molar mass), P, T, and g(P,T) and its the pressure for each of the stable assemblages. Use pressure-temperature derivatives from Eqs 4.45- your results (together with invariant point condi- 4.47 without making use of Eq 4.48; i.e., use one or tions obtained in problem 2.1) to sketch P-T and v- more of the rules given by Eqs 4.40-4.43 to derive T phase diagrams for the aluminosilicate system. ∂∂ an expression for ( Pv/ )s . This problem is Balance the reaction that occurs in the v-T phase partially solved in Maple script ../thermo_course diagram. /chapter4/problem_4_5 _setup.zip).

4.4) Thermodynamics deals with all reversible 4.6) The heat capacity of a material is the amount processes, elastic deformation is reversible and of heat that must be added to it to raise its tempera- therefore described by thermodynamics. In the ture 1 K. The isobaric (constant pressure) heat elastic limit, acoustic (sound) velocities through a capacity c=∂∂( hT) is different from the crystal are entirely a function of thermodynamic P P properties, e.g., sound velocity is isochoric (constant volume) heat capacity cv =∂∂( uT) because in the isobaric case the KS ρ 4.45 v where KS is the adiabatic bulk modulus system can do work (expand) during heating. a) ∂P Beginning from the relations dddu= Ts − Pv and Kv≡− , 4.46 S  dh= Ts dd + vPderive cP − cV as a function of P, T, ∂v S and g(P,T) and its pressure-temperature derivatives. ρ is density ρN= v , 4.47 b) Do you expect cV to be larger or smaller than cP? Why? The Maple setup script for problem 4.5 is and N is the molar mass. In geophysics texts KS is useful for this problem as well. usually expressed in terms of the isobaric expansiv- ity α and heat capacity cP, the isothermal 4.7) The convecting portion of the Earth’s mantle KT, and the Gruneisen thermal is, to a good approximation, adiabatic (i.e., parameter γT as isentropic). Consequently, given the temperature of KKS =TT(1 + Tαγ ) 4.48 the mantle at the base of the lithosphere, where where convection begins, the mantle temperature at ∂P Kv≡− 4.49 greater depth can be obtained by integrating the T ∂v adiabatic gradient (∂∂Tz) =∂∂ρ( TP) g , where 1 ∂v ss α ≡ vT∂ z is depth, the ρg term is the hydrostatic pressure α Kv gradient ∂∂Pz, ρ is the density (Eq 4.47), and a γ = T T 2 cPT− Tα Kv bold-faced g is used to distinguish gravitational ∂h acceleration from the specific Gibbs energy. Derive c ≡ . P ∂∂ ∂T the adiabatic gradient ( Tz)s as a function of N, Derive an expression for the sound velocity through P, T, and g(P, T) and its pressure-temperature a crystal in terms of the N, P, T, and g(P, T) and its

21

derivatives. The Maple setup script for problem 4.5 The resulting expression can be simplified with is useful for this problem as well. alpha := simplify(alpha);

4.8) In Chapter 3 we deduced that for an equation of state of the form u(s), a requirement for stability is that u(s) has positive curvature, i.e., ∂22us ∂ =∂ Ts ∂>0 . Likewise, states of matter represented by a two-dimensional equation of state u(s,v) are stable only if u(s,v) has positive curva- ture. It can be shown that this is true if the determinant of the “Stiffness” matrix (Tisza, 1977)  ∂22us ∂ ∂∂( uv ∂) ∂ s C =   22 ∂∂( us ∂) ∂ v ∂ uv ∂  is greater than or equal to zero. Use this condition, i.e., detC ≥ 0 , to deduce a relationship between the pressure-temperature derivatives of g(P, T) for stable states of matter. Although not required for this problem, it is possible to simplify the relation- ship to constrain the relationship between the isobaric expansivity, 11∂∂vg α≡ = , v∂ T ∂ g ∂ P ∂∂ PT the isobaric heat capacity, ∂∂hg2 cT≡=− , P ∂∂TT2 and the isothermal bulk modulus ∂∂Pg1 Kv≡− =− T ∂v ∂∂ P22 gP ∂ of the material.

DERIVATIVES IN MAPLE In problem 4.4 and 4.5 you do not know the functional form of g, therefore to indicate that g is a function of P and T in maple you must type g(P, T). You can then indicate a derivative of g(P, T) by typing “diff(g(P, T),v1,v2…)” where v1, v2 are, in sequence, the variables of differentiation (i.e., P or T). For example, ∂g = diff (gPT ( , ), P ) ; ∂P and since ∂g = v ∂P the thermal expansivity 1 ∂v α≡ vT∂ can be written in maple as alpha := 1/diff (gPT ( , ), P )*diff ( gPT ( , ), PT , );

22

5: CALCULATION OF G(P, T) FOR AN a ISOCHEMICAL SOLID constant T a =vc/ n/3 For various reasons, most prominently that P and T repulsive 1 are the variables most commonly controlled in experimental systems, empirical thermodynamic data bases are designed for computation of the specific (molar) g (3.4, 3.5). To v this end, the differential dg=−+ sT dd vP 5.1 a=arepulsive+aattractive, n> m , ∂∂av/= − P must be integrated from a reference condition Pr, Tr (normally 1 bar, 298.15 K) at which g, s and v is tabulated to the condition of interest. Here we are m/3 aattractive =v−c2/ concerned with the properties of an isochemical phase, i.e., a phase with fixed composition, Fig 5.1 Schematic of the Mie-Grueneisen therefore the µdn differentials are zero by defini- model of the specific isothermal Helmholtz tion. By splitting the integration into an isothermal free energy. The model assumes the energy can be expressed as the sum of a long-range (i.e., and an isobaric component it is possible to reduce weak dependence on volume, m→ 0) attractive the amount of empirical information on the pressure term and a short range repulsive term (i.e., or temperature dependence of either s or v. Since strong dependence on volume, n >m ). The derivative of the net function with respect to calorimetric measurements are more complicated volume is the pressure, thus physically real the integration path states correspond to specific volumes less than the volume at the minimum value of a (i.e., at PTr ,,PT =−+ higher volumes P <0). g(,) P T g (, Prr T )∫∫ s (,) Pr T dT v (,) P T dP PTrr,,PTr For high pressure, or precise calculations, KT and α 5.2 are given in tabulations as polynomial functions of which requires only s as a function of T, but v as a pressure and temperature. Given these functions it function of both P and T is invariably preferred. is thus possible to formulate an equation of state The immediate problem is to choose the empirical (EoS), i.e., functions used to represent v(P, T) and s(T). 1 ∂∂vvln  α≡ =  vT∂∂PP T  V(P,T) ergo Experimental measurements, e.g., X-ray diffraction, PT, vPT(,) r provide a direct means of extracting absolute r =exp α (P , T ) dT 5.5 vPT(,) ∫ r volumes (e.g., v(Pr, Tr)) as well as the temperature rr PTrr, and pressure derivatives and likewise ∂P PT, KvT ≡−  5.3 v(,) P T −dP ∂v = exp. 5.6 T vP(,) T∫ K (,) PT rTPTr , 1 ∂v α≡  5.4 However, as the integrals in 5.5 and 5.6 are small vT∂ P numbers, in which case a Taylor series expansion which are known, respectively as the isothermal gives exp(xx )≈+ 1 , the combined integral is bulk modulus and the isobaric expansivity. In usually approximated (where it is also assumed that some older tabulations the KT is replaced by its as each integral is small, the product of the two inverse integrals is negligible) as 1 ∂v β ≡− T  vPT(,)≈ vPT (rr , ) vP∂ T PT,,PT r dP 5.7 the isothermal compressibility. Because KT and α 1+α (,)P T dT − ∫∫r KT (,) PT for minerals are relatively small, to a good first PTrr,,PTr 4 approximation at below 10 bar, mineral In practice it is difficult to fit KT and α to arbitrary volumes can be considered constant. polynomials without leading to an EoS that violates

23

the Maxwell relations. This difficulty can be 5.13 is that is not explicit in volume, therefore to avoided, or at least minimized, by adopting the evaluate the integral it is necessary to change structure of a theoretical equation of state. Such variables as in equations were a major focus of solid state physics PT,,vT v(,) P T dP= P (,) v T dv − Pv Pv, 5.14 in the early part of the 20th century. The formula- ∫∫ Pvrr PTrr,,v T tion used in Geoscience follows Mie-Grueneisen The complication in evaluating the in this form is it theory, which assumes that the energy of a solid at first necessary to solve (by iteration) for the volume constant T (the specific Helmholtz energy a, 4.30) at the pressure and temperature of interest, before can be expressed as the sum of an attractive the integral can be evaluated. This is, in fact, what potential that holds the atoms together and repul- is done for high pressure calculations, but for m=−3 sive potential that prevents the solid from 5.10 gives a volume explicit form collapsing V(,) PT= V( Pr , T) cc12 =−+ 1 a mn33 5.8 K′( PT, ) 5.15 vv K′( PT, ) P rr 1− rr where c1 and c2 are, respectively, constants ′ KT( PT r,,) + K( PTrr) P characteristic of the attractive and repulsive that is adequate at pressures below about 105 bar potentials (Fig 5.1). Because the attractive forces are expected to be long range in comparison to the and is identical to Murnaghan equation. To use 5.15, it is necessary to have KT and v as functions of repulsive forces and interatomic distances are T, the former is usually supplied as a polynomial, proportional to v1/3, we expect n > m. Thus, as pressure is whereas the latter is obtained by integration, i.e., PT, ∂ r a ≈ +α P = − 5.9 VPT(,)r VPT (,)1rr (,) PTdTr 5.16 ∂ ∫ v T PTrr, and, from 5.9 and 5.3, differentiation of 5.8 yields 3KTr ( PT ,)(mn++33) ( 33) S(T) P=  ff− 5.10 mn− The experiment used to obtain s(T) is to measure K( PT, ) K(,) PT =Tr (m+3) f(mn++33) −+( n3) f( 33) the energy required to raise the temperature of the T mn−  material of interest as a function of temperature. 5.11 Such calorimetric experiments may be done where constant volume or constant pressure; the latter case f= vPT( ,/) vPT( ,) r is of direct relevance. Reversible heat at constant is volumetric strain. Additionally differentiation of pressure is enthalpy (4.28), therefore the isobaric 4.9 with respect to pressure at Pr (i.e., for f = 1) experiment yields the temperature dependence of gives the enthalpy which is known as the isobaric heat

∂KPTr( ) capacity, i.e., KP′( ) = =(mn ++6) /3 5.12 r ∂ ∂ P T h 1 cp ≡  5.17 ∂ Experimentally it is found that for solids K′ is, for T P practical purposes (i.e., for P < 106 bar), a constant which is usually provided as a polynomial in T. The which is usually near 4; thus for n > m, 3 < n < 6. isochemical differential of enthalpy is For various choices of the exponents m and n, 5.10 dh= Tds − vdP 5.18 has the same low pressure limiting behavior as therefore at constant pressure dh semi-empirical equations of state derived from ds = 5.19 finite-strain theory. In particular, for m = 2 and n = T 4 3 = 53 23 − P KTr( PT ,) f( f 1) 8 5.13

(3Kf′′23 − 12) +− 16 3K ( ) 1 Heat capacity (specific heat) is formally defined is identical to the widely used (truncated) form of as an intensive, i.e., molar or specific (e.g., per unit the Birch-Murnaghan EoS. A complication with mass or volume) property.

24

and context as what is now known as the Debye ∂∂sh1 c Θ = = P 5.20 temperature , which for most minerals is on the ∂∂T TT T order of 1000 K. While this greatly simplifies The third law of thermodynamics (also known as calculations for the inner portions of the earth, the

Nernst’s law) states that at 0 K the entropy of a isobaric heat capacity which is related to cV by 2 perfect crystal is zero, thus the absolute or “third cpV= c +α vK T T 5.25 law” entropy of a solid is, in principle, obtained as does not have a limiting value, although for the PT, r c s0 ( P, T) = P dT 5.21 conditions likely within the earth it is unlikely to r ∫ T Pr ,0 exceed 3NoR by more than 10%. Eq 5.25 has the where the superscript 0 is used to denote specifical- important consequence that the calorimetric ly the “third law” entropy obtained by this component of an EoS is not independent of the integration. Unfortunately, crystals are not perfect volumetric component. An interdependence that is and often are not perfectly ordered at 0 K because often neglected in low pressure geological data the kinetics of ordering (particularly displacive bases. transformations that involve moving atoms within the crystal structure) is too slow to occur on the PERVERSE FORMULATIONS OF 5.2 time scale of experimental observation so that at 0 For traditional reasons, in the geosciences the K the crystal contains residual entropy (disorder). isobaric portion of 5.2 Moreover some potentially important ordering PTr , gPT(,)= gPT (,) − sPTdT (,) 5.26 transformations occur near 0 K, where it is not r rr ∫ r PTrr, possible to make heat capacity measurements. Thus is often perversely written by making use of the the true entropy of a crystal is identity g= h − Ts as PTr , c PT, =conf + P r s( Pr , T) s∫ dT 5.22 T gPT(,)r =+− hPT (,)rr c P d T TsPT (,)r Pr ,0 ∫ PTrr, where sconf represents the contribution from which, in combination with 5.17 and 5.23, be- irreversible disorder, a subject we will return to comes: later. Eq 5.2 requires only integration from the gPT(,)= hPT( ,) − TsPT( , ) reference temperature, i.e., r rr rr PTrr,,PT PTr , c 5.27 c +−P = + P ∫∫cTTPdd T sPT( r,,) sPT( rr) ∫ dT 5.23 T T PTrr,,PTrr PTrr, Both 5.26 and 5.27 are valid, but depending on the and therefore it is to be hoped that s(Pr, Tr) includes any non-third law contributions to the entropy. conventions (see section 4.5) used to define the Sadly, this pious hope is not always justified, thus reference state constants g(Pr, Tr), h(Pr, Tr) and configurational entropy represents a potential s(Pr, Tr) they may yield different values for g(Pr, source of inaccuracy in thermodynamic calcula- T), therefore the two integration schemes should tions. not be mixed.

NON-DISPLACIVE PHASE TRANSFORMATIONS We have treated entropy without regard to micro- scopic Mie-Grueneisen theory which has many Crystalline phases may undergo subtle phase interesting implications for temperature dependent transformations that involve only slight distortions properties. In particular, in combination with Debye of the crystal lattice, most notably the α/β trans- (an enthusiastic Nazi) theory, it justifies the formation in quartz. These transformations may be Dulong-Petit limiting law that states that the gradational, i.e., occur over a range of P-T condi- isochoric heat capacity tions, but are kinetically so fast that their effects ∂u cannot be isolated experimentally. Thus, it is c ≡ 5.24 V  essentially impossible to measure the properties of ∂T V α-quartz in the stability field of β-quartz, and worse reaches a limiting value of 3NoR at high tempera- still, near the “transition” α-quartz consists of a tures. Debye theory defines high temperature in this microscopic mixture of α- and β-quartz like

25

domains. There is a vast literature devoted to such ∆=ggclinoenstatite − g enstatite ff transformations in solid-state physics; however in =ggclinoenstatite − enstatite geological data bases there are only two types of Because databases may have different conventions treatment. The first, and most common, is to (e.g., g of formation from oxides vs elements) it is approximate such transformations as discontinuous. wise to be cautious when mixing data from The second, more elegant, approach consists of different sources. adding a term to the base function 5.2, that accounts for the local effects of the phase transformation. For The “Gibbs energy of formation from the elements” example, for the stable state of quartz at any convention does not imply that the and pressure and temperature Holland & Powell (1998) volumes of the elements are arbitrary. Thus, to use obtain gf(P,T) for a phase from 5.2 it would be g′(,) PT= gPT (,) +λ (,) PT necessary to integrate the volume and entropy where λ accounts for the rapid, but continuous changes for the reaction forming the phase from its effects of the α/β transformation. Because λ constituent elements, rather than the intrinsic involves conditionals, and is relatively small, we volume and entropy of the phase. There is no will not incorporate the term in the exercises of this reason to add this complexity and therefore, course, but the term should be incorporated for real provided it is done consistently, 5.2 leads to applications of this database, which is currently the consistent relative energies. To emphasize that such most widely used thermodynamic database for relative energies are not equivalent to gf(P,T), even 5 minerals at pressures below 10 bar. if they are computed from the integration constant

gf(Pr,Tr), the molar Gibbs energy obtained from is G(PR, TR) - CONVENTION sometimes referred to as an “apparent Gibbs The final issue to be resolved for the evaluation of energy of formation from the elements”. 5.2 is reference state energy. In thermodynamic calculations, absolute energies are never used (or G(PR, TR) – EXPERIMENTAL BASIS I almost never, a possible exception being first The classical experiment for direct determination of principles calculations in mineral physics). Thus, in relative energies involves the measurement of the practice energies are always relative energies amount of energy required to decompose the phase defined via an arbitrary convention. The most to its constituent elements. Because such calorimer- widely used convention is the “Gibbs energy of ic experiments often require extreme conditions formation from the elements” gf in this conven- they are usually done adiabatically rather than tion the stable forms of the pure elements are isothermally, consequently the property obtained is assigned a g of zero at all pressures and tempera- the enthalpy of formation, which is then related to tures. The molar gf for a phase is thus in reality the the Gibbs energy via change in g for the stoichiometric reaction between gff= h − Ts f 1 mol of the phase and its constituent elements in For geological materials, calorimetric measure- their stable states at the pressure and temperature of ments are extremely difficult to make with any interest, e.g., for enstatite (MgSiO3) accuracy and instead indirect measurements from enstatite enstatite Mg Si 3 O2 gf = g − gg −− g 5.28 phase equilibrium experiments are preferred. A 2 Such conventions always involve the arbitrary subject reserved for the next chapter. assignment of energies to a set of c phases (or EXTRACTING OTHER FUNCTIONS FROM G(P,T) entities) the compositions of which are capable of forming a basis for the chemical composition space Although 5.2 is based solely on s(Pr, T) and v(P, T) of interest. That these assignments have no it is valid for all P-T conditions. Thus, properties influence on the computational results is apparent that we do not know as general functions (e.g., s(P, by considering the energy change for the reaction T), α(P, T)) which we do not “know” from our of enstatite to clinoenstatite calibration are readily obtained from elementary thermodynamic relations, e.g.,

26

∂g( PT, ) to compute g(P,T). Calculate the g of forsterite at P s( PT, ) = − 5.29 ∂ = 40000 bar, T = 1000 K; if your script is correct T P you should obtain a g = −2,001,593.99 J/mol. Use 1 ∂v( PT, ) α=(PT, )  the script to compute the following thermodynamic v( PT, ) ∂ T P properties for forsterite at P = 40000 bar, T = 1000 −1 ∂g( PT, ) ∂g( PT, ) K: a) v, b) s (5.29), c) density (ρ, in kg/m3, the = ∂∂T molar mass of forsterite, Nfo, is 0.14073 kg/mol), ∂P ∂P d) T T P isobaric heat capacity (cP, 5.20), e) isobaric 5.30 expansivity (α, 5.30), f) isothermal compressibility indeed a useful trick for testing the correctness of (β, i.e., the inverse of the isothermal bulk modulus the integrations in 5.2 is to verify that the values for KT, 5.3). s(Pr, Tr) and v(Pr, Tr) derived by differentiation of g(P, T) are in fact identical to the values used as 5.2) Using your results from problems 5.1 and 4.5- input for the calculation. 8: a) Compute the isentropic and isothermal speed of sound through forsterite, whose properties are an Additionally, the values of any state function excellent proxy for those of the upper mantle, at P derived by Legendre transformations are trivially = 40000 bar, T = 1000 K (the isothermal speed is obtained as, e.g., computed using the isothermal bulk modulus, KT, u=+− g Ts Pv rather than the adiabatic modulus, KS, in 4.45). b) ∂∂g( PT,,) g( PT) 5.31 =−−g( PT, ) TP The heat capacity ratio (cP/cV) of forsterite at P = ∂∂TP PT40000 bar, T = 1000 K. c) The adiabatic geothermal Such constructions are perfectly valid, but have the gradient in forsterite at P = 40000 bar, T = 1000 K. unfortunate consequence that they yield explicit d) Use the relation derived in problem 4.8 to test functions in terms of the natural variables of G, i.e., whether the equation of state for forsterite from {P, T}. Thus, to obtain the value of the derived problem 5.1 is valid at 2800 K and 1400 kbar, functions in terms of its own natural variables conditions representative of the core-mantle iteration is required. For example, 5.31 gives us boundary. u(P, T), so to obtain u at a specific value of s and v, it would be necessary to vary P and T to obtain the Be sure to indicate the units where appropriate. desired values of s and v at which u is required. Maple can be made to assign and propagate units through calculations to give dimensional results. As REFERENCES I do not use this functionality myself, I certainly do Anderson, D.L. (1989) Theory of the Earth. (Mie- not expect you to use it. However, you may find the Grueneisen theory, available on-line at resolv- convert/units commands helpful, e.g., convert(1, er.caltech.edu/CaltechBOOK:1989.001). units, (kg/mol)*(m/s^2)/(J/K/mol), K/km) converts Holland, T.J.B., and Powell, R. (1998) An internal- 1 (kg/mol)(m/s2)/(J/K/mol) to K/km. ly consistent thermodynamic data set for phases of petrological interest. Journal of Metamorphic Geology, 16, 309-343. Stixrude, L., Lithgow-Bertelloni, C. (2005) Thermodynamics of mantle minerals - I. Physi- cal properties. Geophysical Journal Internation- al, 162, 610-632. (A first principles EoS data base for ultra-high pressure calculations).

PROBLEMS 5.1) Complete the Maple script provided in thermo_course/chapter5/problem_5_1_setup.mws

27

6: COMPONENTS, COMPOSITION, upon environmental conditions), a useful composi- ω− X DIAGRAMS AND FREE ENERGY tion must be defined in terms of these attributes. MINIMIZATION We have established that d0Ω> is a general The foregoing suggests the following thermody- condition for stability in a system in which the c namically rigorous definitions of components and extensive properties Ψ1...Ψc are conserved and the composition in the context of phase equilibria: k+2–c environmental properties θc+1...θk+2 are held Thermodynamic components are the c independ- constant. Further, we have deduced that for an ent extensive properties of a system Ψ1...Ψc that are arbitrary choice of the environmental properties conserved for all processes that occur at constant θ θ θc+1...θk+2 that p, the maximum number of phases c+1... k+2, where the c independent extensive possible in a system, is identical to c, the number of properties of a system are those extensive proper- conservative independently variable extensive ties that are capable of independent variation properties. Since Ψ1...Ψc are conservative proper- among the phases of the system. At a given ties they are the intrinsic physical attributes of the environmental condition θc+1...θk+2 the stable state system and it is natural to regard these properties as of the system is uniquely determined by the the systems components. For example, we can proportions of the components X1...Xc. These imagine composing an isolated system by mixing a proportions define the thermodynamic composi- specific amount of S2, V, and M (for simplicity tion of the system. Recalling the definition of consider chemically homogeneous matter). The specific properties as discussed in Chapter 2, it is stable state of the system (e.g., P and T) is deter- evident that the compositional variables X1...Xc are mined entirely by the proportions of S, V, and M in identical the specific variables ψ1...ψc defined by the initial mixture, thus these properties are the equation 2.8 provided the amount α is defined c components of the isolated system. In contrast, α≡ Ψ e . 6.1 ∑ i Ψi suppose we are interested in a system where P and i T are determined by the environment (e.g., by the systems location in the lithosphere) but the masses MOLAR PROPERTIES AND COMPOSITIONS IN HETEROGENEOUS SYSTEMS M1...Mk are conserved by all processes that occur in the system. In this case M1...Mk are components, but Until now, we have used molar properties to S and V are determined by the amounts of the characterize the properties of a phase rather than components and the environmental parameters P the system itself. This makes sense because it is and T. S and V are thus no longer intrinsic proper- possible to define more or less rigorously what one ties of the system. It is apparent that a logical mole of a phase is (i.e., Avogadro’s number of definition of components depends upon the molecules of the phase, though the definition of a character of the system of interest. Moreover, since molecule is not unambiguous when it comes to the concept of composition is useful only if the solids and liquids), but in a system where phase composition defines the intrinsic attributes of a changes are possible this definition becomes system (i.e., those attributes that do not depend impossible. Consider as an example the system H2O, at low temperature most of the atoms are

associated in H2O molecules, but at high tempera-

ture an appreciable amount of the H2O is

dissociated to O2 and H2, what then is a mole of the 2 In chapter 2, for isolated systems, we identified system? Ψ1...Ψc with S, V, M1, …, Mk because conventional- ly the Gibbs differential is written with U as the A simple, but arbitrary, definition of a mole of dependent variable. This formulation is unfortunate system is that it the amount of a system that from a logical point of view because S is not contains one mole of its thermodynamic compo- conserved in an isolated system, but U is. However nents. Regardless of whether this definition is in practice we are primarily concerned with adopted a mole of phase may not contain a mole of reversible processes and in an adiabatic system the systems components. This creates an annoying reversible processes are isentropic.

28

_ h inconsistency because the molar properties depend _ _ _ 1 2 3 _T= on the choice of components. For example, suppose hX()S hX()S hX()S h XS=1/eS that the molar formula unit of enstatite is chosen to be MgSiO3, then (neglecting the unit quantity divisors e in the formal definition of amount, Eq Ψi 6.1, for brevity) gen= GN en en 6.2 en corresponds to G at NMgSiO3 = 1 (Fig 6.1), but if _µ= h /e we choose NMgO and NSiO2 as the components of a XS=0 M 1 sys 2 0 XS XS XS 1 system composed entirely of enstatite, then XS en en en en gGN=( MgO + N SiO ) 6.3 Fig 6.2 ω − X diagram for the states of an isobaric- 2 en adiabatic system with one chemical kind of matter corresponds to G at NMgO + NSiO2 = 1, which is (i.e., k=1, c=2) and three states of matter described by evidently half the value obtained by Eq 6.2. This 1 2 the equations of state h ( X S ), h (X S ) and inconsistency is inescapable because it is impossi- 3 h (X S ). For this system ω − X and the ble to know in advance what components will be thermodynamic components correspond to S and M. chosen to describe a particular system. In view of rd The 3 law requires that all states degenerate at XS=0 this difficulty, a bar character is introduced to (i.e., at T=0 K), but otherwise the illustration is denote specific properties and amount defined in generic to c=2 systems. If a tangent to the terms of a systems components, i.e., ω − X surface for any state of interest is extrapolated Ψ U to the Xi=1, the ω coordinate of the tangent is identical ψ ≡i ω= i , 6.4 to the value of the dependent potential θi. At αα 1 X S ≤ X S , the stable (minimum enthalpy) state of the Specific properties defined for one definition of system a single phase described by the equation of amount (α) to another ( α ), by observing that 1 state h ( X S ) in which T increases continuously with 1 2 XS. For X S ≤ X S ≤ X S , the system consists of 2 phases 1 2 N with fixed specific entropies (i.e., X S and X S ) whose MgO MgO MgSiO SiO 3 2 proportions vary according to the lever rule, but whose 0 1 X T is independent of XS as indicated graphically by the SiO2 tangent construction.

In view of the fact that ψψ1.... c represent thermody- 1 MgSiO namic composition, it is helpful to distinguish these 3 properties with the notation

Xii=ψ=, ic 1... . 6.6 Mg Si O 0.5 0.5 1.5 Thus, the integral and differentials of the specific form of Eqs 4.24 and 4.19 are, respectively, N c SiO2 ω= θX 6.7 0 1 ∑i=1 ii + ω=ck θ −2 ψ θ dd∑∑= iX i = + ii d 6.8 Fig 6.1 Relationship between 2-dimensional i11ic or, making use of the constraint (Eq 2.19) component space NMgO-NSiO2 and 1-dimensional c composition space for an isobaric-isothermal system. Xe = 1, ∑i=1 i Ψi For such a system ω = g and if specific phase c−1 eΨ θ properties are defined for an arbitrary molar unit, then ω= θ −c θX + c 6.9 ∑i ci = ee = + , where and are the i 1 ΨΨic g g (nMgO nSiO2 ) nMgO nSiO2  ck−+12 specific quantities of the components. Thus if the eΨ dω= θ −c θ ddX + ψ θ . 6.10 molar quantity of Enstatite is MgSiO , then ∑∑i c i ii 3 i=11e ic= + Ψi g En = g En 2. For completeness, in terms of this notation the α dependent potentials of a system composed of p = c ψ=ψii. 6.5 α phases satisfy

29

XX1 11θω θ 11c ω= i 6.13 =  6.11 eΨi Xi =1 p ppθω XX1  cc For the remainder of this chapter we restrict and the lever rule for the fractions of the phase is consideration to phases that cannot change their 1 p sys X1 Xx 11 X 1 thermodynamic composition, i.e., compounds. It is = . 6.12 doubtful that true compounds exist, but the energy 1 p sys of many phases rises so rapidly away from a Xc Xx cp X c particular composition that these alternative compositions are not observed, thus such phases are well approximated as phases with fixed composi- tions and specific free energies. Quartz and aluminosilicates are examples of phases that are commonly regarded as compounds in chemical systems (i.e., as “pure” phases), but via problems 2.1 and 4.3 that they can be treated as compounds in systems that are described in terms of physical components (S, V) as well.

The accessible states of a system composed entirely X 2 of compounds are those defined by positive linear Fig 6.3 ω − X diagram for a system in which all combination of the ω− X states of the compounds possible phases are stoichiometric compounds, i.e., possible in the system. It follows that the accessible ω − X points. Whether true compounds exist is a states will be separated from inaccessible states by metaphysical issue, but many phases have such a piecewise linear c-dimensional hull, the mini- strongly curved ω − X surfaces, a behavior illustrated mum free energy surface of the system as a by the thin solid curves, that they can be regarded has function of its composition (Fig 6.3). Each facet of having fixed compositions. The minimum free energy the hull is defined the ω− X coordinates of c surface separates the accessible states of the system from inaccessible states and defines the stable states of compounds that may stably coexist. From the the system. Because the surface of the system is a c- tangent construction (Eq 6.13) and the lever rule, or dimensional piecewise linear hull, the potentials of the formally from Eqs 6.11 and 6.12, the state of the system vary discontinuously and the state of the system is completely defined for any assemblage of system is undefined for p

30

relevance because many phases are assumed to be in the Fe-O2 system at constant pressure, tempera- true compounds in phase equilibrium calculations. ture and µO2 as illustrated by Fig 6.4. For this system, the Legendre transform of g − X µ O2 O2 gX= µ+ X µ 6.14 ω = = µ Fe Fe O22 O 1− X Fe O2 where I neglect the unit divisors eFe and eO2 to make the equations easier to follow, yields the function to

be minimized at constant µO2 ω=′ gX − µ 6.15 OO22 but ω′ is for a specific amount of each phase

containing XFe moles (or mass units) of iron, thus to obtain the free energy per mole of iron, ω′ must be

divided by XFe = 1-XO2, i.e., ω′ gX−µ ω= = OO22 6.16 X XX1− O2 Fe O2

Fig 6.4 Graphical illustration that the Legendre However, rearrangement of 6.14, shows that transform of g(X ,P,T ) (c=2) for the Fe-O (k=2) gX−µOO O2 2 22 µ=Fe . system to ω(µ , P,T ) (c=1) is analogous to the 1− X O2 O2 petrologic projection of phase relations through the Therefore the function ω obtained by the Legendre

chemical component O2. For an externally imposed transform is identical to the µFe (neglecting the unit µO2, the stable phase is that which minimizes ω, ergo divisor) obtained for each phase in a gX− dia- for the illustrated condition mt is the stable phase. If gram (Fig 6.4) by the petrologic projection of its µO2 was varied continuously, the reactions in the gX− coordinate from the imposed µO2 onto the XFe projected system would involve c+1=2 phases and correspond to the assemblages along the faces if the = 1 axis. Thus, for an imposed µO2 the stable phase minimum g − X surface. The unit quantity divisors O2 (mt, in Fig 6.4) can be found equivalently by are omitted for clarity. finding the phase (or more generally the phase assemblage) that minimizes ω , or doing the graphical projection to find the phase with the

Petrologic Projections and the Legendre lowest µFe. Since the minimization of ω is a 1- Transform dimensional problem, and the projection is a 2- While the notion that the thermodynamic composi- dimensional problem, from a computational tion of a system may not be identical to its chemical perspective the Legendre transform is advanta- composition may seem unfamiliar, such distinctions geous. are entirely commonplace in petrology. For example, if quartz is present in excess (i.e., FREE ENERGY MINIMIZATION FOR SYSTEMS saturated) in a rock, the compositional phase COMPOSED ENTIRELY OF COMPOUNDS relations are often simplified by “projection” In a c-component system in which the only possible through the chemical component SiO2. Such phases are compounds the stable phases assemblag- projections consist of nothing more than simply es always consist of p=c phases. Thus, the stability removing SiO2 from the composition space. of any non-degenerate assemblage can be deter-

Similarly it is commonly assumed that the redox mined by first computing the potentials θ1...θc state of isobaric-isothermal geological systems is from Eq 6.11, and then testing that the inequality c externally controlled, implying that the system is jj ω −∑ θψii >0 6.17 open with respect to O2. This external control is i=1 usually specified by derived thermodynamic for all phases that are not in the original assem- quantities redox potential (Eh) or oxygen fugacity blage. Graphically, this procedure corresponds to (fO2) that are simply alternative expressions for determining the ω− X plane defined by an oxygen chemical potential (µO2). Consider then the assemblage of p=c phases and then checking that problem of identifying the stable phase assemblage the ω− X plane lies below the coordinates of all

31

other phases in the system. This latter requirement components in the system, the optimization is follows from the convex geometry of the minimum subject to linear equality constraints Π free energy surface. This procedure establishes the system jj Ψii =∑ αψ,1ic =  . 6.21 stability of an assemblage, but does not establish j=1 whether the assemblage is possible for a given As formulated here by Eqs 6.19, 6.20 and 6.21, the composition. Thus, given an assemblage of known phase equilibrium problem is a linear optimization stability and a bulk composition, it is necessary to or linear programming (LP) problem. The LP esatablish that amounts α1…αp of the phases are problem was famously solved by George Dantzig greater than zero by solving with the Simplex algorithm in 1947, which is ψ1 ψαp ψsys 1 11 1 widely used for resource allocation problems and is  =  6.18 considered by many to the most important mathe- 1 p sys ψc ψα cp ψ c matical advances of the twentieth century. From before concluding that the assemblage is possible our perspective, this has the advantage that LP for the bulk composition of interest. programming routines are available even in primitive (e.g., Excel) mathematical computer The strategy outlined above for determining the packages. Thus, we have no need to understand the stable phases for a system of specified composition details of the algorithm in order to use it. is a trial and error method that is cumbersome in complex systems. Fortunately, the free energy Up to this point in this section, free energy minimization problem belongs to a category of minimization problem has been formulated in terms problems known generally as optimization of arbitrarily defined specific properties. This is problems that are studied intensively in fields such possible we are merely concerned with the amounts as operations research (industrial engineering) and of the compounds that add up to make the proper- economics. In essence, optimization problems ties of the system. Thus amounts in Eqs 6.18-6.21 consist of finding the values of a set of variables need not be defined consistently. It is of course that optimize (i.e., minimize or maximize) a possible to use specific amounts defined consistent- function subject to a set of constraints, which may ly in terms of the amounts of a systems themselves be functions of the variables. The components. In this case, the minimization function to be optimized is referred to as the variables are the fractional amounts xj of the objective function. For the phase equilibrium compounds subject to the constraint Π version of the optimization problem, the variables ∑ x j = 1 , 6.22 are the amounts (αj) of the phases and the objective j=1 function is the free energy of the system expressed and Eqs 6.19-6.21 become Π as ω=system jj ω Π ∑ x 6.23 Ωsystem = αωjj j=1 ∑ 6.19 j j=1 0≤≤xj 1, = 1 Π 6.24 Π where Π is the total number of compounds that may system jj Xii= ∑ xX,1 i=  c. 6.25 occur in the system, as opposed to those which are j=1 actually stable. From thermodynamic argumenta- If all c constraints from Eq 6.25 are imposed tion we know that of these Π compounds only p=c explicitly for optimization then Eq 6.22 is satisfied can be stable and if a compound is stable its amount automatically, but if only c−1 constraints from Eq must be greater than zero, while the amounts of the 6.25 are implemented, then Eq 6.22 must be unstable compounds must be zero. Thus the for the implemented as an explicit constraint. Thus, either optimization problem the variables αj are subject to formulation (Eqs 6.19-6.21 or 6.22-6.25) involve the linear inequality constraints Π+c explicit constraints, and the only reason to j α≥0,j = 1 Π. 6.20 favor one over the other is whether relative or Additionally, as the amounts of the components in absolute amounts are of greater interest. the stable phases must sum to the amounts of the

32

The free energy minimization strategy outlined here is strictly applicable only to systems composed entirely of compounds, in which case

ω≠f ( ψ1 ... ψc ) and the optimization problem is truly linear. In general, the thermodynamic composition of a phase may vary in which case it is a solution phase for which ω=f ( ψ1 ... ψc ) . In this case, Eqs 6.19-6.21 (or 6.22-6.25) still hold but the compositions of the phases must also be determined by optimization. Optimizations of this type are non- linear, a subject we will return to later after consideration of the thermodynamics of solutions.

PROBLEM 6.1) Use the maple script at thermo_course/ chapter6/problem_6_1_setup.zip to construct the phase diagram for the CaO-Al2O3-SiO2 system (i.e., the ternary chemography) at 1000 bar and 1073 K considering the phases quartz, gehlinite, grossular, sillimanite, anorthite, rankinite, larnite, corundum, lime, and wollastonite. The script loads the requisite thermodynamic data into the function for g(P,T) that you should have obtained from Problem 5.1. It then solves the optimization problem as expressed by Eqs 6.19-6.21.

I will accept a qualitative sketch of the phase diagram for this exercise, i.e., you do not need to plot the diagram in maple (but you are welcome to do so).

If you want to prove to yourself that you actually understand the script you might try modifying the script to solve Eqs 6.22-6.25.

33

7: REACTIONS and when multiplied by −1 become positive Having just gone to great lengths to argue that the quantities, while the coefficients of the products are only meaningful statement of the phase rule is p = defined convention to be positive). This convention c, we will now consider violations of this rule that is of little value in geoscience and in essence may occur when the environmental variables are amounts to defining the products of a reaction as not arbitrarily fixed, but rather chosen or varied in a those phases that have positive coefficients in Eq special way. For illustration imagine that for the 7.1. system shown in Fig 6.3 the only phase whose energy depends on θc+1 is phase ε. Clearly then, if The change in any extensive property of the system th as a consequence of forming νc+1 moles of the c+1 θc+1 is decreased or increased eventually the ω− X coordinate of ε will become co-linear with phase is c+1 −jj ψ = ∆ψ 7.4 the β − γ ω− X line, this co-linearity defines a ∑ j=1 v ii singularity in which the system has an infinite where, by definition, ∆ψj = 0 for j = 1… c, but, in number of possible states that are energetically general, the ∆ψj for j = c+1… k+2 are finite. The indistinguishable. In the special case that the reaction equation relates two possible states of the singularity is defined by c + 1 phases, as in the system that are compositionally and energetically present illustration, all the states are related by a indistinguishable at the equilibrium condition for reaction equation. The reaction equation is deduced the reaction. It follows that equilibrium is defined from the consideration that the formation of new by the condition c+1 phase from the existing phases of a system by a −jj ω = ∆ω = 7.5 ∑ j=1 v i 0 phase transformation must satisfy the conservation and that if ∆ω is computed at any other condition, constraints on the components. It follows that the then ∆ω < 0 if the reaction products are stable, and composition vector of the new phase must be a greater than zero otherwise. linear combination of the composition vectors of the existing phases 11 cc c++1 c 1 Because ∆ψ1...∆ψc are zero by definition the −v φ− − vv φ− φ =0 7.1 complete differential of ∆ω is where + ∆ω = −k 2 ∆ψ θ j dd∑ ic= +1 ii. 7.6 ψ1 j  Eq 7.6 has two special uses. The first is that if we φ= 7.2 ψ j are at a condition at which we do not have equilib- c rium we can estimate the amount we need to j and ν is the reaction coefficient of phase j. Since change a particular potential by to attain equilibri- 7.1 remains valid if multiplied by an arbitrary um as constant, one coefficient in 7.1 may be arbitrarily ∆ω c+1 ∆θ ≈ − specified. Thus, setting ν = 1, Eq 7.1 may be i , 7.7 ∆ψi written which is exact if ∆ψii ≠f () θ and approximate ψ11 ψcc  −ν   ψ +  11  1   1  otherwise. Alternatively, if we consider an assem-    =    7.3 blage at equilibrium, if ∆ψi > 0, then we know that ψ11 ψcc  −ν   ψ +  cc  c   c  an increase in θi will stabilize the phases character- and solved for the unknown reaction coefficients. ized by positive reaction coefficients. The second use of 7.6 is to predict the trajectory of the equilib- Conventionally in chemical thermodynamics Eq 7.1 rium of a reaction as a function of two variables. is not written in the form of a conservation Along this trajectory d0∆ω = , therefore 7.6 implies equation, but is rearranged to obtain a balanced dθ ∆ψ cc++12= − 7.8 reaction, where phases with positive coefficients dθcc++21 ∆ψ (the products of the reaction) are moved to the which is the general form of the Clausius- right of the equals sign. This form results in an Clapeyron relation. equation in which all the coefficients are positive (i.e., the coefficients of the reactants are negative

34

PROBLEMS θ The Maple script in thermo_course/chapter7/ c+2 problem_7_1_2_setup.zip may be helpful for the problems listed below.

7.1) At a prestigious mineralogical institute hidden high in the Swiss Alps, a brilliant, but mysterious, experimental petrologist named Elena Melekhova makes the following observations:

P(bar) T(K) Stable minerals 8000 825 forsterite + phase-A 8000 840 phase-A + chondrodite 18000 860 forsterite + phase-A 18000 880 phase-A + chondrodite θc+1

Sadly, before she can analyze her results she succumbs to Po-210 poisoning. a) Use her observa- Fig 7.1 Schreinemakers projection of univariant (c+1) phase fields, labeled by their associated reactions, tions to estimate the Gibbs energy of formation and about a c+2 phase invariant point in a θc+1−θc+2 third law entropy of chondrodite from the thermo- coordinate frame for a system with c=2 components. dynamic data in ../chapter5/thermo_data_ The stable and metastable portions of the univariant comma_delimited.txt (in this file the abbreviations fields are indicated, respectively, by solid and dashed for Chondrodite, phase-A, and forsterite are, lines. respectively, chond, phaseA, and fo. b) Calculate the temperature of the reaction between forsterite, (Fig 7.1) of the univariant fields about an invariant phase-A and chondrodite at 1000 bar. Hint #1: The field for a two component system (c = 2) in which the α, β, γ and δ are possible compounds. To solve compositions of forsterite (Mg2SiO4), chondrodite this problem you must first deduce the relative (Mg5Si2O8(OH)2), and phase-A (Mg7Si2O8(OH)6) compositions of the compounds. Which univariant lie along a line in the MgO-SiO2-H2O composition space (i.e., they are compositionally degenerate) equilibria can occur in a system that has a bulk such that they can be described by 2 independent composition intermediate between β and δ? chemical components. If you do not want to reformulate their compositions in terms of 2 7.3) a) Compute and plot the univariant curves components, then you can balance the reaction emanating from the sapphirine (spr7) + enstatite between these phases by considering only the MgO (en) + sillimanite (sill) + pyrope (pyr) invariant point in the MgO-Al2O3-SiO2 system as a function and SiO2 components. Hint #2: If you evaluate ∆g for the reaction of chondrodite to forsterite + phase- of pressure and temperature assuming that the A) you obtain an expression of the form: system is saturated with respect to quartz (i.e., quartz ∆=g f(,) PT µ=SiO2 g( PT, ) ). b) These curves divide the

+ν − − pressure temperature diagram into 8 sectors, sketch chondg chond ( PTrr,,) ( T T r) schond ( PT rr) qualitatively (or plot quantitatively if you prefer) which must be zero at the equilibrium conditions of the ω− X diagram for each sector, where the reaction, where νchond is the reaction coefficient ωµ(n , n , PT , , ). c) Sketch the (3) stable of chondrodite. Thus each pair of isobaric experi- MgO Al23 O SiO2 mental observations constrains the equilibrium univariant curves of the pressure-temperature phase temperature and gives you a linear equation in the diagram section for this system if XMgO= nnn MgO/( MgO += AlO ) 0.8 and compute unknown parameters gchond ( PTrr,,) schond ( PTrr,.) 23 the amounts of the phases that are stable in each of 7.2) Sketch the ω− X diagram topology for each of the three sectors. the eight sectors of the Schreinemakers projection

35

excess 8: SIMPLE SOLUTION MODELS g Phases of variable composition, i.e., solution phases, are invariably described as a mixture of s real or hypothetical endmembers, for which data is tabulated (i.e., as in the Holland & Powell ’98 data g base). The problem is then to formulate a solution ° ° model that describes the Gibbs energy of such a g1 mech g2 g solution phase in terms of these endmembers. Such models consist of three components mech conf ex gg= ++ g g 8.1 gconf where gmech is the energy arising from mechanically 0 y2 1 mixing of the endmembers, gconf is the energy expected to arise from theoretical entropic consid- Fig 8.1. Schematic gy− diagram illustrating the ex three components of the specific Gibbs energy erations, and g is a component that accounts for of a binary solution, the diagram is constructed the energetic effects caused by distortions of the with the assumption that the Gibbs energies of atomic structure (e.g., strain) of the chemical the endmembers are zero. mixing process or, in some cases, simply error in sol an immiscible solution), that the equality of gi in conf g . all phases is an auxiliary condition of Gibbs’ stability criterion. NOTATION

In general, we can expect that there may be several THE GIBBS ENERGY OF MECHANICAL MIXING potentially stable solution phases in any given The Gibbs energy of mechanical mixing for a system. Additionally the possible compositions of solution with t endmembers is these solutions may not span the entire range of t mech =  compositions possible for the system, i.e., the t g∑ ygii 8.4 i=1 endmember compositions chosen to represent a where the superscript o denotes a property of the particular solution may not be the same as the c pure endmember. components chosen to represent the systems composition. To avoid ambiguity from these For a binary (t=2) solution, Eq 8.4 defines a line formalities we adopt notation to discriminate (and for t>2 a plane) connecting the Gibbs energies between properties of a system in general and a of the endmembers in a g−y diagram (Fig 8.1). specific solution. To wit, for systems we have Such a diagram is the exact analogy of a gX− di- written agram. As the mechanical mixing process is c sys sys ∂G gX=∑ µii, µ≡ i 8.2 energetically neutral (i.e., there is no gain or loss in = ∂N i 1 i energy caused by mechanical mixing), if we are whereas for a solution we will write only concerned with the internal phase relations of t sol sol sol sol ∂G a solution we may arbitrarily assign the Gibbs g= ∑ gyii, g i≡ 8.3 i=1 ∂Ni energies of the endmembers to be zero, in which th mech thus yi is the mole fraction of the i endmember and case g is likewise zero. sol gi is its specific Gibbs energy defined relative to a mole of its endmembers. As it is always possible to CONFIGURATIONAL GIBBS ENERGY/ENTROPY define a system so that it consists of a single The loss of order, and hence increase in entropy, solution, any relationship that can be written in arising from mixing two or more chemically terms of the gX−µ− coordinates of 8.2 remains distinct endmembers is responsible for stabilizing valid in the gg−−i y coordinates of 8.3. In solution phases. Because the configurational particular, it is pertinent to observe that in consider- component by definition does not account for any ing the phase relations of a single solution, which distortions of the endmember structures during may nonetheless involve more than one phase (i.e., mixing it can be expressed as

36

conf conf M T2 T1 g=−∆ Ts . 8.5 anorthite (Ca) (Al2) (Si2) O8 endmembers, where Classical thermodynamics does not provide a the superscripts indicate the identsites on which the means for quantifying the entropy arising from various elements occur. Three complications mixing, but from it can be resulting from this complexity are: i) mixing may shown that configurational entropy associated with occur independently on different identisites, i.e., Ca a distinguishable object is and Na mix on the octahedral M-site, whereas Al SWconf = k ln 8.6 and Si mix on the tetrahedral T2-sites; ii) there may where W is the number of distinguishable configu- be more (or less) than one identisite of any given rations created by mixing a total of  objects and k type for the formula unit chosen to define the endmember, i.e., the standard 8-oxygen formula is named Boltzman’s constant in honor of the unit for plagioclase has 1 M-site, but 2 T2 sites; iii) Austrian physicist Ludwig E. Boltzman (1844- the endmembers may themselves have configura- 1906), who was the student of Josef Stefan and tional entropy, i.e., in albite Al and Si are Arrehenius and Nernst where among his many disordered on T2. To account for these complica- famous students. Boltzman suffered from bouts of tions we need only recognize that the entropy extreme depression and hung himself in 1906, arising from independent mixing the n identisites of equation 8.6, Boltzman’s equation, is engraved on a solution, by the same logic used to derive 8.9, is his tombstone in Vienna. n mi conf s = −R∑∑qzi ij ln z ij 8.10 If there are t kinds of distinguishable objects, so ij=11 = th that where qi is the number of sites of the i type, mi is the number of kinds of atoms mixing on the ith type, =1 ++... t th and zij is the atomic fraction of the j type of atom the number of distinguishable configurations is th ! mixing on the i identisite. The only difficulty in =  W . 8.7 applying 8.10 is that it is necessary to express the 1 !...t ! atomic site fractions (zij) as a function of the In thermodynamics, we are concerned with independent compositional variables of the ensembles involving large numbers of objects (e.g., solution, i.e., the endmember mole fractions (yi). In atoms or molecules), in which case substituting Eq plagioclase, taking yab as the independent composi- 8.7 in Eq 8.6 and applying Stirling’s approxima- tional variable (i.e., yan= 1− yab), these site fractions tion are lnx !≈− x ln xx 8.8 zy= yields M,Na ab t zyM,Ca =1 − ab conf  S = −kii ln ∑ zy= 2 i=1  T2,Si ab th where i is the mole fraction yi of the i object. zyT2,Al =12 − ab

Thus, in the simplistic limit of mixing between and nm=2,M = 2, m T2 = 2, q M = 1, q T2 = 2 , thus molecular endmembers, the molar entropy ( = NA, 8.10 simplifies to conf Avogadro’s number, and kNA=R, the Universal gas splag =−R[2 yyab ln ab +−( 1 yab ) ln( 1 − yab ) . 8.11 constant) of mixing is +−(2yyab ) ln( 2 −ab ) − ln 4] t conf s= −R∑ yyii ln 8.9 Eqs 8.10 and 8.11 yield the total configurational i=1 entropy of a solution, but our concern is the entropy Since 01≤≤y it is apparent from 8.9 that s conf is i change from the unmixed to the mixed state. To always greater than zero and therefore that for any obtain this change we compute the configurational conf finite temperature g stabilizes mixing (Fig 8.1). entropy that is present in a mechanical mixture of Eq 8.9 is appropriate for truly molecular solutions the endmembers t (e.g., some gases and liquids), but crystals tend to mech conf s= ∑ ysii 8.12 be somewhat more complex. To illustrate this i=1 complexity let us consider plagioclase as a binary and subtract this from the total entropy to obtain M T2 T1 mixture between albite (Na) (AlSi) (Si2) O8 and

37

ntmi conf conf these expectations are often, but not invariably, ∆=−s R∑∑qzi ij ln z ij − ∑ ys i i 8.13 ij=11 = i=1 met. Thus, it is to be expected, for example, that conf Mg2+ and Fe2+, which have comparable radii, mix where si is the configurational entropy of pure endmember i, which is computed from 8.10 for the nearly ideally; and that in general the excess energy component of a solution model is positive as endmember composition. In plagioclase, this conf conf illustrated in Fig 8.1. However, in liquids and gases procedure yields san = 0 and sRab = ln 4 , thus atomic interactions tend to be more complex and 8.13 evaluates to conf negative excess energies are commonplace (e.g., ∆splag =−R[2 yyab ln( ab ) +−( 1 yab ) ln( 1 − yab ) 8.14 H2O-NaCl). This complexity leads to yet another + − − −− (2yab ) ln( 2 yyab ) ( 1ab ) ln 4] type of solution model formulation known as The g conf component of solution models is some- speciation models, which we will consider later. times referred to as the “ideal” component, this However for present purposes, we will confine terminology, while appropriate when the solution is ourselves to a macroscopic formulation appropriate a simple mixture of non-interacting molecules, is for solids and simple fluids. While there are grotesquely misleading when applied to crystals primitive theoretical models to predict the excess because the assumptions regarding individual site properties of trace-element solutions (e.g., Blundy ex populations are so uncertain as to be, in some cases, & Wood, 1994), by and large, models for g are almost arbitrary. Taking plagioclase as an example, empirically calibrated and expressed as polynomi- at low temperatures Al preferentially occupies only als in the endmember fractions yi. There is an one of the T2 sites (low-albite), whereas at high enormous variety of polynomial forms (cf. Ganguly temperatures Al may be present on all four of the 2001), but for most purposes the simplest possible tetrahedral sites (i.e., both the T1 and T2 sites, as in model, known as the regular model is adequate. analbite). Moreover the transitions between these The regular model expresses g ex as tt−1 ordering schemes may be continuous. Worse still, ex g= ∑∑ Wyyij i j 8.16 in the low-plagioclase scenario, it is conceivable i ji= +1 that Si on the T2 site is always located near an M- the sum of pair-wise interactions between the site Na so as to maintain local (ionic) charge endmembers, where the interactions are character- balance. In this scenario, the “ideal” plagioclase ized by the Wij coefficients (Margules model, there is no independent tetrahedral disorder parameters), which are usually taken to be and the model reduces to the molecular model 8.9 constants, but in some cases are allowed to vary as ∆conf =− +− − splag R yyab ln ab ( 1 yab ) ln( 1 yab ) , 8.15 a function of pressure and temperature. For a binary which is drastically different from 8.11. It is also solution, such as our plagioclase model, Eq 8.16 important to recognize that the endmembers chosen yields a symmetric excess function (i.e., ex for a given solution model are derived using the g= Wab-an yy ab an as depicted in Fig 8.1) that is too same ordering scheme as assumed for the solution simple to explain phase equilibrium observations. model. Thus, e.g., Holland and Powell (1998) In such cases, higher order (subregular) expan- provide three different endmembers for the albite sions such as ex composition, “ab” which is fully ordered, “abh” g= Wab-an-an yyy ab an an+ W ab-ab-an yyy ab ab an 8.17 which has Al-Si disorder on T2 and “hab” in which are desirable. The necessity for high order excess Al and Si are disordered over both T1 and T2. functions is often an indication of a poor model for g conf and because high order models may extrapo- THE EXCESS GIBBS ENERGY late poorly they are best avoided whenever In theory the excess Gibbs energy term of a possible. crystalline solution model accounts for strain induced by of foreign cations into an endmember SOLVII crystal lattice. Intuitively, it is to be expected that We have shown previously that the condition for this strain will be proportional to the difference in the stability of a homogeneous isobaric-isothermal the foreign and endmember cation radii and that the system is that its gX− surface must be convex or strain energy will destabilize the solution. Both linear with respect to the g coordinate, i.e.,

38

solvus spinode

spinodes a′ b′ a′′ solvus

b′′

0 1 Fig 8.3 Temperature-composition diagram for Fig 8.2 g− y diagram for a binary system with a an immiscible binary solution. The spinodes and partially miscible solution. Any composition of the solvus coincide only at the critical point and the solution between a'' and b'' is metastable at 0 K. with respect to a mixture of a'' and b'' , these compositions define the limbs of the solvus of ∂2 g the solution. The g− y surface is only convex = 0 ∂ 2 between a' and b' , these points (or, in multidi- yi mensional solutions, curves or surfaces) are the at exactly one composition. This composition and spinodes of the solvus. temperature are the critical composition and ∂2 g temperature of the solutions solvus. Determina- ≥=−0,it 1... 1 2 tions of the critical conditions for a solvus are the ∂X i most accurate method of constraining the parame- where, the equality applies when the system is ters and structure of a solution model, because at heterogeneous (i.e., consists of more than one this condition all the high order derivatives must phase). For the three components of the Gibbs vanish, i.e., energy of a solution in 8.1, for any general (i.e., not ∂n g pure) composition by definition crit crit n =0 forn ≥= 2 at TT , yii = y 8.18 2 mech ∂y ∂ g i ≡ 0 2 Thus critical conditions provide a set of equations ∂yi that can be solved for the unknown parameters of a and 2 conf model. ∂ g ≥∝ 2 0 T ∂yi Above the critical temperature the g−y surface is where the equality applies only at T = 0 K. There is always concave and all compositions of the solution no rigorous constraint on g ex , but to a first are stable. Below the critical temperature the approximation for many solutions surface is convex in the vicinity of the critical ∂2g ex <≈0 constant . composition, and such compositions are metastable ∂ 2 yi with respect to a mixture of compositions that can conf Additionally, g is a weak logarithmic function be determined from the stability criterion, these of composition, whereas g ex varies as at least a compositions are said to define the limbs of the quadratic function of composition. Thus we can solvus (Fig 8.2−3). The stable compositions of a expect that at non-zero temperature, g conf will solvus are always on the convex portion of the g−y dominate (and stabilize) dilute solution behavior, surface of the solution and therefore include the but that the destabilizing role of g ex will become conditions for which the surface is concave. The points (or curves or planes for multidimensional more significant at intermediate solution composi- conf solutions) at which the surface changes from tions. Since the stabilizing effect of g vanishes convex to concave are referred to as spinodes. at T = 0 K, but must be infinite at T = ∞ K, then if There is no reason to ascribe spinodes any special temperature is reduced continuously from a high significance from classical thermodynamics, but in value there must be a finite temperature at which

39

statistical mechanics it is argued that between the could be used to establish P-T the conditions of spinodes any microscopic fluctuations within a equilibration for the rock containing the phases. homogeneous phase will lower the bulk G and From 8.3, for any given composition of a solution, hence favor exsolution of immiscible phases. In gi for an endmember is obtained by extrapolating contrast, such fluctuations for compositions the tangent to the g−y surface at the solutions between the spinodes and the solvus raise the bulk composition to the endmember composition (Fig energy and inhibit exsolution. This argument has 8.2). Thus for a solution with two endmembers been confirmed experimentally, in that phases ∂g gg11=+−(1 y) within a spinode decompose rapidly to the spinodal ∂y 1 8.23 compositions, but equilibration beyond the spinode ∂g gg22=+−(1 y) to the solvus compositions is often extraordinarily ∂y2 slow. where ∂∂gg = − By analogy with gX− diagrams, a tangent to the ∂∂yy12 g−y surface of a solution extrapolated to the pure and more generally for a solution with t endmem- endmember composition defines the partial molar bers Gibbs energy of the endmember in the solution (Fig t ∂g gg= − y . 8.24 8.2). Thus if A and B are two phases of a solution ij∑ ji≠ ∂y j that coexist across a solvus the equalities Given that gAB= gi = 1... t 8.19 ii ∂Gsol sol ≡ can also be used to constrain the parameterization gi 8.25 ∂Ni and/or structure of a solution model. Alternatively, 8.23 and 8.24 may seem unnecessarily complex, if the solution model is known, then the numerical however because solution models are formulated in solution of 8.19 offers a means of computing the terms of constrained compositional variables yi solvus, although for our purposes graphical differentiation with respect to the general variable methods may be preferable, and in the case of Ni is not at all straightforward. heterogeneous systems Gibbs energy minimization is always superior for this purpose. The concept of thermodynamic activity separates the thermodynamics of the pure endmember from PARTIAL MOLAR ENERGY AND ACTIVITY that of the solution model by defining the activity The partial molar Gibbs energy of an endmember of an endmember as increases monotonically with the endmembers  ggii− concentration for any stable composition (i.e., not ai ≡ exp 8.26 RT within a solvus), and can thought of as a measure of  the thermodynamic “concentration” of the Activity is often closely to the endmember endmember. Thus, if it is possible to write a concentration yi, and in the limit of a pure −= = = reaction between the endmembers of various phase ggii0 and ayii1. Activity models solutions, e.g., can thus be derived from a complete solution model grossular + 2 kyanite + quartz = 3 anorthite 8.20 by substituting 8.24 into 8.26, e.g., in the case of a necessary condition for the equilibrium of this the plagioclase solution model (8.14, 8.17) derived reaction is that the change in the partial molar earlier this procedure yields the activity of albite as energies for the reaction is zero, i.e., 2 2 (1− yWab) an-an-ab Plag Grt Ky Qtz ayab= ab(2 −⋅ y ab ) exp⋅ ∆=g03 = gan − g gr − 2 gg ky − qtz , 8.21  RT 8.27 or in general 2 21y( −− yW) ( W ) expab ab an-ab-ab an-an-ab ∆=g0 =∑ vgii 8.22  RT Recognition of this fact in the geosciences led to Rearranging 8.26, the partial molar energies in Eq the idea that, under the assumption of equilibrium, 8.22 are  measurements of the composition of impure phases gii= g + R Ta ln i 8.28

40

Thus 8.22 is formulated in terms of activities as

 Hint 2: you can solve for the solvus conditions by ∆=g0 =∑ νii( g +R Ta ln i) 8.29 at least four methods: i) plotting the g−y surface; ii) or solving for the conditions at which  Ab-Plag An-Plag Ab-Plag An-Plag ∆=g0 =∆+ g R TK ln 8.30 ggan = an and ggab = ab ; or iii) Plag Plag where the equilibrium constant K is defined as constructing a plot of aan vs aab , although in this

νi Ka≡ ∏ i , 8.31 case you’ll have to figure out for yourself how to which in the specific case of reaction 8.21 is interpret the plot. a 3 K = an 8.32 2 8.2) A rock contains garnet (yalmandine=0.67, aagr ky a qtz ypyrope=0.16, ygrossular=0.17), plagioclase (yab= 0.64) where given that quartz and kyanite are likely to be and pure kyanite, sillimanite and quartz. a) Derive very nearly pure we may reasonably as- activity models for anorthite in plagioclase and sume aaky= qtz ≈ 1 . grossular in garnet from 8.26. For plagioclase use the solution model from problem 8.1, for garnet PROBLEMS assume a regular model (8.16) with only one non- 8.1) An experimentalist establishes that the critical zero term Wpy-gr = 33000 J/mol and assume that point of the Na-Ca plagioclase solvus is at yab = 1/3 mixing occurs only on the A-site (i.e., and T = 900 K. a) Use this observation to determine A (Ca,Mg,Fe)3 Al2 Si 3 O 12 ). b) Assuming that the Wab-ab-an and Wan-an-ab for a subregular excess temperature of equilibration for the rock is known function (8.17) in combination with the configura- tional entropy model of 8.14 (the 2nd and 3rd to be 1000 K by some independent means (for derivatives of the solution model must be zero at example, coexistence of kyanite+sillimanite), use these activity models to compute the pressure for the critical condition. b) Use these parameters to compute and plot the plagioclase solvus from 900 the grossular-kyanite-anorthite-quartz equilibrium (i.e., the conditions for which 8.30 is true for the to 400 K. c) As a petrographer, you observe observed plagioclase and garnet compositions). anorthite rich plagioclase (yab=1/10) coexisting with a more albite-rich plagioclase in a metapelite, what NOTE: The thermodynamic data for the pure temperature did the metapelite equilibrate at? phases are loaded for you in the maple script thermo_course/ chapter8/problem_8_2_setup.zip.

Hint 1: use conditions 8.18 to determine Wab-ab-an and Wan-an-ab, but be sure to express gplag entirely as a function of yab.

41

9: THE EVIL OF THERMOBAROMETRY Bio Bio Pphl Pann AND THE TRUE PATH TO WISDOM “Phase diagrams are the beginning of wisdom...” phlogopite – Sir William Hume-Rothery (1899-1969) annite

For historical reasons (a euphemism for irrational Biotite reasons) thermodynamic formalisms in the geoscience focus more on activity models, then the Bio solution models from which activities are derived. P Because of this, albeit unwarranted, attention the g present chapter enlarges on the activity concept as introduced in chapter 8. gphl°( P,T ) RTa ln HYSICAL ODEL FOR CTIVITY UGACITY phl A P M A /F bio gphl (PT , )= To obtain a more intuitive understanding of activity gphl°( Pphl ,T ) as introduced in the previous chapter, consider the osmotic system of Fig 9.1. Here the main portion of RTf ln phl the system contains a 2-component solution such as biotite (endmembers annite and phlogopite), which is separated from two, initially empty, chambers by gphl°( Pr ,T ) rigid membranes that are permeable with respect to phl ann only one of the endmembers, i.e., osmotic mem- yann branes. Since the pressures in these chambers is Fig 9.1. An isothermal system consisting of independent of the pressure on the main portion of biotite with endmembers phlogopite and annite. the system, we can arrive at an equilibrium The system is connected to 2 compartments by rigid osmotic membranes that are permeable condition in which the chambers are filled with with respect to only one endmember. In each pure endmembers at pressures dictated by the compartment pressure is dictated by the constraint that the partial molar Gibbs energy of the constraint that the partial molar energy of the pure endmember in the compartment must be endmembers must be equal in all parts of the equal to that of the endmember in biotite. It system where they are possible, e.g., follows that only 1 of the 3 pressures is Bio  independent. The conventional definition of the g( PT,, y) = g P , T 9.1 phl phl( phl ) activity of a solution endmember is related to the where although PP≠ , the partial pressure Pphl is a difference in the partial molar energy of the phl endmember in the solution and in its pure state at function of the total P on biotite through 9.1. the same PT and . Alternatively, activities may Activities are simply a means of accounting for this be defined relative to the partial molar energy of the pure endmember at a different pressure, in difference, this can be done explicitly in terms of which case the activity is usually referred to as a pressure in which case the activity is referred to as fugacity. a fugacity as in Bio  to define the properties of the pure endmember at gphl ( PT,, y) = gphl( P phl , T) the pressure and temperature of interest, i.e.  = + Bio  gphl( PT r ,) R T ln fphl g( PT,, y) = g( PT ,) phl phl where Pr is the arbitrary reference pressure, i.e.,  =gphl ( PT,) + R T ln aphl g( PT,, y) − g ( P , T) f ≡ expiir 9.2 i.e., i RT  g( PT,, y) − g ( PT ,) a ≡ exp ii a convention often used for fluids. This convention i  RT corresponds to the activity of the endmember Such definitions are arbitrary and introduce defined relative to a standard state for the pure artificial complexity into thermodynamic theory. endmember at the reference pressure and the temperature of interest. For solids it is conventional

42

g MORE TECHNICALITIES ON ACTIVITY ° The Gibbs energy of a solution can be expressed in g1 ideal g terms of activities as:  tt RT ln a1 sol  g=∑∑ ygii + y iR T ln a i ii=11= µ1=g1 where comparison with 6.4 identifies the first term to be gmech term of a solution model. The relation- RT ln a* ship of the second summation to the remaining 1 components of a solution model (gconf + gex) is made clear splitting the activity into an “ideal” g1* factor (aideal) related to the configurational entropy 0 1 and non-ideal factor known as the activity coeffi- y2 cient, γ, related to the excess energy, in which case tt t sol  ideal g=+∑∑ ygii yTa iR ln i +γ ∑ yTi R ln i 9.3 ii=11= i=1 where, in the special case of a molecular solution Henry’s law region ideal model (6.9) ayii= . Comparison of 6.1, 6.24 and 9.3 requires a1 t conf ideal conf ∂g RTa ln ij= g − ∑ y 9.4 ≠ ∂y ji j Raoult’s law region t ex ex ∂g RT ln γ=ij gy −∑ . 9.5 ji≠ ∂y j − The geometry of g y diagrams (Fig 9.2), or 0 1 alternatively a Taylor series expansion of 9.4 and y2 9.5, suggests two important limiting behaviors for − actvities Fig 9.2 gy diagram showing the difference ideal between the activity of endmember 1 for an yi→→1 aa ii intermediate solution composition measured 9.6 o yk→0 γ→ relative to the solvent (g ) and solute (g* ) iiH reference states. The solvent reference state is known respectively the Raoult’s and Henry’s the pure endmember (or species) at PT and . The solute standard state is the (hypothetical) limiting laws, where kH is the Henry’s law pure species at infinite dilution and the P and constant. From application of the Gibbs-Duhem T of interest. The lower diagram shows the relation (1.9) it can be shown that if one endmem- activity of component 1 in the solution as a function of y . There are 2 limiting behaviors, ber of a solution obeys Raoult’s law, then the other 2 in the Raoult’s law region the activity is ideal endmembers must behave according to Henry’s if measured relative to the solvent standard law. Henry’s law behavior is sometimes used, state, in the Henry’s law region the activity is ideal if measured relative to the solute particularly in aqueous chemistry, to formulate a standard state for a pure endmember at infinite The application of this model to melts and solid- dilution, also known as a solute standard state, solutions has been popularized by Holland & that is derived from the pure endmember, or Powell (1998), who essentially refer to the solute solvent standard state, as standard state as a standard state that has been *  gii≡+ gR Tk ln H . 9.7 corrected according to Darken’s quadratic formalism (DQF), and the term RTk ln as a The virtue of adopting the Henry’s law standard H state for a dilute solute is that the contribution of DQF-correction. For any stable solution it is evident that > must be positive and therefore the excess (“non-ideal”) component of the solution kH 0 *  model becomes implicit and the energy of the ggii> . solution has the pseudo-ideal form * ideal gii=+→ gR Ta ln( yi 0) . 9.8

43

THERMOBAROMETRY AND ACTIVITY-CORRECTED ω EQUILIBRIA ω alm PT11,  The use of activities in thermobarometric calcula- ωann tions as briefly outlined in chapter 6 is seductively St RT ln ( a ) simple. One merely measures the compositions of Bio alm the minerals in some rock, identifies equilibria RT ln ( a ) involving endmember compositions of the mineral, ann and computes the P-T location of the equilibria. In many cases, the equilibria are constrained for the Grt Bio pure endmembers by experimental measurements ωalm = ωann ω so that the free energy change of the pure endmem- PT32, ber reaction is a known function of pressure and temperature, i.e., ∆=g f( PT, ) Bio Grt Bio Grt Bio ωalm = ωann and therefore it is only necessary to compute the ωpy = ωphl St equilibrium constant for activity models for each equilibrium and solve 0=f( PT ,) + R T ln K for the conditions of equilibrium. Unfortunately, ω PT, this method, while simple, can lead to solutions that 21 are in fact totally inconsistent with equilibrium. The origin of this problem is that while the condition ωGrt = ωBio ∆=g 0 is a necessary condition for equilibrium it is py phl St Bio not a sufficient condition. Moreover even if a system is in equilibrium, it is not necessarily stable.

py MgO FeO p X hl P2 Fig 9.4 Free energy composition diagrams corresponding to conditions on Fig 9.3, the

free energyhere is ω = g −nnSiO2 µSiO2- −Al2O3 µAl2O3 −

nnK2Oµ−K2O H2O µH2O.. The "observed" composi- tions of Grt and Bio are indicated by filled t r o i G black circles on the corresponding ω-X P 3 B + curves, the true equilibrium compositions of t S coexisting Grt+Bio are indicated by open circles. P 1 n an To understand the consequences of using a alm necessary but not sufficient condition for equilibri- T1 T2 um, consider the “GASP” barometer defined by

Fig 9.3 P-T diagram projection for phase reaction 6.20 in the CaO-Al2O3-SiO2 system, the relations in the system K2 O-FeO-MgO-Al23 O - condition ∆=g 0 for this reaction implies that

SiO2 with µµSiO2 , Al2O3 , µ K2O and µH2O determined by external (e.g., saturation constraints) so that activity corrected partial molar energies of anor- the system has two thermodynamic compo- thite, grossular, quartz and kyanite are coplanar in nents (FeO and MgO). The univariant the CaO-Al2O3-SiO2 gX− space and therefore that equilibrium biotite (Bio) = Staurolite (St) + Garnet (Gt) defines the true limit for the the chemical potentials of CaO, Al2O3 and SiO2 are stability of Bio. The two thin solid lines equal in the observed garnet, plagioclase, etc. represent "activity corrected" univariant curves for specific (e.g., observed) compositions of However, both garnet and plagioclase contain other garnet and biotite as depicted in the ω-X components as well and these are not constrained to diagrams of Fig 9.4. be equal at the conditions of the GASP equilibrium;

44

indeed, in general they will not be equal. To illustrate this problem, consider garnet-biotite equilibrium in the K2O-FeO-MgO-Al2O3-SiO2-H2O system. To reduce the dimension of the composi- tion space, assume that the system contains water, quartz, kyanite and muscovite at all conditions of interest, thus we can make the Legendre transform ω=gn − µ − n µ − n µ − n µ H2 O H 2 O K 2 O K 2 O SiO2 SiO 2 Al 23 O Al 23 O 9.9 and we are left with MgO and FeO as thermody- namic components, so we can now analyze the phase relations in a two dimensional ω− X dia- gram (Fig 9.4). Both biotite (a solution between annite and phlogopite) and garnet (a solution between almandine and pyrope) are continuous solutions in the reduced composition space. The necessary conditions for equilibrium are (assuming we have chosen the formula units of garnet and biotite so that each has a total of 1 mole of FeO+MgO, so that the stoichiometric coefficients are one) Grt Bio Grt Bio ω=ωalm ann (i.e., µ=µFeO FeO ) 9.10 ω=ωGrt Bio µ=µGrt Bio py phl (i.e, FeO FeO ) 9.11 These equalities can be reformulated as the conditions for independent “activity corrected” univariant equilibrium in the MgO and FeO subsystems, i.e., from 9.10 ∆ω = ωGt − ω Bio 1 alm ann a Gt 9.12 it establishes a range of conditions that include the = ∆ω + T ln alm 1 R Bio aann equilibrium condition, but which, in general, are and from 9.11 not themselves possible conditions. Gt Bio ∆ω2 = ωpy − ω phl A more insidious problem is that even if simple a Grt 9.13 = ∆ω + RT ln py inverse thermobarometry is done in such a way as 2 a Bio phl to establish the conditions of a true equilibrium, it which for any arbitrary, e.g., observed composi- cannot provide information about the stability of tions, can be solved to define the P-T conditions of the observed phases relative to those that are not activity-corrected univariant equilibria. Taken observed. In our Grt+Bio example, we have a individually these curves do not define the equilib- system with c=2 components and 3 phases must rium conditions for Grt+Bio, but rather the coexist along any real univariant curve such as that conditions at which the tangents of the ω− X corresponding to the equilibrium of Grt + Bio + surfaces of each phase at the specified compositions Staurolite, which limits the stability of Biotite (Fig extrapolate to the same intercept (Fig 9.4). Thus, in 9.3). Thus, in this example, the thermobarometric general the curves do not indicate equilibrium results are inconsistent with the phase relations conditions, the only true equilibrium condition predicted from essentially the same data. Often being the condition at which the two “activity- such inconsistencies are dismissed by statements corrected” univariant curves intersect (Fig 9.3). Use such as “the assemblage is observed, therefore it of the activity corrected GASP barometer is exactly must have been stable”, but such casual argumenta- analogous to considering only one of these curves, tion belies the complexity of thermobarometry. The

45

sad truth is that observed mineral compositions To simplify the formulation of Free energy rarely represent anything approximating equilibri- minimization we will describe all the phases of the um conditions (after all, if they did system as solutions in all the components of the thermobarometry would always yield ~298 K and 1 system. If a phase has a fixed composition, or bar) and therefore it is not possible to find a single otherwise limited solution behavior, then it can be P-T condition at which all the observed phases of a treated as a special case by imposing appropriate rock are in equilibrium. In recent years, in recogni- constraints. The problem can be stated as follows: tion of this complexity petrologists have developed minimize Π more sophisticated inverse methods such as sys jj j j ω =x ω yy 9.14 WEBINVEQ (Gordon 1992) and “Average P-T” ∑ ( 11c− ) j =1 (Powell 1985, Holland & Powell 1998) that attempt subject to the physical constraints to obtain identify the conditions at which an Π sys =jj = − observed rock is closest to equilibrium. However, Xii∑ xy i1... c 1 9.15 j=1 to date there are no inverse methods that also 01≤≤x j 9.16 account for thermodynamic stability. j 01≤≤yi 9.17 THE TRUE PATH: FREE ENERGY MINIMIZATION Π−1 j ≤ The fundamental flaw in thermobarometry is that it ∑ x 1 9.18 j =1 involves prescribing mineral compositions and then c−1 j ≤ asking what conditions yield the prescribed ∑ yi 1 9.19 (observed) compositions. The flaw being that we i=1 cannot prescribe mineral compositions and expect where Π is the total number of phases that may to find a thermodynamically consistent solution. occur in the system, as opposed to those which are j th Given that thermodynamics provides us with a actually stable; x is the molar proportion of the j means of predicting all the properties of a rock as a phase and i indexes the c-components of the system unique function of its environmental variables, the and the c-endmembers of each solution. Constraint alternative is to use such predictions (i.e., forward 9.15 is a restatement of the lever rule, which models) to find the closest match between observed requires that the amounts of the components in the and predicted phase relations. Phase diagrams phases of the system must sum to the total amounts embody thermodynamic these predictions and are of the components in the system; and constraints themselves a consequence of the geometry of the 9.16-9.19 are necessary to ensure all amounts and minimum ω− X for a system as function of its compositions remain positive. As formulated the environmental variables. The method by which this Free energy minimization problem is the classical surface is determined graphically is straightfor- non-linear minimization problem, wherein 9.14 is ward, but to compute such surfaces numerically is the objective function to be minimized, subject to surprisingly difficult. Here we will only briefly the linear constraints 9.15, 9.18 and 9.19; and the outline the formulation of the simplest method, linear inequalities 9.16 and 9.17; to solve for the j j constrained free energy minimization, which amounts (x ) and compositions ( yi ) of the stable enables us to predict the amounts and compositions phases. In the context of the optimization problem, of the stable phases of a system as a function of the p ≤ c stable phases are those phases with non- variables{XX1,..., c ,θθ c+1 ,..., k+2} . The construction zero amounts. A maple script to solve the general of a phase diagram using constrained free energy non-linear problem is provided in the course notes minimization, amounts essentially to a mapping for the solution of problem 9.1. Drawbacks of non- program in which one samples the variable space, linear optimization strategies are that they often records the stable phases, and maps the fields in require initial guesses for the phase compositions. which the various phase assemblages are stable (Fig In contrast, the case in which the possible phases all 9.5). have fixed compositions can be solved by linear optimization methods that are numerically robust. The robustness of the linear optimization, suggests a third method in which the non-linear problem is

46

linearized by replacing approximating each possible solution phase by a series of stoichiometric compounds. The linearized problems is illustrated by the maple script at www.perplex.ethz.ch/simplex.html.

PROBLEMS 9.1 The maple script in ../problem_9_setup.zip derives ω−X functions, where X =nK/(nK+nNa), for white mica and alkali-feldspar for a system in which these phases coexist with water, kyanite and quartz (i.e., the Legendre transform is equivalent to a petrological “projection” through water, kyanite and quartz. a) Use the ω−X functions to construct a T-X phase diagram (at P = 3000 bar, as set in the scrupt) for temperatures from 800-980 K. The topology of the diagram is determined entirely by the location of: two 2-phase reactions (pure albite reacting to pure paragonite, pure sanidine reacting to pure muscovite), two 3-phase reactions, and the sanidine critical condition (at T = 961 K, X = 1/3). Locate univariant equilibria and critical points within +/-1 K, the rest of the diagram can be sketched qualitatively. b) Balance the 3-phase reactions, do you expect the reaction stoichi- ometries to vary as a function of pressure?

Hint: you can do this problem graphically by plotting the ω−X or by making use of the optimization command in the script, the latter is more practical for determining the phase relations in the vicinity of the 3-phase reactions (to use the script for this purpose you will need to vary the constraint on bulk composition specified in the NLPSOLVE command).

47

10: MOLECULAR FLUIDS There is nothing unique to the thermodynamics of fluids (or, for that matter, any phase), but fluids, and in particular gases, are microscopically simpler than solids and therefore have a longer experi- mental and theoretical history. This history has led to a situation in which, from a practical perspective, fluids are treated differently than solids in two distinct ways: fluid equations of state tend to incorporate more theory and the thermodynamic reference state for fluid species is defined different- ly than for solids.

WHAT IS A FLUID? Fluids are defined by concepts that are themselves poorly defined, but a reasonable starting point is to define a fluid as any phase that lacks long-range order on an atomistic scale. Thus, fluids are clearly distinguished from crystals in which atoms are arranged periodically on an effectively infinite structure, excepting the occasional defect. By this definition, glasses qualify as fluids, which may be a little counterintuitive, but is certainly less troubling than the rheological definition of a fluid, which is anything that has time-dependent deformation, e.g., the earth’s mantle. The only legitimate definition of a gas (or vapor) is that it is a fluid that may undergo a discontinuous transformation to a fluid with higher density, which is thereby defined to be a liquid fluid. Ignoring the complexities introduced by impure fluids, the necessity for this unsatisfying definition is that the pressure-temperature trajectory of the liquid-gas equilibrium phase boundary of any one-component system terminates at a point at which the properties of gas and liquid become indistinguishable, i.e., a critical point. For example, the critical point of water is at 647 K and 220 bar, at a temperature below this point if the pressure of a water gas is continuously increased at some point the gas will undergo a discontinuous transformation to liquid water (Figs 10.1 and 10.2). However, because of the existence of the water critical point, it is possible to vary pressure and temperature in such a way as to go from the gaseous to the liquid state continuously, thereby making the distinction between liquid and gas meaningless. Another vague non-thermodynamic distinction is that of molecular and ionic fluids. The former

48

supposedly being composed entirely of covalently bonded molecular species, and the latter consisting of ionized species. To a first approximation, fluids composed of volatile elements (i.e., gaseous elements at room pressure and temperature) tend to be molecular, whereas fluids composed of non- volatile elements tend to be ionic. Thus, C-O-H-N- S-Cl-F fluids are dominantly molecular, while aqueous fluids with dissolved solids (e.g., electro- lytes) or melts tend to be ionic. Unfortunately, the validity of these generalizations is limited, in that pressure tends to decrease the ionic character of both electrolytes and melts. The critical points of molecular fluids are rarely at more extreme conditions than the critical point of water, thus within the earth molecular fluids are typically supercritical.

ELEMENTARY THEORY The basis for the early theoretical treatments of fluids was Clapeyron’s (1834) combination of Boyle’s (1662) and Charles’s (1787) laws as the PV= nR T 10.1 or intermolecular distance in a gas is V1/3, Van der Pv= R T 10.2 Waals argued that the pressure predicted by 10.3 the first complete equation of state (EoS). This could be corrected for the effect of intermolecular equation of state is theoretically justified in the forces as limit that the phase of interest is composed of zero- RT a P = − 10.4 dimensional non-interacting particles, which is not vb− v2 an entirely unreasonable approximation for low- where a is a constant characteristic of the gas, pressure gases. However, at higher pressure the which must be positive if the force are attractive. I condensation of the gas to a liquid provides do not know enough history to know if Van der evidence of both attractive intermolecular forces Waals motivation for the 1/r6 (i.e., 1/v2 ) depend- and that gases cannot be compressed to zero ence for interatomic forces was physical or volume, i.e., the existence of short range repulsive mathematical, but I do know he was interested in intermolecular forces. obtaining an equation of state that could reproduce

boiling. The mathematical requirement for such an Van der Waals (1873) improved on the ideal gas equation is that for a given isotherm: i) it must be model by postulating the molar volume of a gas able to predict two states with different molar should approach a limiting value at 0 K, i.e., volumes at a single pressure, and ii) on either side RT bv= − 10.3 of this pressure the molar volume must decrease p with pressure (∂∂

49

v PT least third order in volume to predict boiling, and vPT***=,, = = . this is exactly what results from Van der vccc PT 2 Waals av/ term, as by rearrangement of 10.4 This realization was the origin of the theory of 23 av−+( PbR T) v += Pv ab . 10.5 corresponding states, which essentially postulates With suitable parameters 10.4 (or 10.5) has 1 real that all gasses in the same reduced state have the root for T > Tc, and 3 real roots at T < Tc. Of the 3 same reduced properties (reduced compressibility, roots, only the minimum and the maximum have etc.). There is nothing wrong with this theory in ∂∂vP< 0, and therefore meet the second require- principle, but in practice, it requires an equation of ment for the equation of state, the intermediate root state with perfect accuracy. In this regard, the Van has no physical significance. der Waals EoS has been found wanting.

Experimental and theoretical models have shown Although the van der Waals equation is not that the there are 4 major categories of intermolecu- accurate, its formulation established the importance lar forces in fluids: nuclear repulsion ~ 1/rn, n = 6- of cubic equations of state for the prediction of 12, short range; electrostatic (or Coulombic) due phase transitions and it has served as a basis for to permanent polar moments, net attractive forces, virtually all subsequent theoretical and semi- ~ 1/r2, comparatively long range; empirical equations of state. In earth science, inductive, induced by interaction of polar and empirical modifications of the Redlich-Kwong nonpolar molecules, attractive, ~ 1/r6; and disper- (1941) EoS RTa sive, (or London) caused by fluctuations by in P = − 10.10 − nonpolar molecules, attractive, ~ 1/r6. Thus, as vb T( v+ bv) electrostatic forces are absent in true molecular are the most popular descendent of the van der fluids, dispersive and inductive forces are indeed Waals equation. Typically these modifications are consistent with van der Waals’ model. Moreover, to the covolume (b) and/or dispersion term (a) is because all the intermolecular forces decay rapidly taken to be a function of pressure and/or tempera- with intermolecular distance, all gases must behave ture. as ideal gases in limit P → 0 (this is a law named after some dead physicist). GIBBS ENERGY AND FUGACITY OF PURE FLUIDS The typical reference state used for solids is the The choice of the covolume b and dispersion a term Gibbs energy of the pure solid at the pressure and in the Van der Waals equation of state are easily temperature of interest. In contrast, for molecular derived by differentiation of 10.4 fluid species, the reference state is the temperature

∂PR2 Ta of interest and an arbitrary reference pressure (Pr). =−+3 10.6 ∂−v vb v Thus, the Gibbs energy of a pure fluid at the ∂2 P=2R Ta − 6 10.7 pressure and temperature of interest must be ∂vv2 34 (vb− ) evaluated from Equating these derivatives to zero at the critical P gPT(,)= gPT (,) + vdP  10.11 condition (Fig 10.2) and solving for the constants in r ∫ Pr terms of the critical temperature (Tc), pressure (Pc) For an ideal gas, substituting 10.2 into 10.11yields and volume ()vc gives P vT93R g,ideal (,) PT= g (,) P T + Rln T . 10.12 b=cc, a = R, Tv P = . 10.8 r  cc c Pr 38 8vc Van der Waals realized that by scaling the pressure, Historically, because non-ideal fluids are described temperature and volume of a gas relative to the by complex equations of state, for practical respective critical properties, 10.4 yields a “univer- applications the value of the integral in 10.11 was sal” equation of state tabulated. However, rather than tabulate the volume T * 3 integral directly a proportional function known as * = − P * *2 10.9 was defined 31vv− fugacity where the scaled or “reduced” variables are

50

P f≡ exp∫ vdP R T 10.13 0 the virtue of this function is that it parallels that for the ideal gas, in that the Gibbs energy of a pure is then f (,) PT = + gPT(,) gPT (,)r Rln T  . 10.14 f( PTr , ) Such manipulations are anachronistic in that the integral P f (,) PT vdP= Rln T =gPT(,) − gPT (,) ∫ f ( PT, ) r Pr r can be evaluated analytically and efficiently by computer. Nonetheless, as fugacities persist in geoscientific literature as relicts of the past, it is worthwhile to clarify a complication in explicitly evaluating 10.13, namely that the volume of a gas tends to infinity in the limit P → 0. To avoid this A minor complication in evaluating 10.19 is that it difficulty a simple trick is to evaluate integral of the is necessary to compute the volume at the pressure  difference real gas volume from its ideal volume of interest to obtain the integration limit v . P  ,ideal f(,) PT ∫ (v−= v) dPR T ln . 10.15 Another seemingly peculiar feature in the thermo- 0 P dynamic treatment of molecular fluids is that the From the dead-physicist limiting law, the volume of reference state provided in virtually all tabulations a real fluid must approach that of an ideal gas in the limit P → 0, thus the integrand of 10.15 is finite for of thermodynamic data for gaseous species is not for a pure real gas, but rather for a hypothetical any real pressure. This formulation suggests an pure ideal gas consisting entirely of the species of alternative expression for fugacity as interest. The reasons for this convention are two- fP= ϕ 10.16 fold: i) the thermodynamic properties of an ideal where ϕ is known as the fugacity coefficient, and gas can be computed from theory and ii) the subject to the limiting law ϕ→1,P → 0, thus properties of the corresponding real gas are 10.15 is P dependent on the particular equation of state used ∫ (v−=ϕ v,ideal ) dPR T ln  10.17 to evaluate the pressure dependence of the fluid. 0 Combining 10.12 and 10.14 A potential complication that arises using complex f ( PT, ) g(,) PT−= g,ideal (,) PT Rln T r , equations of the general form P=f(v), is that it may rr Pr not be possible to obtain the equivalent volume 10.20 explicit form, i.e., v=h(P), which is necessary to where the superscript “ideal” denotes the hypothet- evaluate 10.17 analytically. This complication can ical ideal gas standard state and using 10.20 to be circumvented by the rule of integration by  eliminate g(,) PTr from 10.14, the Gibbs energy of parts (Fig 10.4), which states that if b is a continu- a real gas is ous function of a, then  ab f(,) PT g(,) PT= g,ideal (,) P T+ Rln T . badd=−− abab ab. 10.18 r ∫∫00 Pr ab00 10.21 Applying this rule to the volume integral in 10.17 P ,ideal RT (v− v) dP =−− P v IMPURE FLUIDS ∫ P 0 10.19 With ever greater frequency in recent literature, v RT impure fluids are treated identically to solid −−∫ P() v dv . ∞ P solutions as outlined in chapter 6, with the minor

51

technical difference that the partial molar energy of pure species or endmembers are computed from 10.14 (or its approximation 10.21). However, many fluid equations of state include empirical mixing rules that relate the parameters of an impure gas to the parameters of the pure constituent species. For example, in the case of van der Waals and the unmodified Redlich-Kwong equations, the following mixing rules are considered appropriate 2 amix = (∑∑ yi a i) , b mix = ybi i Given such a rule, it can be shown by the same type of graphical argumentation as we used to define activities, that the fugacity of a species in an impure fluid is ∞  ∂PTR  v = −− RT ln fi ∫  dvR T ln  v ∂nvi R T Tvn,, j Algebraically, the thermodynamic activity of an impure molecular species is fPϕ = γ=i = ii ayi ii  10.22 fPiiϕ which leads to a bewildering variety of ways in which equilibrium constants may be expressed for reactions involving both fluids and solids.

PROBLEMS 10.1 The critical temperature, pressure and molar volume for water are, respectively, 647 K, 220 bar, 9.16 J/bar (the critical volume has been faked here to make the van der Waals EoS give the correct critical pressure and temperature, the real critical volume is 4.787 J/bar). Calculate the covolume (b) and dispersion (a) terms for water in the van der Waals EoS (10.8). Evaluate the molar volume and fugacity of water 700 K and 2000 bar.

Hint: after determining which of the three roots of the van der Waals equation for volume is real at the conditions of interest, this root (i.e., vP ( ) ) can be used explicitly in 10.17 to obtain the fugacity coefficient of water.: Error! Bookmark not defined.

52

H O 11: SPECIATION AND ORDER-DISORDER 2 SOLUTION MODELS Solution models as discussed in the previous

3 2 / / 1 chapter are “simple” in that the models have as 1 = = O O X many endmembers as the solution has independent- X ly variable chemical components. It is, however, H2 O2 possible to formulate a solution model as a mixture H O of t chemical “entities” where t > c, the number of XO components necessary to describe the chemistry of H2 O the mixture. By definition, t − c of the entities are compositionally dependent as it must be possible to

X

O write t − c reactions among these “entities”. There = X

1

O

/

3 = are two distinct limiting model types: pure specia- 1

/ tion models in which the thermodynamic 2 CH CO properties of all t entities are independent and pure 4 2 H O reciprocal models in which the thermodynamic XO properties of t − c of the entities are described in Fig 11.1 Relation between the ternary terms of the remaining c independent entities. Such speciation composition space for H222 -O -H O and CH-CO-HO mixtures and the macro- models combine as reciprocal solution models with 4 22 scopic H-O composition space. The dashed speciation. Reciprocal and combined solution lines in the speciation composition space are models involve more complex bookkeeping and are lines of constant bulk composition. The speciation problem amounts to identifying the described in chapter 12. speciation that yields the lowest free energy

for a given bulk composition. For CH42 -CO -

SPECIATION MODELS H2 O these speciations are not at constant C- content, but the dimensionality of the true C- Thermodynamic theory neither requires nor treats O-H composition space can be reduced by species within a solution. Such models are simply a projection through carbon (as would be possible for graphite-saturated fluids). means of introducing atomistic models to explain macroscopic behavior. The success of such a model g fluid =11 ⋅µ + ⋅µ HO22 H2 O2 depends on the accuracy of the atomistic model. fluid fluid =ggHO + Suppose as an example we wish to model macro- 22 =++g go R T ln aa scopic H-O fluids (Fig 11.1). If we recognize that HO2 2 ( HO22) on the atomistic scale H-O mixtures consist The activity of H2O2 is then ggfluid −  dominantly of H2O, H2, and O2 molecules, we may fluid ( HO22 HO 22) aeHO ≡ R T succeed in predicting the macroscopic properties of 22 the fluid as a nearly ideal mixture of three mole- and if the relation between H2O2 activity and cules. However if we describe the fluid as a mixture composition is known we can obtain the concentra- tion of H2O2, e.g., in the ideal limit we equate of only H2 and O2 molecules we can expect that the model will need a large excess term to account for activity with concentration. If we find that this concentration is significant, then we may expect the special stability of the fluid at the H2O composi- tion (Fig 11.2). that the ideality of our macroscopic model will improve if we model the macroscopic H-O mixture Once we have a macroscopic model we can of as consisting of four species: H2O, H2, O2, and course calculate the thermodynamic properties of H2O2. Thus, the success of a speciation model, i.e., any species k in phase j through the relation its ability to explain macroscopic behavior without c invoking non-ideal interactions, is critically gnjk= µ . 11.1 k∑ ii dependent on the choice of species; models that i=1 incorporate insignificant species are unnecessarily Thus in our example we can calculate the partial complex. molar free energy of H2O2 in our macroscopic H2-

O2 mixture as

53

0

0 1 0 1 Fig 11.2 g−X diagram for an H-O fluid at a O Fig 11.3 Schematic H-O speciation as a temperature and pressure at which a water-rich function of bulk composition. It can be shown phase may coexist with hydrogen- and by application of the Gibbs-Duhem relation oxygen- rich fluids. The equilibrium free that the maximum in the activity of any energy of the fluid as a function of bulk species must occur when the fluid and the composition would be determined by solving species have the same composition (Connolly, for the stable speciation of the fluid at each 1995). There is no thermodynamic requirement − bulk composition. The gXO surface lies below that the maximum in the species concentration the free energy of pure H2O because the fluid occurs at this composition, but generally at the water composition (XO =1/3) is a mixture maxima in species concentration and activity of all three species. Because speciation models are in close proximity to each other. often predict a strong stabilization of the solution when the bulk composition is coincident with a species, such models are tion from the variation in mass it is desirable to often referred to as "compound formation" define a normalization function that keeps the mass of the solution constant for a given bulk composi- There little new about the formulation of speciation tion as a function of its speciation. For any models; the models consist of three components mech conf ex speciation the molar mass of the solution is gg' = ++ g g 11.2 c = where the prime is used to indicate the formulation NN∑ yii 11.6 i=1 as in chapter 8. However, in speciation models where Ni is the molar mass of species i. Defining a there are t − c independent reactions between the reference mass species of the form c 11 cc kk NN00= y 11.7 vkφ+ + v kk φ+ v φ=0 kc = + 1... t 11.3 ∑ ii i=1 for which the free energy change of reaction 00 where yy1 ,..., c are the mole fractions of c chemi- ∆≠g 0 . 11.4 k cally independent species when the mole fractions Therefore for purposes of computing homogeneous of the chemically dependent species are zero (i.e., phase equilibria, in contrast to the simple solution 1 k yyct+1 =... = = 0 ). Observing that vvkk=∂∂ n1 nk models of chapter 8, we cannot assign all t species is the change in the number of moles of species 1 arbitrary energies; although, as before, we can per mole of species k formed by speciation reaction arbitrarily assign c chemically independent species k; Eq 11.3 requires that a variation at both constant energies of zero, in which case 8.4 simplifies to t mass and composition must satisfy mech  t i g=∑ ygkk ∆ . 11.5 ν =0 + k kc= +1 nyii∑ k n k = + ν A more subtle problem is that the molar mass of the kc1 k solution may vary as a function of speciation, e.g., where ni is the un-normalized molar amount of species i. The total number of moles of the species for the H2O bulk composition, if the fluid consists after this variation is then entirely of H2O molecules (yH2O = 1) if molar mas t ttν is ~18 g/mol, whereas when it is composed entirely nn= =1 +  ni ∑ik ∑∑ν of H2 and O2 molecules (yH2 = 2/3 yO2 = 1/3) its i=1kc =+= 11 i k molar mass is ~12 g/mol. Anticipating that we will and defining want to separate the effects of variation in specia-

54

t ν i ≡ k δνk ∑ k i=1 ν k Homogeneous Fluid the normalized compositions of the solution are t νi 0 + k ynik∑ k n = + ν y =ik = kc1 . i n t 1+ ∑ nkkδν kc= +1 "H2 O" Substituting these normalized compositions into "H O" + 2 "O " 11.6 and rearranging the order of summation + 2 t ttνi 0 + k "H2" ∑NNiiyn ∑∑ kk i = = + ν = i11kc i k N t . 11.8 0 1 1+ ∑ nkkδν kc= +1 Fig 11.4 Schematic isobaric phase diagram illustrating paired solvi around the H O As the first summation in the denominator of 11.8 2 composition (XO =1/3). The species names is N0 (11.7) and the final summation in 11.8 is the indicate the dominant species in the coexisting change in mass caused by speciation reaction k, fluids. Aqueous molecular fluids typically show this type of behavior at temperatures in which is zero by definition, 11.8 simplifies to the vicinity of the water critical point (647 K). t NN=0 1 + n δν . 11.9 ∑ kk gmech = yg ∆  . 11.13 kc= +1 HO22 HO Thus, g', the specific free energy per mole of If the fluid is truly a molecular mixture, then from species, can be renormalized to the specific free 8.9 conf energy of the mass of a mole of the reference g=R Ty( H ln y H ++ y O ln y O y HO ln y HO) . 2 22 22 2 speciation as 11.14 gg= Ξ ' 11.10 As in simple solution models, the excess compo- Where the normalization factor nent of speciation models is largely ad-hoc. In H-O 0 t N fluids at temperatures below ~650 K paired solvi Ξ= =11 + ∑ nkkδν 11.11 N kc= +1 separate the H2O-rich fluids from both H2- and O2- is commonly referred to as the Helffrich-Green rich fluids (Fig 11.4). This behavior is reproduced Stuffed moles normalization factor. Although the with an excess function of the form derivation of Ξ is notationally impenetrable, it is gex = W yy+ W yy 11.15 HO-H222 HO H 2 HO-O 222 HO O 2 not a complicated term. For example, in the present if the interaction energiesWH O-H and WH O-O are case of the three species H-O fluid, taking 22 22 0 2 0 1 assigned positive values to counteract the stabiliz- yH = 3 , yO = 3 , vvH HO = −1, 2 2 22 ing entropic effects of 11.14. It is noteworthy, that 1 vv= − 2 , and Ξ=11 −n 2. In fact, in O22 HO ( HO2 ) if H-O fluids were treated as a simple binary solid solutions, speciation reactions typically mixture the existence of paired solvi would require, conserve the number of moles of the species, in at least, a fifth order excess function. Thus, the ease which case Ξ=1. with which complex behavior such as paired solvi are explained by speciation models is a virtue. Returning to our 3-species water model (c=2, t=3), Although the formulation of speciation models in general 11.4 is involves no new features, they introduce additional t mech  degrees of freedom in that for any general bulk g= ∑ ygii i=1 11.12 composition the species fractions are not uniquely =yg ++ yg y g  determined. From thermodynamics, the stable H22 H O 22 O HO 2 HO 2 th speciation must be that which minimizes the energy Choosing H2O as the c+1 species, there is one of the phase for the bulk composition of interest. To independent speciation reaction determine this speciation it is necessary to convert −H2 − ½ O2 + H2O = 0 the solution model in terms of the independent and assigning H2 and O2 energies of zero, 11.5 species fractions gy,..., y , y ,..., y (where yc becomes ( 1cc−+ 11 t)

55

has been eliminated by the closure constraint) to a exploits the fact that derivative of a function must function of the bulk solution composition vanish at the functions minimum. Thus given − { XX11,..., c− } and the fraction of t c chemically gX( 1,..., Xcc−+ 11 , y ,..., y s) , the stability criterion dependent species, i.e., gX( 1,..., Xcc−+ 11 , y ,..., y t) . provides t − c additional constraints of the form This trivial, albeit sometimes messy, conversion is ∂g = = +  0jc 1... t, 11.22 accomplished by noting that the total amount of any ∂y j XX11 c− chemical component in a phase must be the sum of which comprise a system of t − c simultaneous non- that component in the species present in the phase linear equations that can be solved, usually weighted by the species fractions, i.e. tt numerically, for the unknown species concentra- jj Xi= ∑∑ yn ji yn jtotal ,1 i=  c 11.16 tions. The disadvantage of such an approach is that jj 11.22 is a necessary, but not sufficient condition, where n j is the number of moles of component i in i for a global minimum, i.e., 11.22 may have more j species j, ntotal is the total number of moles of the than one solution, thus in general it is not certain components in species j, and that a particular speciation is the speciation with the t lowest possible free energy. Indeed, 11.22 may also yyci=1 − ∑ . 11.17 ic≠ correspond to maxima. Returning again to our H-O fluid model and taking the atomic fraction of oxygen as the independent A Less Rational Analytical Solution: bulk compositional variable, 11.16 and 11.17 are The equilibrium constant method is a less rational yy+ 2 n HO22 O means of solving the speciation problem. It is X =O = 11.18 O nn+23 y ++ y 2 y O H H22 HO O 2 therefore perhaps unsurprising that this method is y=−−1 yy. 11.19 widely applied in earth sciences. The method O2H 22 HO exploits the fact that the equilibrium condition Eqs 11.18 and 11.19 can be rearranged as requires that the free energy change of any reaction yH =−+11( XyO) HO 2 − X O 11.20 22 between the species of a solution must be zero at y=+(12 Xy) +− X y 11.21 O222O HO O HO equilibrium. Thus from 11.3 we have and substituted into 11.13−11.15 to obtain ∆=gj 0 jc =+ 1... t 11.23

gX( O, y HO) , which is minimized at constant XO to and expanding 11.23 as in 8.30 2 , j determine the equilibrium concentration y , the ∆+gR T ln Kj = 0 jc =+ 1,..., t . 11.24 HO2 remaining species concentrations are then obtained The activities that comprise the equilibrium from 11.20-11.21 (Fig 11.3). In this simple case, constants Kj are then derived from 6.24 and, with the minimization can be done graphically by 11.16 and 11.17, recast as functions of

XX11,...,c− , yyct+1,..., so that, as in 11.22, a system plotting gX( O, y HO) vs yHOat constant XO. 2 2 of t − c simultaneous non-linear equations in the unknown species concentrations is obtained. The ANALYTIC SOLUTIONS OF THE SPECIATION reason this method is “less” rational is that it has all PROBLEM the disadvantages of the direct solution of 11.22, The speciation problem can be formulated analyti- but involves many more mathematical operations. cally to avoid the necessity of numerical or graphical energy minimization. Such a formulation

56

COMPOUND FORMATION AND ORDER-DISORDER MODELS Speciation models are widely applied in the Mag description of mineral solutions, but because people Do seem to be uncomfortable with the idea of a Cc Do Do “species” in the context of a solid phase, these + + models often referred to as compound formation Mag Cc or order-disorder models. The origin of this terminology arises from the fact that in some solutions it is possible to achieve particularly 0 1 favorable energetics if different cations can be regularly ordered on a particular crystallographic site. A famous example of this is dolomite, which Mag has two octahedral identisites, M1 and M2, per Do M2 M1 Ca0.5 Mg 0.5 CO 3 formula unit. At low temperature, Cc the energy of stoichiometric dolomite is minimized Do Do + if Ca is entirely on M2 and Mg on M1. Any + Cc deviation from this perfectly ordered scheme causes Mag a rapid increase in the energy, but such a deviation is inescapable if there is solution of either magne- M2 M1 M2 M1 0 1 site, Mg0.5 Mg 0.5 CO 3 , or calcite, Ca0.5 Ca 0.5 CO 3 , in dolomite. This effect destabilizes calcite and Fig 11.5 Schematic isobaric phase diagrams illustrating the two types of paired solvi that magnesite solution, or viewed from another may result from a speciation model of calcite- perspective, stabilizes the formation of stoichio- magnesite solutions with an ordered dolomite species. In the upper diagram the solvi metric dolomite, i.e., compound formation. With connected by a "tricritical" line, which marks a increasing temperature dolomite structure expands discontinuous ordering transition between so the energetic penalty of putting Ca on the M1 dolomite and fully disordered calcite- magnesite solution. Solutions with that exhibit site decreases and, as there is an entropic benefit to this type of behavior are said to have disordering, the extent of solution between "convergent" ordering. In such solutions the distinction between an ordered and disordered stoichiometric dolomite and both calcite and phase is legitimate. In the non-convergent case magnesite increases and ultimately becomes (lower diagram) there is a continuous complete, at which point there is no distinction transition between ordered and disordered states. The nature of the transition is controlled between dolomite and a calcite-magnesite solution by the parameter values for the speciation M2 M1 (Ca,Mg)0.5 (Ca,Mg)0.5 CO 3 . tions. In the case of calcite magnesite, mixing Chemically the dolomite solution is analagous to occurs on M1 and M2 with site fractions the H-O fluid speciation discussed earlier, in that zy= we can write a reaction between calcite (cc) and M1,Ca cc zM1,Mg= yy mag + odo magnesite (mag), in both of which no octahedral 11.25 = + site order-disorder is possible, and ordered dolomite zM2,Ca yy cc odo

(odo) zyM2,Mg= mag ½ mag + ½ cc = odo required by the species site populations, deduced as Thus, we can formulate a speciation model of previously for plagioclase (chapter 8). From Eqs calcite-magnesite solution that accounts for 8.31 and 11.25, the change in configurational dolomite formation with cc, mag and odo as entropy of our calcite-magnesite model (n=2, species. However, in contrast to the H-O fluid, mM2 =mM1 = 2, qqM2= M1 = 12) is where our atomistic model of the fluid consisted of conf s=−R 2 [ yycc ln cc +− (1 ycc ) ln(1 − ycc ) true H2O, H2, O2 molecules, in a crystal structure 11.26 +yymagln mag +− (1 ymag ) ln(1 − ymag )] the species influence individual identisite popula-

57

note that in this case the configurational entropies tion gX( 1,..., Xcr− 11 , Q ,..., Q) , whereas with of the endmembers are zero, since in cc and mag speciation gX( 1,..., Xcc−+ 11 , y ,..., y t) . The relation there is only one octahedral cation, and in odo, although there are two cations, they are perfectly between the independent order parameters and ordered onto M1 and M2 (i.e., smech =0). As in the species concentrations is recovered by substituting H-O fluid, the species fractions of calcite and the site fraction definitions in terms of species magnesite can be expressed from 11.16 and 11.17 fractions into the definition of the order parameters in terms of bulk composition and the fraction of the (or vice versa). Thus, for calcite-magnesite from ordered dolomite species 11.25 and 11.28 we have the trivial result Qy=mag +− y odo y mag = y odo . nCa Xcc = =yycc + odo 2 nnCa+ Mg Not uncommonly, the relationship between order- parameters and species fractions is more complex, ymag =−−1 yycc odo but in essence order parameters are species and rearranged to obtain fractions, and just as in a speciation model, the yXycc= cc − odo 2 11.27 equilibrium state of order in an order-disorder y=−−1 Xy2. mag cc odo model is determined by finding values of the order mech ex Combining suitable expressions for g and g parameters that minimize the free energy of the with 11.26 and substituting 11.27 in the result solution at a given bulk composition. Or analogous- would yield gX( cc, y odo ) which is minimized at ly to 11.22 constant Xcc (and, of course, P and T) to determine ∂g = =  0jr 1, , . 11.29 the equilibrium free energy and speciation of the ∂Q j XX − solution phase. As discussed previously, this may 11c Mathematically there is no distinction between be done by free energy minimization, or, as is more speciation models and order-disorder models, but common, by solving 11.22 or 11.24. the latter formulations are more popular in the

geological literature. I personally prefer the In effect speciation models are temperature (and in speciation formulation because mass balance some cases pressure) dependent models for the constraints and the physical meaning of terms, such configurational entropy of a solution. Recognition as of this aspect of speciation models had led to W yy , alternative formulations of the speciation problem cc-odo cc odo in which the free energy of solutions are expressed are relatively transparent. In contrast, in the order- disorder formulation the solution model must be in terms of bulk composition and microscopic expressed as a function of macroscopic composi- order parameters that specify crystallographic site populations. For example, in our calcite-magnesite- tional variables and microscopic ordering parameters, e.g., the above term is ordered-dolomite model, we may define a micro- W( X− QQ) , scopic order parameter cc-odo cc

Qz=M1,Mg − z M2,Mg 11.28 and as there is no closure constraint on microscopic that describes the degree to which Mg is ordered ordering parameters, mass balance constraints tend between the M1 and M2 sites. For the endmember to be less straightforward than in speciation models. site populations, Q is 0 for calcite and magnesite ORDER-DISORDER PHASE TRANSFORMATIONS species and 1 for ordered dolomite. Since the microscopic site fractions of a speciation model are Order-disorder in solids is intimately associated completely determined by the macroscopic with polymorphic phase transformations. This fractions of the independent species, it is mathemat- association arises because ordering may result in a ically evident that the number of independent lowering of point group symmetry. For example, in ordering parameters in an order-disorder solution high temperature dolomite the M1 and M2 sites model r = t − c the number of independent reactions have the same average site populations and are thus that can be written between the species of the chemically indistinguishable and do not constrain solution. Thus, in the order-disorder formula- symmetry; whereas at lower temperatures the ordering of Ca and Mg onto M2 and M1, respec-

58

tively, makes the sites chemically non-equivalent transformation is an additional example of a and distinguishable; an effect that can only decrease transformation that is appropriately by speciation. symmetry. The transformation from ordered to disordered point group may be smooth in that the PROBLEMS order-parameter that characterizes the transition 11.1) Holland & Powell (1996b) suggest a specia- varies continuously towards zero (i.e., toward the tion model for omphacitic clinopyroxene M2 M1 disordered high symmetry state), in which case the [(Ca,Na) (Al,Mg) Si2O6] in which they propose phase is said to have convergent ordering (Fig that at low temperature the M2 and M1 sites split 11.5). Alternatively, the ordering parameter may into four non-equivalent octahedral sites M2a, jump from finite values to zero across a boundary M2b, M1a, and M1a with the following species: in pressure-temperature-composition space that is said to define the critical line of non-convergent Site ordering transformation (the boundary between M2a M2b M1a M1b low-symmetry dolomite and fully disordered qi 1/2 1/2 1/2 1/2 calcite-magnesite solution in Fig 11.5). Speciation di Ca Ca Mg Mg models are capable or reproducing both convergent Species: jd Na Na Al Al and non-convergent phase transformations. The om Na Ca Al Mg relationships between the model parameters necessary to produce a particular behavior is Additionally they specify that the free energy discussed in detail by Holland & Powell (1996ab). change for the speciation reaction om − ½ jd − ½ di = 0 is −3500 J/mol and independent of pressure SPECIATION MODELS FOR PURE PHASES and temperature and specify the following interac-

As the speciation problem is solved at constant bulk tion energies: Wdi-jd = 26000 J/mol, Wom-jd = 16000 composition, it may be self-evident that the J/mol, Wom-di = 16000 J/mol. At 773 K, for a bulk thermodynamic properties of pure phases, i.e., composition identical to that of the om species, use phases that can have only one bulk composition, the model to compute: a) the equilibrium specia- can be described by speciation models. A pure tion; b) Holland & Powell’s order parameter Q = oxygen fluid is a trivial example of this application. zM1a,Al – zM1b,Al; and c) the thermodynamic activity Ordinarily we assume such an oxygen fluid to be of diopside. composed of the dimer O2, but spectroscopic measurements show that while O2 is usually the dominant species, ozone O3 and monatomic oxygen O can also be significant species if not dominant species in oxygen fluids. For such a system, minimization of the free energy of a speciation model analogous to our H-O fluid model would yield the equilibrium speciation and true free energy for oxygen.

A more geological example of the application of speciation models to pure phases is the prediction of order-disorder transformations in pure minerals. A prominent example is albite, where at high temperature Al and Si are disordered on the T2 T2 T1 site, Na( AlSi) ( Si)2 O8 , while at low temperature the T2 site splits to T2a and T2b with Al and Si confined to a single site, i.e., T2a T2b T1 α β Na( Al) ( Si) ( Si)2 O8 . The / quartz

59

12: RECIPROCAL SOLUTIONS KBr CsBr “What is a reciprocal solution?” -R. Powell, 2006 “The answer my friend is blowing in the wind” -Puff, the Magic Dragon, ca 1966 If a solution model is described in terms of t > c chemical entities (e.g., endmembers or species), it NaBr is always possible to describe t − c of these entities as a chemical mixture of c compositionally KCl CsCl independent endmembers. In speciation models, although the compositions of t − c entities are compositionally independent, the remaining thermodynamic properties of the entities are w independent. Thus, the stable speciation can only be x y determined by finding the mixture of these entities that minimizes the total energy of the solution subject to mass balance constraints. Logically there NaCl can be only one other limiting case of a solution Fig 12.1 Triangular prismatic composition space of the reciprocal (Na,K,Cs)(Cl,Br) model with t > c endmembers; to wit, reciprocal solution, formed by joining the 2-dimensional solutions in which the thermodynamic properties of composition space of the cation site (i.e., the − triangular faces) with the 1-dimensional t c of the endmembers are dependent. However, if composition space of the anion site (the the t − c endmembers are both compositionally and vertical axes). Although the solution has 6 thermodynamically dependent, then why not endmember compositions, only 4 of the endmembers are independent (i.e., the describe the solution in terms of c independent composition space is 3-dimensional and endmembers? The answer to this question is that a therefore can be defined by 4 components). reciprocal formulation provides a simple means of Na1−x−yKxCsyCl1-wBrw = relating real and model solution compositions. For NaCl + x KNa−1 + y CsNa−1 + w BrCl−1 this reason, reciprocal solution models are common 12.1 in petrology. Unfortunately, the failure to recognize where the exchange operators, KNa−1, CsNa−1 and that the endmembers of reciprocal solutions cannot BrCl−1, are imaginary chemical entities that be independent is a widespread and fundamental combine linearly with a real composition to obtain thermodynamic error. another real composition, i.e., to affect an ex- change. NOTE: as currently written, this chapter may not account for the possibility that the Helffrich Green Exchanges are mutually dependent if the stoichio- Stuffed moles normalization factor Ξ is not metric coefficients of the corresponding exchange constant (Eq 11.11). operators are dependent. For halite solution 12.1, charge balance requires WHAT IS A RECIPROCAL SOLUTION? 01≤+≤xy 12.2 A reciprocal solution is a phase that exhibits more 01≤≤w 12.3 than one independent chemical exchange. The thus, the exchanges of K and Cs for Na are classical example being ionic salts such has halite, mutually dependent, but independent of the BrCl−1 in which the exchange of alkali metal cations is, at exchange. Each group of dependent exchanges least to a good approximation, independent of the defines an orthogonal composition space, which exchange of halogen anions. Taking solution of K, may be thought of as defining the composition of a Cs and Br in halite as an example, we may write chemical mixing site. If the number of independent any general halite composition as the sum of a basis species mixing on the ith chemical mixing is site is composition (e.g., NaCl) and three exchange ci, the dimension of the corresponding composition operators space must be a di-dimensional simplex, where di =

60

ci −1. The (orthogonal) conjunction of these In combination Eqs 12.7 and 12.9 provided a composition spaces defines the complete composi- perfectly admissible means of calculating the tion space of a reciprocal solution which is a properties of a reciprocal solution in terms of t > c polygon with endmembers that are defined by the stoichiometric M limits of a reciprocal solution. However, in tc= ∏ i 12.4 i contrast, to speciation problems where the fractions vertices and dimension 0 M d=∑ dci = −1 , 12.5 i where M is the number of mixing sites, correspond- ing to the endmember compositions of the reciprocal solution. In halite (Fig 12.1), taking the cation site as site 1, c1 = 3, c2 = 2, the number of endmembers is t = 6 from 12.4, and the dimension P of the composition space (a triangular prism) is d = 0 1 3 and therefore c = 4, from 12.5. The result that M cc< ∑ i 12.6 i is a peculiar consequence of the fact that the composition spaces share a common origin (e.g., 12.1). 1 Adopting the notation that the composition of NaCl independent species j on chemical mixing site i of a Fig 12.2 Reciprocal salt composition space as in Fig 12.1, showing the barycentric coordi- solution is Zij; and that yklm… is the fraction of the nates ZZ1,Na and 2,Cl for the composition endmember with species k on the mixing site 1, indicated by point P. The vertical projection of species l on mixing site 2, species m on site 3, and P onto the triangular face of the prism is Z1,Na so forth, geometric argumentation gives which is identical to the area of the heavily shaded triangle; the horizontal projection of P y= ZZZ  12.7 klm... 1 k 2 l 3 m onto the vertical axis defines Z2,Cl ; thus the For halite this is illustrated in Fig 12.2, in which it shaded volume is identical to barycentric fraction yNaCl , as given by eq 12.7, for the salt at is seen that if a general point is placed in the composition P. interior of the composition space it is possible to fill of the chemically dependent species yy,..., are the composition space with t non-overlapping ct+1 triangular prisms that share this vertex. The volume determined by thermodynamics, 12.7 implies the of the triangular prism opposite the NaCl vertex is fractions of all species are determined by stoichi-

Z1Na Z2Cl which, from 12.7, is yNaCl. Eq 12.7 permits ometry. two equivalent expressions for any extensive or molar property ψ of a solution, i.e., In petrology, it is not uncommon that this is where cc12c3 the story of reciprocal solutions ends, i.e., the ψ= ...ZZZ ψ 12.8 ∑∑∑ 1k 2 l 3 m klm ... properties of the t reciprocal endmembers are klm which, I think, is called a Bragg-Williams determined, and configurational entropy and excess summation, or, dropping the multiple subscripts, energy models are formulated in terms of the the form employed previously for simple solutions fractions of these endmembers. Sadly, this simple t ending is also an unhappy ending, because although ψ= ψ ∑ yii 12.9 12.7 assigns endmember fractions that satisfy mass i balance, the assignment is not unique, i.e., any as in, e.g., 6.4 t general composition of the solution in an infinite mech =  g∑ ygii number of ways from the t endmembers. This i=1 statement follows from the observation that we may

61

vary the fractions of t − c endmembers arbitrarily reciprocal solution model as a function of inde- provided the remaining fractions conserve mass, pendent endmembers and the formulation of simple i.e., solution models as outlined in chapter 8. Thus, the 11 cc i viiφ + + v φ +φ =0 ic = + 1... t. 12.10 answer to Powell’s introductory, and probably These variations are rhetorical question, question is that from a mathe-

δy1i + +δ yci +δ y i =0 ic = + 1... t 12.11 F where δyi is an arbitrary variation of the fraction yi and δyji is the variation in the fraction of yj required in response to δyi by mass balance (12.10). Equating 12.10 and 12.11 gives BC BD j δyyji = −ν i δ i . 12.12

Since the variations δδyyct+1,..., are arbitrary, any composition yy1,..., t from 12.7, can be converted to a composition involving only c independent endmembers by setting E AC AD G δ=−yii y ic =+1... t. 12.13 Designating the fractions of these endmembers in Fig 12.3. Possible representations of the composition space of a reciprocal solution this new composition pi, 12.11-12.13, give t (A,B)xy (C,D) . All possible compositions of the j solution can be respresented as a positive pyi=−ν i∑ ij yi =1 c. 12.14 jc= +1 linear combination of E, F, and G. Alternatively, compositions can be represented Thus if we have a reciprocal solution model in terms of any three endmember compositions of the reciprocal solution if negative composi- gy( 1 ,..., yt ) , 12.14 provides a mathematically tions are allowed. For example, taking AC, BC and BD as independent endmembers, then the equivalent model gp( 1 ,..., pc ) . Furthermore, as the composition of the AD endmember is yAC =− 1, models are mathematically equivalent, it follows yyBC=1, BD =1; and the thermodynamic that the thermodynamic properties of the t depend- properties of the dependent endmember AD must be given by the solution model at this ent endmembers of the reciprocal solution must correspond to those of the model gp( 1 ,..., pc ) at matical perspective there is no such thing as a the endmember compositions. reciprocal solution.

Until now, we have implicitly assumed that Before going on to consider the complexities that compositions are positive, but there is no funda- may arise in silicate solutions, it may be useful to mental requirement for this. A simple make the algebra of 12.10-12.14 less abstract by demonstration of this follows from the choice of revisiting the halite (c=4, t=6) example. Taking

H2O and O2 as components for an H-O system. H2 KBr and CsBr as dependent endmembers, 12.10 is

= H2O – ½ O2, thus for H2, nH2O = 1 and nO2 = – ½, NaCl − KCl − NaBr + KBr = 0 and XH2O = 2 XO2 = – 1. While, the fractions defined NaCl − CsCl − NaBr + CsBr = 0. by 12.7 are positive, reciprocal solution composi- tions defined by p1,…, pc are negative over half the For a composition Na1−x−yKxCsyCl1-yBry with x = y = composition space of the reciprocal solution. The ⅓ and w = ½, from 12.7, each endmember has a 1 reason for these negative compositions are that the fraction of 6 , thus 12.14 gives th 1 composition of the j dependent endmember, from pNaCl = yNaCl − yCsBr − yKBr = − 6 1 12.14, is pKCl = yKCl + yKBr = 3 j 1 pCsCl = yCsCl + yCsBr = pii= −ν ic =1 . 12.15 3 1 . As at least one stoichiometric coefficient in 12.10 pNaBr = yNaBr + yCsBr + yKBr = 2 must be positive, at least one composition from 12.15 must be negative. Other than this technicali- A SILICATE EXAMPLE: CHLORITE ty, there is no difference between the formulation of An important chemical exchange in silicates is the coupled Tschermaks exchange

62

Mg4 Al 4 Si 2 O 10 (OH) 8 Fe4 Al 4 Si 2 O 10 (OH) 8 clinochlore composition. The compositions of the amesite Fe-amesite remaining endmembers (Fig 12.4) can be deduced from 12.16 with {x=1, y=0}, {x=0, y=1}, {x=1,

Z y=1}. Thus the chlorite has a quadrilateral composi- 2 , ( Z M g 2 , A

x tion space (c=3, t=4) and one dependent F e l 2 1 ( − M x endmember. ) g S i A l x F e 1 − −

2

y x ) − 1 S i Table 12.1 Endmember site populations for − 1 chlorite. The table shows only sites on which solution occurs, chlorite has two additional sites: clinochlore x daphnite one M4 occupied solely by Al, and two T1 sites Mg5 Al 2 Si 3 O 10 (OH) 8 Fe5 Al 2 Si 3 O 10 (OH) 8 occupied solely by Si. Z1,Fe

Z1,Mg Site: M T Fig 12.4 Reciprocal composition space of a chlorite model, although the solution can be qi 5 2 described in terms of two independent Clin Mg Al1/2Si1/2 exchanges, the chemical identity of the second (Tschermaks) exchange is dependent on the Species: Daph Fe Al1/2Si1/2 first. Moreoever, in contrast to the salt example Ames Mg4/5Al1/5 Al of Figs 12.1-2, the chemical mixing sites do not correspond to crystallographic sites (Tab fames Fe4/5Al1/5 Al 12.1).

VI IV VI IV Table 12.2 Endmember site populations for a Al AlM Si or simply Al2M−1Si−1, in which -1 -1 chlorite model to depict the tendency of Al to order 3+ Al replaces a divalent cation M on an octahedral preferentially on one octahedral site (M1). This site and a tetrahedral Si. For a solution with more model is incorrect because the atomic site fraction than one kind of divalent cation mixing on octahe- of Al on the M1 site of ordered Fe-amesite dral sites, the Tschermaks exchange is independent (ofames) cannot be formed from a linear combina- if the exchange does not alter the relative propor- tion of the M1 site populations of ordered amesite tions of the divalent cations. Thus for an Fe-Mg-Al- (oames), daphnite, and clinochlore that satisifies silicate, the Tschermaks exchange is conservation of mass (i.e., 12.18). Al2(FexMg1−x)−1Si−1. As written, the Tschermaks exchange operator is only dependent on the number Site: M1 M2 T2 of divalent cations, i.e., 1+x−x, and as this number qi 1 4 2 is constant, the Tschermaks exchange is independ- Clin Mg Mg Al1/2Si1/2 ent. The combination of Tschermaks with FeMg-1 Species: Daph Fe Fe Al1/2Si1/2 exchange therefore results in a reciprocal solution, Oames Al Mg Al and as both exchanges are typical of Fe-Mg-Al- ofames Al Fe Al silicates, reciprocal solutions are an omnipresent feature of the petrologic landscape. To illustrate the This example reveals a common complexity of complete formulation of a reciprocal silicate silicate reciprocal solutions, in that the chemical solution consider a chlorite model mixing sites bear little or no relation the crystallo- (FexMg1−x)5−yAl2(1+y)Si3−yO10(OH)8 = graphic sites (Table 12.1), a consequence of the fact Mg5Al2Si3O10(OH)8 + x FeMg-1+ y that the Tschermaks substitution operates across Al2(FexMg1−x)−1Si−1, 0≤≤xy 1, 0 ≤≤ 1 12.16 sites. Additionally, although the exact chemical identity of the Tschermaks exchange cannot be where the basis endmember (x=0, y=0) specified without knowledge of the FeMg-1 Mg5Al2Si3O10(OH)8 is known as clinochlore, as x exchange, we can specifiy the composition of the and y are independent it is evident that the Chlorite chemical mixing sites representing Tschermaks is a reciprocal solution with two binary chemical solely as a function of the stoichiometric coefficient mixing cites, therefore the composition space is the y deduced from the chlorite formula. Thus, we may conjunction of two orthogonal 1-d simplexes at the

63

4 express the species fractions on the two independ- zppM,Mg= clin + 5 ames ent mixing sites as: zpM,Fe= daph

ZxM,Fe = 11 zM,Al=55 pp ames + fames 12.22 Zx=1 − M,Mg z=++11 p pp 12.17 T,Al22 clin daph ames ZyT,Al( Fe Mg) Si = 11 211xx−−−1 zT,Si=22 pp clin + daph Zy=1 − T,( Fexx Mg12−−) SiAl Evaluating 12.22 at the composition of pure conf conf Given 12.17 and 12.7, the barycentric endmember daphnite and clinochlore yields ssRclin= daph = ln 4 conf fractions are and for amesite sRames = ln(3125/ 256) . Thus, from

yclin= ZZ M,Mg =−−(11x)( y) T,( Fexx Mg12−−) SiAl 6.13, ∆=conf conf −conf −conf − conf . ydaph= ZZ M,Fe T,( Fe Mg) SiAl = x(1 − y) sspspspsclin clin daph daph ames ames xx12−− y= Z Z =(1 − xy) Evaluating this model at the composition of Fe- ames M,Mg T,Al211( Fexx Mg−−) Si −1 44 amesite (i.e., pclin =−==55, ppdaph ,ames 1), the yfames= Z M,Fe ZT,Al( Fe Mg) Si = xy 211xx−−−1 free energy of pure Fe-amesite is: Selecting Fe-amesite as the dependent endmember, 44   gfames =−++55 gclin gg daph ames and making use of the mass balance relation . −16 −+44 4 4 20WWWclin-daph5 clin-ames5 daph-ames 5 clin − 5 daph − ames + fames = 0. 12.18 The chemical difference 4 4 SPECIATION IN RECIPROCAL SOLUTIONS 5 clin − 5 daph = 4 MgFe-1 thus 12.18 can be rewritten A deficiency of the model just outlined is that in fames = ames − 4 MgFe-1 12.19 natural chlorites Al partitions preferentially, i.e., where MgFe-1 is implicitly the Mg-Fe exchange orders, into one of the five octahedral sites operator for the octahedral site of chlorite. From previously designated M (in addition to the M4 site, 12.14, the independent endmember fractions are which was assumed to contain only Al). A first 4 pyclin= clin − 5 y fames guess as to the model necessary to describe this 4 pydaph= daph + 5 y fames 12.20 behavior would logically be a reciprocal solution with endmember site populations as proposed in pyyames= ames + fames Table 12.2. Unfortunately, the model is incorrect For the sake of completeness, the components of a because the site occupancy of the ordered Fe- complete chlorite model, assuming a regular excess amesite endmember cannot be formed by a linear function are: combination of the site populations of the remain- gmech =++ pg p g p g  clin clin daph daph ames ames ing endmembers that satisfies conservation of mass: ex 4 4 g= Wclin-daph pp clin daph+ W clin-ames pp clin ames ofames = oames − clin + daph 12.23 12.21 5 5 +Wdaph-ames pp daph ames This difficulty follows from the fact, that if the M1 gconf =−∆ Tsconf and M2 sites are energetically distinct for Al, then conf they also must be energetically distinct for Fe and where ∆s is necessary because the clinochlore Mg. Microscopically, this means that chlorite is and daphnite endmembers have Al-Si disorder on characterized by two thermodynamically independ- the T2 site. From the site populations in Table 12.1 M1 ent MgFe−1 operators, one for M1 ([MgFe−1] ) and and 6.10 M1 − conf another for M2 ([MgFe 1] ), so the chemical s=−+R[5( zM,Mg ln zzz M,Mg M,Feln M,Fe 4 4 difference 5 clin − 5 daph becomes + 4 4 4 M1 16 M2 zzM,Alln M,Al ) 5 clin − 5 daph = 5 [MgFe−1] + 5 [MgFe−1]

++2(zT,Al ln zzz T,Al T,Si ln T,Si )] 12.24 where which cannot possibly yield the ofames site population in combination with the oames site populations because oames does not have Mg on the M1 site. The only means of resolving this dilemma is to introduce an additional endmember that allows us to discriminate between the energet-

64

ics of the two substitutions. The anti-ordered However the complete formulation of such a model amesite (aames) and Fe-amesite (afames) endmem- is a little too much for these notes. bers of our previous model serve this purpose (Table 12.3). In this case, Table 12.3 Endmember site populations for a valid M1 M2 aames – afames = [MgFe−1] + 3 [MgFe−1] reciprocal chlorite model that accounts for Al- 12.25 partitioning of Al preferentially on M1. The model which can be combined with 12.24 to isolate is a reciprocal solution with speciation. M2 [MgFe−1] is possible to isolate the two octahedral exchanges as: Site: M1 M2 T2 M2 (clin – daph) – (aames – afames) = [MgFe−1] qi 1 4 2 M1 4 − 3 (aames – afames) – 3(clin – daph) = [MgFe 1] clin Mg Mg Al0.5Si0.5

12.26 daph Fe Fe Al0.5Si0.5 Since aames and afames are chemically equivalent oames Al Mg Al to oames and ofames, it is evident that the model ofames Al Fe Al must be a reciprocal solution with speciation. aames Mg Mg3/4Al1/4 Al

afames Fe Fe3/4Al1/4 Al

65

13:AN INDEX OF SORTS equilibrium, 7 activity, 40 equilibrium constant, 41 coefficient, 43 Euler’s chain rule, 20 adiabatic bulk modulus, 21 Euler’s criterion, 4, 5 amount, 9 exchange operators, 60 relative, 12 extensive, 9 Birch-Murnaghan EoS, 24 first law, 1 Boltzman, Ludwig E, 37 Fowler, 7 Boyle, 49 free energy, 17 Bragg-Williams summation, 61 free energy function, 18 bulk modulus fugacity, 50 isothermal, 23 fugacity coefficient, 51 Caratheodory, 7 Gibbs differential, 6 Charles, 49 Gibbs energy chemical mixing site, 60 apparent, 26 Clapeyron, 49 Gibbs energy Clausius, 5 formation from the elements, 26 Clausius-Clapeyron relation, 34 Gibbs energy components, 28 mechanical mixture, 36 compound, 30 Gibbs energy compound formation models, 57 configurational, 36 compressibility Gibbs energy isothermal, 23 excess, 36 conjugate, 7 Gibbs function, 16 conservative properties, 9 Gibbs-Duhem relation, 7 continuity, 9 Goldschmidt, 17 convergent ordering, 59 Gruneisen thermal parameter, 21 Corresponding State theory, 50 heat capacity Coulombic force, 50 isochoric, 25 covolume, 49 heat capacity critical conditions, 39 isobaric, 24 critical line, 59 Helmholtz energy, 19 critical point, 48 Henry's law Dantzig, George, 32 constant, 43 Darken's quadratic formalism, 43 heterogeneous system, 11 Debye, 25 homogeneous function, 9 Debye temperature, 25 ideal gas law, 49 diathermal, 16 inductive forces, 50 dilational work, 1 inexact differential, 1 dispersive forces, 50 infinite dilution, 43 displacive transformation, 25 integration by parts, 51 DQF, 43 intensive, 9 Dulong-Petit limiting law, 25 internal, 11 electrolytes, 49 internal energy, 1 electrostatic force, 50 internal equilibrium, 7 endmembers, 36 inviscid, 16 enthalpy, 19 ionic fluid, 48 entropy, 5 isobaric expansivity, 23 EoS, 23, 49 Legendre transformation, 18 lever rule, 12

66

linear programming, 32 solution model Lomonosov, 9 regular, 38 London forces, 50 subregular, 38 LP, 32 solution phase, 36 Maple solvus, 39 derivatives, 22 sound velocity, 21 plotting points, 13 specific variables, 9 solving matrix problems, 13 spinodes, 39 Margules parameters, 38 spontaneous process, 14 Maxwell relations, 20 stable equilibrium, 14 Mie-Gruneisen, 24 standard state minimum free energy surface solute, 43 compounds, 30 solvent, 43 mobile components, 20 state, 9 molar variables, 10 state function, 1 mole, 10 Stefan mole fractions, 10 Jozef, 1890, 18 molecular fluid, 48 Stirling's approximation, 37 Murnaghan equation, 24 supercritical, 49 natural variables, 19 third law, 7 Nernst, 7 third law entropy, 25 non-convergent ordering, 59 total differential, 4, See complete differential nuclear repulsion, 50 Tschermaks exchange, 63 objective function, 32 Van der Waals EoS, 49 optimization, 32 variance, 12 order parameters, 58 volume, 1 order-disorder solution models, 57 work function, 18 partial differential notation, 19 zappy do law, 5 path dependent, 1 zero’th law, 7 phase, 11 phase rule Khorzhinskii, 20 phase rule, 12 Goldschmidt's, 17 mineralogical, 17 potential functions, 6 potentials, 2, 6 projection, 31 Raoult's Law, 43 reaction coefficient, 34 reaction equation, 34 reciprocal solutions, 60 Redlich-Kwong EoS, 50 relative amount, 12 reversible, 5 simplex, 11 algorithm, 32 size, 9 solution model, 36

67

Since t−c = 1, there is only one independent EXAMPLE: C-O-H FLUID SPECIATION reaction between the species of the form EQ REF!

For geological conditions, graphite-saturated fluids CO242+= CH 2H O 12.30 can to a good first approximation be considered (after projection through carbon); thus Ωmech =y ∆Ω 12.31 mixtures of three molecular species water, carbon HO2 dioxide (CO2) and methane (CH4). Since the system where ∆Ω is the free energy change of 12.30. is saturated in a pure carbon phase (graphite), the Given that the model is for a molecular fluid, a Legendre transform simple molecular configurational entropy model Ω= − o,graphite G nGC (6.9) is appropriate in which case allows us to eliminate carbon as a component with S conf the result that the fluid has c=2 components (H,O)  =−RyH Oln ( y H O) ++ y COln ( y CO) y CHln ( y CH ) and t=3 species with the carbon-free compositions 2 2 2 24 4

H2O, O2 (carbon dioxide) and H4 (methane). 12.32

Choosing XO as the independent macroscopic and, for the sake of simplicity, neglecting non- variable, EQ REF yields ideality sol mech conf HO HO CO Ω=Ω−TS y n22++ n yn2 ( H22 O H O CO O ) X O = which after substituting the expressions for ynCH4+ y nHOHO 22 ++ n ynCO2 ( CH42 H H O H O CO2 O ) y and y as functions of XO and yH2O from  CO2 CH4 12.27 12.29 and 12.28 can be differentiated with respect and from EQ REF to yH2O and equated to zero to solve for yH2O as a y=−−1. yy 12.28 CO2H 24 O CH function of XO , pressure and temperature. This Substituting 12.28 into 12.27, and rearranging the expression is too ugly to produce here, but it is result gives methane concentration easily generated and solved in maple (Problem 9.1).

CO22 H O H 2 O CO2 (nO−− n O ny H) HO − nX O O 2 PROBLEM H O CO CO +−n22 ny + n 2 O O HO2 O 11.1 Using the graphite-saturated COH fluid yCH = 4 CO2−− CO 24 CH nO n OHO nX speciation model as formulated in the preceding 12.29 example to calculate the speciation at 1000 K and 1 kbar for XO = 0.1, 1/3 and 0.8. At these condi- as a function of the macroscopic composition XO tions ∆Ω for reaction 12.30 is -80000 J/mol H2. and yH2O, the concentration of ordered species, a result can also be substituted into 12.28 to obtain

CO2 concentration as a function of XO and yH2O.

68