DOCSLIB.ORG

Thermodynamic Potentials and Natural Variables

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Thermodynamic Potentials and Natural Variables Revista Brasileira de Ensino de Física, vol. 42, e20190127 (2020) Articles www.scielo.br/rbef cb DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2019-0127 Licença Creative Commons Thermodynamic Potentials and Natural Variables M. Amaku1,2, F. A. B. Coutinho*1, L. N. Oliveira3 1Universidade de São Paulo, Faculdade de Medicina, São Paulo, SP, Brasil 2Universidade de São Paulo, Faculdade de Medicina Veterinária e Zootecnia, São Paulo, SP, Brasil 3Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brasil Received on May 30, 2019. Revised on September 13, 2018. Accepted on October 4, 2019. Most books on Thermodynamics explain what thermodynamic potentials are and how conveniently they describe the properties of physical systems. Certain books add that, to be useful, the thermodynamic potentials must be expressed in their “natural variables”. Here we show that, given a set of physical variables, an appropriate thermodynamic potential can always be defined, which contains all the thermodynamic information about the system. We adopt various perspectives to discuss this point, which to the best of our knowledge has not been clearly presented in the literature. Keywords: Thermodynamic Potentials, natural variables, Legendre transforms. 1. Introduction same statement cannot be applied to the temperature. In real fluids, even in simple ones, the proportionality Basic concepts are most easily understood when we dis- to T is washed out, and the Internal Energy is more cuss simple systems. Consider an ideal gas in a cylinder. conveniently expressed as a function of the entropy and The cylinder is closed, its walls are conducting, and a volume: U = U(S, V ). fluid at fixed temperature T surrounds it. The cylinder Experimentalists frown at this dependence, for en- contains a fixed number N of particles. Its volume V can tropies are more difficult to measure than temperatures. nonetheless be varied, by means of a plunger. We push Moreover, they may have to let the plunger move back the plunger and do work on the gas. Heat flows out to and forth and adjust the cylinder pressure to that of the maintain the gas at the temperature T . If, instead, we environment. They would therefore prefer the variables slowly pull out the plunger, heats flows inwards. P and T , instead of S and V . In experiments, the pressure P applied to the plunger At this point, it would seem natural to say “Very well, can be controlled, and the fluid temperature T can also let us make the experimentalist happy. Determine the be controlled. The variables P , T and V are not indepen- entropy and volume as functions of T and P . Since U is dent. It has been known for centuries that this system known as a function of S and V , all we have to do, then, approximately obeys an equation of state, which binds is to substitute S(T,P ) and V (T,P ) for S and V . The the three variables: result will be U[S(T,P ),V (T,P )], which expresses U as a function of T and P .” PV = Nk T, B (1) But then we would be at odds with good textbooks, which prefer to define another extensive quantity, on where kB is Boltzmann’s constant. equal footing with the internal energy: the Gibbs Free One might think that Eq. (1) is all we need to describe Energy G(T,P ), a function of the experimentalist’s fa- the ideal gas. The equation of state, nonetheless, tells us vorite variables. The Internal Energy and the Gibbs Free nothing about other important properties of the system. Energy are thermodynamic potentials. As we shall see, a The Internal Energy U rises or drops as the temperature simple expression relates G(T,P ) to U(S, V ). The Gibbs changes. We might also be interested in the entropy S, Free Energy G must be expressed as a function of T and so forth. and P , just like the Free Energy U must be expressed The ideal gas is a mathematical concept: a collection as a function of S and V . The temperature T and the of particles moving independently from each other and pressure P are called the natural variables of G. The bouncing elastically off the cylinder walls. The simple dy- natural variables of U are S and V . namics makes U directly proportional to the temperature Once again, our inner voice interrupts the train of and number of particles. The proportionality to N, a vari- thoughts to argue in favor of the experimentalist: “This able that—except as explicitly indicated—we will keep is all very well, but G is obtained from U. Therefore, in fixed, is an inherent property of the internal energy. The order to compute G(T,P ), we will first have to express *Endereço de correspondência: [email protected]. the entropy and volume as functions of temperature and Copyright by Sociedade Brasileira de Física. Printed in Brazil. e20190127-2 Thermodynamic Potentials and Natural Variables pressure to compute U[S(T,P ),V (T,P )]. Our friend, the and the pressure experimentalist, wants the Internal Energy as a function ∂U(S, V ) ∂U of T and P . So give her U[S(T,P ),V (T,P )]. Why waste P = − ≡ − . (4) ∂V ∂V time computing G(T,P )? Who needs the Gibbs Free S Energy?” Here we have adopted the traditional notation [2, 4], Textbooks on Thermodynamics define the thermody- which explicitly indicates the fixed quantity defining namic potentials and associate them with their natural the partial derivative. Since we work with fixed number variables. A few texts offer indirect answers to the two of particles, to simplify the notation we have avoided questions we have posed. To explain why S and P are the adding a second, redundant subscript N to the partial natural variables of the Enthalpy, Schroeder shows that derivatives. changes in H have simple physical interpretation when In view of Eq. (3), T and S are said to be conjugate the pressure is held fixed [1]. To explain why T and P are variables. Likewise, in view of Eq. (4), P and V are the natural variables of the Gibbs Free Energy, Callen conjugates. In a conjugate pair, one of the variables (T or shows that, for a system in contact with a pressure and P ) is intensive, independent of the number of particles, a temperature reservoirs, the equilibrium value of any of while the other (S or V ) grows in proportion to the its internal parameters will minimize G at the reservoir number of particles and is, hence, extensive. pressure and temperature [2]. Callen also explains that The dependence on extensive variables makes U some- information is lost when a thermodynamic potential is what special among the thermodynamic potentials. That expressed not as a function of its natural variables, but the Internal Energy is a (single-valued) convex function as a function of its derivative with respect to those vari- of all its natural variables, S and V , was first pointed ables [2]. To prop up the argument, the author presents a out by Gibbs. Given the convexity and that S and V are geometrical depiction of this problem and of its solution extensive, it is a simple matter to derive the Second Law by means of Lagrange Transformations. This explanation of Thermodynamics, as recalled by Wightman, in a re- pleases experienced teachers. To the same extent, it dis- markably clear, succinct account of Gibbs’s reasoning [5]. appoints the students, who often regard the illustration The other thermodynamic potentials cannot be directly as an abstract solution to an abstruse conundrum. related to the Second Law because they depend on one or The purpose of this paper is to answer the two more intensive variables; while they are convex functions questions directly, in a physical setting, that is, of the extensive variables, they are concave function of to explain why it is better to work with G(T,P ) the intensive ones [2]. than with U[S(T,P ),V (T,P )]. We will show that Nonetheless, practical considerations often focus one’s U[T,P ] ≡ U[S(T,P ),V (T,P )] conveys less information interest on intensive variables. One must then turn to than U(S, V ), or G(T,P ). In order to preserve informa- thermodynamic potentials other than U(S, V ). We have tion, we have to express each thermodynamic potential already discussed one of them, the Gibbs Free Energy as a function of its natural variables. Conversely, the vari- G(T,P ). Since we have four variables (S, V , T , and P ), ables of interest uniquely determine the thermodynamic there must be more than two thermodynamic potentials, potential. In our example, the experimentalist would like and can ask for the number of alternatives. to work with T and P . She should then ask for G(T,P ), As illustrated by Eq. (2), the differential of a thermo- not for U(T,P ). dynamic potential is a linear combination comprising the differentials of its natural variables; the linear coefficients are the conjugates to the natural variables. Out of the 2. Four thermodynamic potentials two conjugate pairs (S, T ) and (V, P ), we can construct four pairs of natural variables: (i) S and V , (ii) S and P , As explained in Sec. 1, four physical variables are as- (iii) T and V , and (iv) T and P . We have, therefore, four sociated with the gas in the cylinder: T , P , S, and V . thermodynamic potentials. Two of them were introduced When these variables are increased or reduced, the ther- in Sec.
Load more

Recommended publications
  • PDF Version of Helmholtz Free Energy
    Free energy Free Energy at Constant T and V Starting with the First Law dU = δw + δq At constant temperature and volume we have δw = 0 and dU = δq Free Energy at Constant T and V Starting with the First Law dU = δw + δq At constant temperature and volume we have δw = 0 and dU = δq Recall that dS ≥ δq/T so we have dU ≤ TdS Free Energy at Constant T and V Starting with the First Law dU = δw + δq At constant temperature and volume we have δw = 0 and dU = δq Recall that dS ≥ δq/T so we have dU ≤ TdS which leads to dU - TdS ≤ 0 Since T and V are constant we can write this as d(U - TS) ≤ 0 The quantity in parentheses is a measure of the spontaneity of the system that depends on known state functions. Definition of Helmholtz Free Energy We define a new state function: A = U -TS such that dA ≤ 0. We call A the Helmholtz free energy. At constant T and V the Helmholtz free energy will decrease until all possible spontaneous processes have occurred. At that point the system will be in equilibrium. The condition for equilibrium is dA = 0. A time Definition of Helmholtz Free Energy Expressing the change in the Helmholtz free energy we have ∆A = ∆U – T∆S for an isothermal change from one state to another. The condition for spontaneous change is that ∆A is less than zero and the condition for equilibrium is that ∆A = 0. We write ∆A = ∆U – T∆S ≤ 0 (at constant T and V) Definition of Helmholtz Free Energy Expressing the change in the Helmholtz free energy we have ∆A = ∆U – T∆S for an isothermal change from one state to another.
    [Show full text]
  • Math Background for Thermodynamics ∑
    MATH BACKGROUND FOR THERMODYNAMICS A. Partial Derivatives and Total Differentials Partial Derivatives Given a function f(x1,x2,...,xm) of m independent variables, the partial derivative ∂ f of f with respect to x , holding the other m-1 independent variables constant, , is defined by i ∂ xi xj≠i ∂ f fx( , x ,..., x+ ∆ x ,..., x )− fx ( , x ,..., x ,..., x ) = 12ii m 12 i m ∂ lim ∆ xi x →∆ 0 xi xj≠i i nRT Example: If p(n,V,T) = , V ∂ p RT ∂ p nRT ∂ p nR = = − = ∂ n V ∂V 2 ∂T V VT,, nTV nV , Total Differentials Given a function f(x1,x2,...,xm) of m independent variables, the total differential of f, df, is defined by m ∂ f df = ∑ dx ∂ i i=1 xi xji≠ ∂ f ∂ f ∂ f = dx + dx + ... + dx , ∂ 1 ∂ 2 ∂ m x1 x2 xm xx2131,...,mm xxx , ,..., xx ,..., m-1 where dxi is an infinitesimally small but arbitrary change in the variable xi. nRT Example: For p(n,V,T) = , V ∂ p ∂ p ∂ p dp = dn + dV + dT ∂ n ∂ V ∂ T VT,,, nT nV RT nRT nR = dn − dV + dT V V 2 V B. Some Useful Properties of Partial Derivatives 1. The order of differentiation in mixed second derivatives is immaterial; e.g., for a function f(x,y), ∂ ∂ f ∂ ∂ f ∂ 22f ∂ f = or = ∂ y ∂ xx ∂ ∂ y ∂∂yx ∂∂xy y x x y 2 in the commonly used short-hand notation. (This relation can be shown to follow from the definition of partial derivatives.) 2. Given a function f(x,y): ∂ y 1 a. = etc. ∂ f ∂ f x ∂ y x ∂ f ∂ y ∂ x b.
    [Show full text]
  • 3 More Applications of Derivatives
    3 More applications of derivatives 3.1 Exact & inexact di®erentials in thermodynamics So far we have been discussing total or \exact" di®erentials µ ¶ µ ¶ @u @u du = dx + dy; (1) @x y @y x but we could imagine a more general situation du = M(x; y)dx + N(x; y)dy: (2) ¡ ¢ ³ ´ If the di®erential is exact, M = @u and N = @u . By the identity of mixed @x y @y x partial derivatives, we have µ ¶ µ ¶ µ ¶ @M @2u @N = = (3) @y x @x@y @x y Ex: Ideal gas pV = RT (for 1 mole), take V = V (T; p), so µ ¶ µ ¶ @V @V R RT dV = dT + dp = dT ¡ 2 dp (4) @T p @p T p p Now the work done in changing the volume of a gas is RT dW = pdV = RdT ¡ dp: (5) p Let's calculate the total change in volume and work done in changing the system between two points A and C in p; T space, along paths AC or ABC. 1. Path AC: dT T ¡ T ¢T ¢T = 2 1 ´ so dT = dp (6) dp p2 ¡ p1 ¢p ¢p T ¡ T1 ¢T ¢T & = ) T ¡ T1 = (p ¡ p1) (7) p ¡ p1 ¢p ¢p so (8) R ¢T R ¢T R ¢T dV = dp ¡ [T + (p ¡ p )]dp = ¡ (T ¡ p )dp (9) p ¢p p2 1 ¢p 1 p2 1 ¢p 1 R ¢T dW = ¡ (T ¡ p )dp (10) p 1 ¢p 1 1 T (p ,T ) 2 2 C (p,T) (p1,T1) A B p Figure 1: Path in p; T plane for thermodynamic process.
    [Show full text]
  • Thermodynamics
    ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 6. Thermodynamics January 26, 2011 Contents 1 Laws of thermodynamics 2 1.1 The zeroth law . .3 1.2 The first law . .4 1.3 The second law . .5 1.3.1 Efficiency of Carnot engine . .5 1.3.2 Alternative statements of the second law . .7 1.4 The third law . .8 2 Mathematics of thermodynamics 9 2.1 Equation of state . .9 2.2 Gibbs-Duhem relation . 11 2.2.1 Homogeneous function . 11 2.2.2 Virial theorem / Euler theorem . 12 2.3 Maxwell relations . 13 2.4 Legendre transform . 15 2.5 Thermodynamic potentials . 16 3 Worked examples 21 3.1 Thermodynamic potentials and Maxwell's relation . 21 3.2 Properties of ideal gas . 24 3.3 Gas expansion . 28 4 Irreversible processes 32 4.1 Entropy and irreversibility . 32 4.2 Variational statement of second law . 32 1 In the 1st lecture, we will discuss the concepts of thermodynamics, namely its 4 laws. The most important concepts are the second law and the notion of Entropy. (reading assignment: Reif x 3.10, 3.11) In the 2nd lecture, We will discuss the mathematics of thermodynamics, i.e. the machinery to make quantitative predictions. We will deal with partial derivatives and Legendre transforms. (reading assignment: Reif x 4.1-4.7, 5.1-5.12) 1 Laws of thermodynamics Thermodynamics is a branch of science connected with the nature of heat and its conver- sion to mechanical, electrical and chemical energy. (The Webster pocket dictionary defines, Thermodynamics: physics of heat.) Historically, it grew out of efforts to construct more efficient heat engines | devices for ex- tracting useful work from expanding hot gases (http://www.answers.com/thermodynamics).
    [Show full text]
  • Thermodynamic Analysis Based on the Second-Order Variations of Thermodynamic Potentials
    Theoret. Appl. Mech., Vol.35, No.1-3, pp. 215{234, Belgrade 2008 Thermodynamic analysis based on the second-order variations of thermodynamic potentials Vlado A. Lubarda ¤ Abstract An analysis of the Gibbs conditions of stable thermodynamic equi- librium, based on the constrained minimization of the four fundamen- tal thermodynamic potentials, is presented with a particular attention given to the previously unexplored connections between the second- order variations of thermodynamic potentials. These connections are used to establish the convexity properties of all potentials in relation to each other, which systematically deliver thermodynamic relationships between the speci¯c heats, and the isentropic and isothermal bulk mod- uli and compressibilities. The comparison with the classical derivation is then given. Keywords: Gibbs conditions, internal energy, second-order variations, speci¯c heats, thermodynamic potentials 1 Introduction The Gibbs conditions of thermodynamic equilibrium are of great importance in the analysis of the equilibrium and stability of homogeneous and heterogeneous thermodynamic systems [1]. The system is in a thermodynamic equilibrium if its state variables do not spontaneously change with time. As a consequence of the second law of thermodynamics, the equilibrium state of an isolated sys- tem at constant volume and internal energy is the state with the maximum value of the total entropy. Alternatively, among all neighboring states with the ¤Department of Mechanical and Aerospace Engineering, University of California, San Diego; La Jolla, CA 92093-0411, USA and Montenegrin Academy of Sciences and Arts, Rista Stijovi¶ca5, 81000 Podgorica, Montenegro, e-mail: [email protected]; [email protected] 215 216 Vlado A. Lubarda same volume and total entropy, the equilibrium state is one with the lowest total internal energy.
    [Show full text]
  • Vector Calculus and Differential Forms with Applications To
    Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level.
    [Show full text]
  • Notes on the Calculus of Thermodynamics
    Supplementary Notes for Chapter 5 The Calculus of Thermodynamics Objectives of Chapter 5 1. to understand the framework of the Fundamental Equation – including the geometric and mathematical relationships among derived properties (U, S, H, A, and G) 2. to describe methods of derivative manipulation that are useful for computing changes in derived property values using measurable, experimentally accessible properties like T, P, V, Ni, xi, and ρ . 3. to introduce the use of Legendre Transformations as a way of alternating the Fundamental Equation without losing information content Starting with the combined 1st and 2nd Laws and Euler’s theorem we can generate the Fundamental Equation: Recall for the combined 1st and 2nd Laws: • Reversible, quasi-static • Only PdV work • Simple, open system (no KE, PE effects) • For an n component system n dU = Td S − PdV + ∑()H − TS i dNi i=1 n dU = Td S − PdV + ∑µidNi i=1 and Euler’s Theorem: • Applies to all smoothly-varying homogeneous functions f, f(a,b,…, x,y, … ) where a,b, … intensive variables are homogenous to zero order in mass and x,y, extensive variables are homogeneous to the 1st degree in mass or moles (N). • df is an exact differential (not path dependent) and can be integrated directly if Y = ky and X = kx then Modified: 11/19/03 1 f(a,b, …, X,Y, …) = k f(a,b, …, x,y, …) and ⎛ ∂f ⎞ ⎛ ∂f ⎞ x⎜ ⎟ + y⎜ ⎟ + ... = ()1 f (a,b,...x, y,...) ⎝ ∂x ⎠a,b,...,y,.. ⎝ ∂y ⎠a,b,..,x,.. Fundamental Equation: • Can be obtained via Euler integration of combined 1st and 2nd Laws • Expressed in Energy (U) or Entropy (S) representation n U = fu []S,V , N1, N2 ,..., Nn = T S − PV + ∑µi Ni i=1 or n U P µi S = f s []U,V , N1, N2 ,..., Nn = + V − ∑ Ni T T i=1 T The following section summarizes a number of useful techniques for manipulating thermodynamic derivative relationships Consider a general function of n + 2 variables F ( x, y,z32,...,zn+ ) where x ≡ z1, y ≡ z2.
    [Show full text]
  • Physical Chemistry II “The Mistress of the World and Her Shadow” Chemistry 402
    Physical Chemistry II “The mistress of the world and her shadow” Chemistry 402 L. G. Sobotka Department of Chemistry Washington University, St Louis, MO, 63130 January 3, 2012 Contents IIntroduction 7 1 Physical Chemistry II - 402 -Thermodynamics (mostly) 8 1.1Who,when,where.............................................. 8 1.2CourseContent/Logistics.......................................... 8 1.3Grading.................................................... 8 1.3.1 Exams................................................. 8 1.3.2 Quizzes................................................ 8 1.3.3 ProblemSets............................................. 8 1.3.4 Grading................................................ 8 2Constants 9 3 The Structure of Physical Science 10 3.1ClassicalMechanics.............................................. 10 3.2QuantumMechanics............................................. 11 3.3StatisticalMechanics............................................. 11 3.4Thermodynamics............................................... 12 3.5Kinetics.................................................... 13 4RequisiteMath 15 4.1 Exact differentials.............................................. 15 4.2Euler’sReciprocityrelation......................................... 15 4.2.1 Example................................................ 16 4.3Euler’sCyclicrelation............................................ 16 4.3.1 Example................................................ 16 4.4Integratingfactors.............................................. 17 4.5LegendreTransformations.........................................
    [Show full text]
  • Exact and Inexact Differentials in the Early Development of Mechanics
    Revista Brasileira de Ensino de Física, vol. 42, e20190192 (2020) Articles www.scielo.br/rbef cb DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2019-0192 Licença Creative Commons Exact and inexact differentials in the early development of mechanics and thermodynamics Mário J. de Oliveira*1 1Universidade de São Paulo, Instituto de Física, São Paulo, SP, Brasil Received on July 31, 2019. Revised on October 8, 2019. Accepted on October 13, 2019. We give an account and a critical analysis of the use of exact and inexact differentials in the early development of mechanics and thermodynamics, and the emergence of differential calculus and how it was applied to solve some mechanical problems, such as those related to the cycloidal pendulum. The Lagrange equations of motions are presented in the form they were originally obtained in terms of differentials from the principle of virtual work. The derivation of the conservation of energy in differential form as obtained originally by Clausius from the equivalence of heat and work is also examined. Keywords: differential, differential calculus, analytical mechanics, thermodynamics. 1. Introduction variable x. If another variable y depends on the indepen- dent variable x, then the resulting increment dy of y is It is usual to formulate the basic equations of thermo- its differential. The quotient of these two differentials, dynamics in terms of differentials. The conservation of dy/dx, was interpreted geometrically by Leibniz as the energy is written as ratio of the ordinate y of a point on a curve and the length of the subtangent associated to this point.
    [Show full text]
  • Thermodynamic Potential of Free Energy for Thermo-Elastic-Plastic Body
    Continuum Mech. Thermodyn. (2018) 30:221–232 https://doi.org/10.1007/s00161-017-0597-3 ORIGINAL ARTICLE Z. Sloderbach´ · J. Paja˛k Thermodynamic potential of free energy for thermo-elastic-plastic body Received: 22 January 2017 / Accepted: 11 September 2017 / Published online: 22 September 2017 © The Author(s) 2017. This article is an open access publication Abstract The procedure of derivation of thermodynamic potential of free energy (Helmholtz free energy) for a thermo-elastic-plastic body is presented. This procedure concerns a special thermodynamic model of a thermo-elastic-plastic body with isotropic hardening characteristics. The classical thermodynamics of irre- versible processes for material characterized by macroscopic internal parameters is used in the derivation. Thermodynamic potential of free energy may be used for practical determination of the level of stored energy accumulated in material during plastic processing applied, e.g., for industry components and other machinery parts received by plastic deformation processing. In this paper the stored energy for the simple stretching of austenitic steel will be presented. Keywords Free energy · Thermo-elastic-plastic body · Enthalpy · Exergy · Stored energy of plastic deformations · Mechanical energy dissipation · Thermodynamic reference state 1 Introduction The derivation of thermodynamic potential of free energy for a model of the thermo-elastic-plastic body with the isotropic hardening is presented. The additive form of the thermodynamic potential is assumed, see, e.g., [1–4]. Such assumption causes that the influence of plastic strains on thermo-elastic properties of the bodies is not taken into account, meaning the lack of “elastic–plastic” coupling effects, see Sloderbach´ [4–10].
    [Show full text]
  • W. M. White Geochemistry Appendix II Summary of Important Equations
    W. M. White Geochemistry Appendix II Summary of Important Equations Equations of State: Ideal GasLaw: PV = NRT Coefficient of Thermal Expansion: ∂ α≡α ≡ 1 V V ∂T Compressibility: ∂V β≡β≡± 1 V ∂P Van der Waals Equation: RT a P= - 2 V ± b V The Laws of Thermdynamics: First Law: ∆U = Q + W 1 written in differential form: dU = dQ + dW 2 work done on the system and heat added to the system are positive. The first law states the equivalence of heat and work and the conservation of energy. Second Law: dQrev = TdS 3 Two ways of stating the second law are Every system left to itself will, on average, change to a condition of maximum probability and Heat cannot be extracted from a body and turned entirely into work. Third Law: lim S= 0 T → 0 3 This follows from the facts that S = R ln Ω and Ω =1 at T = 0 for a perfectly crystalline pure substance. Primary Variables of Thermodynamics The leading thermodynamic properties of a fluid are determined by the relations which exist between the volume, pressure, termperature, energy and entropy of a given mass of fluid in a state of thermodynamic equilibrium - J. W. Gibbs The primary variables of thermodynamics are P, V, T, U, and S. Other thermodynamic functions can be stated in terms of these variables. For various combination of these variables there are 1 W. M. White Geochemistry Appendix II Equation Summary characteristics functions. The characteristic function for S and V is one of the primary variables: U. Thus dU = TdS + PdV 5 Other Important Thermodynamic Functions What then is the use of thermodynamic equations? They are useful precisely because some quantities are easier to measure than others.
    [Show full text]
  • Basic Thermodynamics
    Basic Thermodynamics Handout 1 First Law of Thermodynamics Basically a statement of conservation of energy which includes heat as a form of energy: dU = ¯dQ + ¯dW; where ¯dQ is heat absorbed by the system, and ¯dW is work done on the system. U is a function of state called internal energy. Zeroth Law of Thermodynamics Suppose A, B and C are distinct thermodynamic systems. If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then A is in thermal equilibrium with C. Systems in thermodynamic equilibrium can be described in terms of physical variables that do not change with time. The Zeroth Law implies that there exists some physical property which systems that are in thermal equilibrium have in common, regardless of their size, shape, substance, etc. We call this property temperature.A thermometer is a device that can be used to assign a numerical value to temperature through measurements of a physical property. Constant-volume gas thermometer. The mercury reservoir is raised or lowered to adjust the difference in height h between the two mer- cury columns M and M 0 so that the volume of gas in the bulb + capillary is constant. Absolute temperature (perfect gas scale) lim ! (pV ) T (K) = 273:16 p 0 T ; limp!0(pV )TP where TP stands for the triple point of water. Right: Readings of a constant- volume gas thermometer for the temperature of condensing steam as a function of the pressure p3 at the triple point of water. Curves for different gases are shown.
    [Show full text]