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...... 17 4.6Euler’stheoremforhomogeneousfunctions...... 18
II Lecture Notes 19
1 Thermodynamics Concepts 20 1.1Whatisitgoodfor...... 20 1.2 Definitions...... 20 1.2.1 systems...... 20 1.2.2 Greeklesson...... 20 1.2.3 Additionalterms...... 20 1.3Thermometry...... 20
1 1.4Pressure+ ...... 21 1.4.1 Pressureunits...... 21 1.4.2 gasconstant...... 21 1.5EoS’s ...... 21 1.5.1 IGL...... 21 1.5.2 Realgases...... 21
2FirstLaw 22 2.1TheInternalEnergy...... 22 2.1.1 Components...... 22 2.1.2 History...... 22 2.1.3 Statement...... 22 2.2Work...... 23 2.3Heat...... 23 2.4HeatCapacities...... 23 2.5StateFunctions...... 24 2.6 Reversibility ...... 24
2.7 W vs W ...... 24 2.8DesignerfxnEnthalpyH...... 24 2.9dUvsdH...... 24 2.10AdiabaticExpansionandCompressionofanIG...... 25
3 U(T), H(T), and Heat Capacities (C)26 3.1Mathematicalpropertiesofstatefxns-Review...... 26 3.1.1 EulerexercisewithIGL...... 26 3.1.2 Some useful coefficients...... 26 3.1.3 Generalrelationships...... 26 3.2 ∆U(T,V)...... 27 3.3 Example ∆U(T,V)...... 27 3.4 ∆H(T,P)...... 27 3.5HeatCapacities-TheTincrementfunctionsforU&H...... 28 3.5.1 Overview...... 28
3.5.2 vs ...... 28 3.5.3 HeatCapacityData...... 29 3.6 ∆u(T) and ∆h(T)...... 29 3.7 Joule-Thomson effect...... 29
4 Thermochemistry 31 4.1EnergyStorageandReference...... 31 4.2ChemicalRxns...... 31 4.3Hess’slaw...... 31 4.3.1 Overview...... 31 4.3.2 BondEnthalpies(energies)...... 32
4.4 ∆( ) ...... 33 4.5 Experimental ∆ and ∆ ...... 34 4.5.1 Calorimetry...... 34
5 SecondandThirdLaws 36 5.1SpontaneousProcesses...... 36 5.1.1 SecondLawStatements...... 36
2 5.2HeatEngines...... 37 5.2.1 Impossible (perpetual motion machines) engines of the 1 kind...... 37 5.2.2 Impossible engines of the 2 kind...... 37 5.2.3 CarnotCycle-reversible...... 38 5.2.4 CarnotCycle-irreversible...... 38 5.2.5 ProofthatSisastatefunctionusingaGeneralizedCarnotcycle...... 39 5.3Another(formal)perspective...... 39 5.4S(T,P)andS(T,V)...... 40 5.4.1 ∆ (Micro/Macroinsight)...... 40 5.5EntropychangesinHeatConduction...... 41 5.6SecondlawReview...... 41 5.7Thirdlaw...... 41 5.7.1 Historyandstatements...... 41 5.7.2 ResidualEntropy...... 42 5.7.3 Adiabaticdemagnetization...... 43 5.8StandardEntropy...... 43
5.9 ∆( ) ...... 43
6 Chemical Equilibrium 44 6.1 Equilibrium Criteria ...... 44 6.1.1 Definitions...... 44 6.1.2 Types of equilibrium ...... 44 6.1.3 Criteriawithvariousconstraints...... 44 6.2Maxwell’srelations...... 45 6.3Curvaturerelations...... 45 6.4Incrementfunctions...... 45
6.4.1 G (P)...... 46 6.4.2 G (T)...... 46 ∆ 6.4.3 Gibbs-Helmholtz ( )...... 46 6.5 FreeenergyandWork...... 46 6.5.1 Helmholtz work ...... 46 → 6.5.2 Gibbs work ...... 47 → 6.5.3 Summary...... 47
6.6 The Chemical potential ...... 47 6.7 ∆S and ∆G ...... 47 6.8ChemicalpotentialofanIG...... 48 6.9 Chemical equilibrium ...... 49
7RealGases 50 7.1EquationsofState...... 50 7.1.1 Criticalpoints...... 51 7.1.2 Correspondingstates...... 51 7.2FugacityofaRG...... 52 7.3 Compressibility Fugacity...... 53 ⇒ 7.3.1 Gross Compressibilities ...... 53 7.3.2 f(P) or (Z)...... 53 ln 7.3.3 f(T) or ( ) ...... 53 7.4MixturesofRGs...... 54
3 8Phasechanges 55 8.1GibbsphaseRule...... 55 8.1.1 Whatcanyoucontrol?...... 55 8.1.2 Whatisacomponent?...... 55 8.1.3 TopologicalAside...... 56 8.2 Geometric interpretation of and ...... 56 8.3First-orderphasetransition...... 58 8.4Clapeyronequation...... 58 8.5Clausius-Clapeyronequation...... 59
8.6 P(inertgas)...... 59 8.7 H( )...... 60 8.8 G( )...... 60 8.9Surfacetension...... 60
9 Solutions and Multicomponent systems 61 9.1IdealSolution...... 61
9.1.1 The IS is like IG ...... 61 9.2 Equilibrium between a S-IS ...... 62 9.3RealSolutions...... 63 9.4 Solvent/Solute ref. states ...... 63 9.4.1 Non-idealities&AzeotropesSurvey...... 64 9.5Gibbs-DuhemEquation...... 66 9.5.1 Exercise ...... 66 → 9.6 Colligative properties - I ...... 67 9.7Activities...... 67 9.8Review-Referencestates...... 67 9.8.1 Gases...... 67 9.8.2 Solvent...... 67 9.8.3 Solute...... 68
9.9 The Eq. Constant ...... 68 9.10*a(P)...... 69 9.11*a(T)...... 69 9.11.1 h ...... 69 9.11.2 a( )...... 69 9.12MoreonBinarySystems...... 70 9.12.1WhysomeliquidsdoNOTmix...... 71 9.12.2Surveyofresults...... 71 9.13DeterminationofActivities...... 72 9.13.1FromVapormeasurements...... 72 9.13.2FromColligativeProperties...... 72 9.14*Thermo.consistency ...... 74 9.15*Partial-MolarQuantities...... 74 9.15.1Methodofintercepts...... 75 9.15.2UsingthePMQ(a)...... 75 9.15.3 Using the difference(b)...... 75 9.15.4 PMQ for ...... 75
10 Electrolytes 76 10.1Basics...... 76
4 10.2 ∆ :Bornmodel...... 77 10.3 Mean activities ( ) & activities coefficients ( )...... 77 ± ± 10.3.1 from solubility measurements ...... 78 ± 10.4 Debye-H¨uckelTheory...... 78
11 Electrochemical Cells 81 11.1 Electrochemical potential ∼ ...... 81 11.2Conventions,NotationandStandardstates...... 82 11.3Reactions&theNernstEq...... 83 11.4Cellpotentials...... 84 11.5Afewpointers&reminders...... 84 11.6 The E( )...... 85 11.7*Anotherexample...... 85
12 Probability 86 12.1 Permutations and Configurations...... 86 12.2 Distinguishable configurationsforFDandBE...... 86 12.2.1 Bos´-Einstein...... 87 12.2.2Fermi-Dirac...... 87 12.3Stirling’sapproximation...... 87 12.4Statecountingandentropy...... 87 12.4.1FD...... 87 12.4.2 Bos´-Einstein...... 88 12.4.3ClassicalMB...... 89
12.4.4 Summary for f andS...... 89 12.5TheBinomialDistribution...... 89 12.6DiscretevsContinuousdistributions...... 89
13 Statistical Mechanics 90 13.1Background...... 90 13.1.1RoleofSM...... 90 13.1.2BoltzmannDistribution...... 90 13.1.3PartitionFunction...... 90 13.1.4StatisticalThermodynamics...... 91 13.2PhaseSpace...... 92 13.2.1HO...... 92 13.2.21d...... 92 13.2.32d...... 92 13.2.43d...... 92 13.2.5PSvol(3D)...... 92 13.3IdealSystems...... 93 13.4AddingmoretermstotheHamiltonian...... 94 13.4.1Logic...... 94 13.4.2Rotation...... 94 13.4.3Vibration...... 95 13.4.4Electronic...... 95 13.5EqualPartition...... 95 13.6Unitpartitionfunction...... 96 13.6.1Logic...... 96
5 13.6.2Crystals...... 96 13.7 Chemical Equilibrium ...... 97
14 Kinetic Theory and Transport 99 14.1Preliminaries...... 99 14.2KineticTheory...... 99 14.2.1 Boltzmann ...... 99 → 14.3TransportPhenomena...... 101 14.3.1Flux...... 101 14.3.2 Fick’s second-law of diffusion...... 101 14.3.3 Statistical understanding of diffusion...... 102 14.3.4Thermalconduction...... 102 14.3.5Viscosity...... 103 14.3.6Viscosityexample...... 103
15 Elementary Rate Theory 104 15.1Rates/overview...... 104
15.2 Activated complex‡ orsaddlepoint(I)...... 105 15.3SimplecollisionTheory...... 105 15.4 Rate Constant ( )...... 106 15.5ActivatedComplexTheory(II)...... 106 15.5.1Background...... 106 15.5.2Rxn...... 107 15.5.3ThermodynamicInterpretation...... 107
III Practice Exams 108
1 402 Practice Exams 109 1.1Prac.Midterm-1 ...... 109 1.2Prac.Midterm-2 ...... 110 1.3PracticeFinal...... 111
6 Part I
Introduction
7 Chapter 1
Physical Chemistry II - 402 -Thermodynamics (mostly)
1.1 Who, when, where LS 250. The material covered by each midterm will also be announced in class before the exam. Exams will con- Lecturer: Professor Lee G. Sobotka sist of conceptual questions (often chosen from those in Lectures: MWF 10-11 am, LS 250 the text), numerical questions (often chosen from those Office: 259 Radiochemistry worked out in the text or assigned) and proofs (also of- TA: Monatrice Lam and Patrick Chan ten chosen from those in the text or worked out in class.) The exam dates are: MHelpsession: 11:15-1:30 in Radiochem 259 or by appointment Midterm # 1: 7:30- 9:00 pm, W, Feb. 15, LS-250 TA Help sessions: TBA in LS-400 (tentative) Midterm # 2: 7:30- 9:00 pm, W, March 28, LS-250 Text: Thermo., Stat. Mech. & Kinetics,2 ed T. Engel Final: (XII) -10:30-12:30 M, May 7, LS-250 and P. Reid. ISBN 13:978—0-321-61503-0 1.3.2 Quizzes Two or three short (10 min.) in-class quizzes will be 1.2 Course Content/Logistics given. You will be told in class that a quiz is coming up 1-2 lectures before they are given. CONTENT: Classical Thermodynamics [Thermo., Ch. 1-11]; Statistical Mechanics [SM, Ch.12-15]; and Kinet- 1.3.3 Problem Sets ics, Transport and Rate Theory [Kin.Ch. 16-17]. After presentation of some introductory material, Problems, selected from those in the text, will be as- not presented in a concise form in the text, we will fol- signed approximately weekly. Only a few a few of the low the text, at a rate of about one chapter per week. assigned problems from each set, will be designated for Lecture notes will be handed out in class. This practice turning in for grading. Each problem will be grading on is intended to free your hands and minds to concentrate a 0, 1, or 2 point basis (2 = perfect.) The solutions will on the material during the lectures. This practice is also be posted on the WEB. intended to encourage questions as you will not have to worry about writing down every single board scratch. 1.3.4 Grading On the completion of each chapter we will go over the conceptual questions (found in the text, just before the Item Midterm 1-2 Quizzes PS Final % 30-30 10 20 40 problems) in class. ∼ The lowest (relative to the mean) Midterm or Q+PS will be dropped. 1.3 Grading
1.3.1 Exams There will be two midterm exams and a final. The midterms will be Wednesday evenings, 7:30- 9:00 pm, in
8 Chapter 2
Constants
1 amu = 931494013 [MeV/c2] = 29.8782458 [cm/ns] = 1.66053873 10 27 [kg] 12 19 − 1[eV]=160219 10− [erg] =160219 10− [J] ∗ 31 × × m = 9.109382 10− [kg] 1 ∗ 4 2 = 8066[− ]= 26.06 [ ] = 5.485799 10− [amu] = 511 [keV/c ] ˜ ∗ 27 12398 10 4[]=1[ 1]= = m = 1.6749272 10− [kg] − − ∗ · ≡ = 1.008664926 [amu] = 939.56533 [MeV/c2] 1[]=4184[] 5 27 1[ ] 104 10− [ ]or1[ ] 104[ ] m = 1.67262158 10− [kg] ⇔ · ⇔ ∗ 2 1 yr = 365.25 d = 315576 107 [s] = 1.00727647 [amu] = 938.271998 [MeV/c ] × m = 1.67353249 10 27 [kg] 3 ( ) ( ) − v( ) =1389 10− () =1389 () ∗ 2 · = 1.007825017 [amu] = 938.78298 [MeV/c ] q q ~ 24 =927400899 10− [J T] ≡ 2 × ~ 27 =505078317 10− [J T] ≡ 2 × 27 = 19135 n.m. = 96623707 10− [JT] − − × 26 =+279275 n.m.= 141060761 10− [JT] × 1 ~ = 2 = 13703608 2 15 = 2 =2817935 10 [m] − 2 · 4~ 11 = 2 =52918 10− [] · 12 2 =885419 10− [ ] 5 · 2 8 = 3 2 =567051 10− [ 2 4 ] 15 − 34 · 27 ~ =10546 10− [ ]=10546 10− [erg s] · · · · 22 16 =6582122 10− [MeV s]= 658 10− [eV s] · · · · =6465 [MeV12 amu12 fm] · · 7 ~ =197.3 [eV nm] =197.3 10− [eV cm] · · · =197.3 [MeV fm] 4 · =1.2397 10− [ev cm]=1239.7 [ev nm] · · · 1973 1973 2 = ( ) [fm] = ( ) [nm] ≡ 1 23 − =138065 10− [ ]=0695[ ] · 4 10 =086171 10− [ ]= 086171 10− [ ] · ∗ =831[ ]=199[ ] · · =008314[ · ]= 008206[ · ] 23 · · N = 6.022 10 [#/mole] · V(298 1)=22711[ ] 2 =1.44[MeVfm] =1.44 [ev nm] =1.44 103[ev pm] · · · · = 96485.309 [C/mole e−] F
9 Chapter 3
The Structure of Physical Science
22 “If nature were not beautiful, it would not be worth = = (the position increment fxn) knowing, and if nature were not worth knowing, life 2 2 would not be worth living.” - Henri Poincar´e (1854-1912) and = ∴ • = = − (the momentum increment fxn.) 3.1 Classical Mechanics You draw the example V and F.
Object To propagate all 6N coordinates in time.
r p =0 = r p { } ⇒ { } (Bold denotes vectors).
Degrees of Freedom (DoF) Stationary states All 6N positions and momenta. The 6N coordinates evolve in time toward stationary Rules states. There are stable equilibrium configurations (like the bottom of a potential valley), unstable equi- Therules(whichcanbeformulated3ways)aretime librium configurations (pendulum “stuck” on top) and reversal invariant: metastable configurations (local but not absolute min- r p =0 r p ima.) { } ⇐⇒ { } Generalized “postions” (i.e. answers to the question “where”) are often given the symbol “q”. Why things happen 1. F = a = p• Application of (classical) forces under constraints such ≡ as E conservation. 2. With ≡ − ≡ − ( ) =0 • − Comments 3. Or with + ≡ 1. We often deal with conservative systems for which • = and • = − the forces are derivable from a scalar potential . This one condition implies several others. These two equations should be viewed as position and momemtum “increment rules”, respectively. To (a) F = or in one dimension, = try to make these equations less abstract, lets consider −∇ − the common case of “conservative” systems (see below) (b) The the work done against this force on any where ()and (). circular path is zero, Fs =0 • • = = and = = (c) The force (field) is irrotational. − H position increment rule and momentum increment rule. Using some very simple examples for and : We will not use the last of these conditions.
10 stationary probability distributions. However the sub- ject is “mechanical” and the rules generate trajectories for the distributions of all 6N variables.
Why things happen
The same as CM (with some conditions that nature has seen fit to apply.)
Comments. 2. The sympletic structure between the conjugate vari- QM is a refinement of CM. While is may seem completely ables representing “where is the system” and “where different, it has the same purpose. QM reduces to CM if is the system going”, each informing the other how the deBroglie wavelengths ( ) are small compared to advance in time, defines the mechanics. to the scale of the objects in the≡ problem. Concept of sympletic structure: Two trains representing r and p, each telling the other how to advance. 3.3 Statistical Mechanics
3. The 6N coordinates evolve in time. Object
To propagate distributions in coordinate and momentum 3.2 Quantum Mechanics space for an ensemble of particles. The 6 coordi- nates are not followed. While CM and QM are most Object oftenemployedwhen is small, SM is most often em- ployed when is large. The trajectories of the r0 ThesameasCM. and p0 are not followed, but the distribution (on) say Desire to: “ r p =0”= “ r p ” the “coordinate” ( ) is. Usually this is still too de- { } ⇒ { } However the best that can be done is propagate manding so just a few moments of this distribution are probability amplitudes for each r and p subjected to analysis.
Degrees of freedom Degrees of freedom
All 6N positions and momenta. Selected distributions are usually characterized by the lowest few moments. Higher moments become more and more important as the system gets smaller of if the dis- Rules tribution is not Gaussian. The rules are time reversal invariant. The probability averages distributions for individual coordinates evolve in time () = = () according to a wave equation. Ψ 2nd moment width of distribution P 2 Ψ = } where 2 ⇒ 2 () H = = () Ψ( )=()() 2 2 2 = If = ()then −P H 6 () ()=}() H } Rules H = = } = and ()=− Laws of probability. One must know if: a) the units are H Therefore, distinguishable (labelled) and b) if the laws of QM allow } for multiple occupancy of quantum states. Ψ( )=()− Systems evolve toward maximum probability and the evolution is not time-reversal invariant: Stationary states (r p) (r p) ⇒ We often focus on stationary states. These states (so- This mechanics does not “know” the individual p’s and lutions of the time independent SE) have well defined r’s.
11 Stationary states 3. Note that one (and only one) coordinate from each sector is extensive (proportional to the amount The stationary states are stationary “distributions”. of “stuff”) while the other is intensive (univalued Individual p’s and r’s are evolving but the distributions throughout matter in equilibrium). This means reach stationary forms defined by statistical arguments. that any function constructed from: sums of prod- These distributions are not the same as the QMical dis- ucts of the elements in a sector (e.g. ()= tributions. The former are for the ensemble, the latter + ) will be extensive with the units of en- for EACH of the 6N variables. ergy.
Why things happen 4. One often chooses to divide through by the number Increase in (QMical) options. of moles to get molar quantities. Texts denote the molar quantities with an overbar ,asubscriptH or (my preference) a lower case, ie = Comments − will read [ ]= (Your text, − 1. The distributions are defined by a) the interactions, like most, is inconsistent. Sometimes it uses the sub- b) statistical theory and c) the constraints. script, to denote a molar quantity, and sometimes the reader must infer the quantity in question is mo- 2. Allows for the generation of macroscopic variables lar by the context.) (P,V, T.....) and their interrelationships.
Examples: Rules
Example 1 No interactions = = The characteristics (topology) of the state functions ⇒ w.r.t. variations in the selected variables are defined Example 2 The most elementary consideration of: by the rules of differential calculus. This science is not dynamical in the sense of the other subjects. It tells you a) attractive interaction which macroscopic states are the most stable and when b) excluded volume equilibrium is achieved and how much of each compo- nent is present at equilibrium. The dynamical sense of c) uniform density Thermo., is that it tells you the generalized forces as- sociated with the macro variables. The relevant time leads to = = ( ) 2 or development rules (which are really the subject of kinet- ⇒ − − ics, see below) are NOT time reversal invariant. ( + )( )= 2 − Stationary states 3.4 Thermodynamics Thermodynamic stationary states are dynamical states Object in terms of the individual p0 and r0 , but the macrovari- ables are time stationary. To describe the macroscopic state of the system with the absolute minimal set of variables. Common elements of this variable set are: { } Why things happen
Degrees of freedom Minimization of energy variables or maximization of the QMical options. The latter encoded extensively by Newtonian sector: taking the natural logarithm of the number of options.
Thermal sector: ln(QMical options). ≡ Chemical sector: Remark 1 The “TS” product has the units of energy 1. These are conjugate pairs, like F dx, with product x (or energy per unit of material) and in the expression units of energy. ln(QMical options), the constant carries the ≡ 2.Generallywewillchooseonefromeachsectortobe units. Also note that because = that we only independent and the other will be dependent. really need two of these three constants.
12 Comments (c) lim 0 = 0 for a perfect (crystalline) state. → = ln(QMical. options) thus at =0 The “Thermo” mindset: “Dear father Newton, we have there is but one option in a “perfect” (crys- come to you on bended knee to present a slight extension talline) state. of your science. We hope you find it to your liking.” The famous epitaph on Boltzmann’s tomb- stone is. 1. Newtonian: = F r =() = • = ln( ). The “ ”standsfor where = p• (=momentum flux) “Wahrscheinlichkeit”. Look it up. Note that ≡ ∴ = Boltzmann’s work proceeded the development − of QM. + compress − + expand 7. From the set of 4 variables (for a 1 component sys- − tem, C=1) two each from the thermal and New- 2. Add thermal sector: tonian sectors, we can define 4 state functions. = + These functions carry IDENTICAL information. − We will use for the thermodynamic internal en- However, as they have different natural variables, ergy of a system. (U is “the mistress of the world.” each facilitates calculations under different condi- But who is her shadow?) tions.
3. Add chemical (association) sector: (a) ==1 + (Internal energy) − = + + =1 − (b) =+ + (Enthalpy) This sector can be viewed as a type of “chemical” P (c) ==1 (Helmholtz free energy) generalized force-response. The work is done when − − atom “associations” are destroyed or created. How- (d) =+=1 (Gibbs free energy) ever this sector is so important for the world around − us that it deserves its own separate sector. (The Remark 2 Note the sign change with swapping of the chemical potential is a “potential”, an imbalance dependent independent variables. What does the of which produces a force to relocate or reassociate stand for?⇔ See section on ”Legendre Transformation” in atoms.) Note that this is a sum of quantities that the chapter on “requisite math”. are extensive, as one (and only one) term in each product is extensive. Remark 3 The internal energy () is unique in that the 4. Other sectors must be added if work is done by the independent variables (in the Newtonian, thermal and application of electric or magnetic fields. (If one is association sectors) are extensive. The complementary concerned with atmospheric chemistry, work done constuction (i.e. only intensive variables are indepen- against a gravitational field, must be added.) dent) for C 1 is equal to 0.
5. ,(andthe0 ), are generalized forces (or po- tentials) corresponding to1, 3.5 Kinetics = ( )=+( ) =+( ) − Object and the 0 are the generalized “responses” to these “forces”. To follow “trajectories” in the macroscopic variable space of thermodynamics. One is most often interested 6. Thermo. laws in the time development of the amount (the n0 )ofevery (a) Energy is conserved: = d + d species. (b) Every isolated system that is left to itself will evolve toward a condition of maximum QMical Rules options. Differential equations (usually coupled, i.e. one equation ∆ = 0 ≥ depends on the result of another) provide a set of “step- ping rules” in time. These “Master equations” take the 1The subscript “o” in a partial means, “other”. If no subscript initial conditions (usually number density or is provided, all other independent variables are assumed to be held ≡ fixed. [concentrations]) down stream in time to determine the
13 conditions later in time Only the time forward appli- dition to a lower (free) energy condition which is itself cation of these stepping rules is physical. The master metastable. The utilitarian use of the idea of an equi- equations are of the form, for a species librium state is that the distribution on the variables we []= gain terms - loss terms. choose to study are stationary. However “stationary” only means unchanging on a time scale which has tested The solution to these equations yield [] [] ⇒ the limits of our patience. I will reenforce this point by two examples. Stationary states
If the master equations provide a means for reac- 1. Consider the diamond in my wife’s wedding ring. At = 1 atm graphite is more stable than dia- tions in both directions ( ¿ ) a stationary condition in terms of the amountsR of eachP material ( =0) mond. The diamond is UNSTABLE wrt conversion will ultimately be reached. When this “chemical”P equi- to graphite. We say the diamond is “metastable” librium condition is achieved, the forward and backward w.r.t. graphite. However, its lifetime will greatly rates will be equal as the gain and loss terms in the Mas- exceed my wife’s lifetime (nature’s patience with ter equations will be of equal magnitude and opposite in her) and certainly her patience with me. As a result sign. we can consider diamond as a stationary thermody- namic state. A pseudo stationary state can be created when the system is not closed and an influx of new material (reac- 2. The most stable nuclei are 56Fe and 56Ni. Given tants) offset the steady loss from other terms in the Mas- enough time, nuclei would interact with one another ter equation This common situation is called “steady and all matter would convert to these nuclei. How- state”. Most manufacturing (including biological) is ever nuclei are essentially isolated from one another of this sort. (The exception is “batch-mode” process- by 1) repelling e− clouds (at low energy and atomic ing.) A steady-state condition for intermediates can be distances) and 2) the like-charged nuclei (at high en- achieved in a closed system for a rather broad window ergy and nuclear distances). Thus once made (and in time, under conditions that are not uncommon. removed from a nucleosynthesis source - star cores or SuperNova) the mass number of each nucleus is Velocity and Energy distributions fixed. On the other hand, “unimolecular” decays do occur. The primary example here is “” decay, 14 14 + The rates of reactions usually depend on the amount of e.g. C N +e− + Wecanthensaythe energy brought into the reaction collision. Therefore the elements we→ have on earth are the result of equili- velocity and energy distributions of the potential reac- bration along an “isobar” (constant A=N+Z) but tants are central issues. notalongthemassdegreeoffreedom.Themass distribution Prob(),ofwhatwediguponearth, Why things happen is not the absolute equilibrated one, but rather one thatwillnotchangeonthetimescalewearegener- Nothing is really at equilibrium. Kinetics is the study of ally concerned with. nature’s response to an “out of equilibrium” condition. We either force a response by taking a system close to equilibrium and nudging it away from equilibrium (e.g. excitation with a laser pulse) or mixing substances for which a different atomic level association has a lower (free) energy than the associations in the reactants. The approach to equilibrium can be “unimolecular” (isolated decay) or the result of reactions. In either case the indi- vidual “atomic level” processes are time reversal invari- ant and must follow all the rules of mechanics (energy and momentum conservation.) However, as the flow to- ward equilibrium is always towards the condition with the most microscopic states, the “arrow of time” is im- pressed on kinetics. Movies of the approach to equilib- rium are valid, in our universe, only when run forward. One more point on the ultimate final state - the equilibrium state - is in order. Essentially all the kinet- ics we observe is from one unstable or metastable con-
14 Chapter 4
Requisite Math
What did you ask at school today? - Richard Feynman. knowthevaluesofthestatevariables( ) we know the The requisite math is multivariable calculus. Euler value of the state function relative to some reference. [1707-1783] is the “father” of a branch of mathematics This brings up two issues. called analytic geometry that encompasses multivariable calculus. The word “geometry” is extremely revealing in 1. Reference states. All energy functions have an ar- that our thermodynamic functions have clear geomet- bitrary reference point. We choose the reference for ric interpretations. For pure (one component C =1) convenience. The entropy is a state function but systems, all our functions of the state (like the internal is not an energy function. S has a natural, but “re- energy ,Enthalpy and the free energies and ) mote” and thus inconvenient, zero at =0 of the system are bivariate (if only one form of work is 2. Variables. What are they, and are some sets more allowed.) The number of truly independent variables in- convenient than others? This question confuses all creases with the number of components and decreases who take Thermo. for the first time. We will gen- with the number of phase (or reaction) equilibria. The erate 4 energy functions. All the functions con- discussion below presumes functions of 2 variables. This tain the same information. They encode the isthecaseforaC = 1, subject to one work form (say and have exact differentials. The difference is Newtonian “”work)inasinglephase. that they have different “natural variables”. The Please think of a function () with 2 independent natural variables are those for which the exact dif- variables and the attendant plot. Put the “ ”Carte- ferential is easy to express. (You will get the idea sian plot in your head with a surface that we want to soon enough.) Section 4.5 deals with how the 4 en- study as it informs us about the state of a system and ergy functions are related to one another. its response to a stimulus. That is, if YOU stimulate the system (change an independent variable) the system (and thermo. function) MUST respond in a prescribed fashion to remain on the (equilibrium) surface. While the exact form of the surface depends on what we will call the Equation of State (EoS e.g. = ) all EoS’s and thus functions, obey the laws Euler derived for multivariable functions. Below we run through the multivariable relations that are valid for all EoS’s - valid for all (equilibrium) thermodynamic functions. (Again we simplify by considering only 2 independent variables.)
4.1 Exact differentials
Consider a function of two variables, ( ). has an exact differential if any path taken [say from the ori- 4.2 Euler’s Reciprocity relation gin to the indicated ( ) point] accumulates the same change in Of course if you go back to the origin the Let = ( ) accumulated change is 0 Our work in Thermo. is built =( ) +( ) around functions with exact differentials. Thus, if we = +
15 2 =[ ( )] = =[( )] = = the relative pressure change with at constant =⇐⇒ ∴ The three quantities listed above are all read- This is the Reciprocity relation and it is a nec- ily measurable. What I mean by this is that essary and sufficient condition for exactness (totality) of we have “meters” for all quantities involved in adifferential expression. It simply “says” that the order these partials (including those held constant.) of differentiation is irrelevant. The important Maxwell’s Have you ever seen an entropy meter? (If you relations are just applications of reciprocity to the ex- see one, please send me a copy of the instruc- pressions for the thermodynamic energy differentials. tion manual. I want to study how it purports to take the measure of the “shadow”.) Remark 4 Subscripts usually indicate that, that vari- able is held constant. Sometimes a partial will be written 4. Therefore, ( ) without indicating what is held fixed. This implies 1 2 that all OTHER variables are fixed. =(+ − ) +( − ) 1 − 2 ( − ) 2 ( − ) [ ]= − =[ − ] 4.2.1 Example − 1. The molar volume of an IG, 4.3 Euler’s Cyclic relation 1 ()[ ] = − Nowgobacktotheexpressionfor and consider vari- 2. The molar volume IS a function of state and has an ations wrt which leave Z unchanged. (Or just operate exact total differential. by ( )) =( ) +( ) 0=( )( ) +( )1 3. The two partials have clear physical meanings. Multiply by ( ) 0=( )( )( ) +1or 1 (a) ( ) =+ − 1 =(© ) ( ) ( ) 1 1 − ( ) = ( ) 0 Which is the cyclic relation. (Note that the variables ≡ = coef. of thermal expansion are just cycled around.) = the relative volume change with at constant 4.3.1 Example (Aside: The text calls the coef. of thermal ex- pansion This usage is highly unusual and out 1. Again consider the molar volume of an IG - NO I of step with both the past and future. Check take that back. Consider the molar volume () Wikipedia.) of ANY C = 1 system! We can rewrite this using the molar density. =( ) +( ) 1 [ ] − Therefore ≡ 1 2. Now apply the differential operator ( ) = − The differential element is 2 = − Therefore, (a) ( ) =( )( ) +( ) ( ) − 1 2 ( ) = ( ) ≡− − 0=( )1+( ) ( ) = ve of the relative density change. − (b) The volume cannot change with fixed volume. 2 (b) ( ) = − − (c) A plot of vs has unity slope. 1 ( ) 0 ≡− 3. Rearranging, = isothermal compressibility. 1=( ) ( )( ) = - the relative volume change with at − constant (If you do not like my use of vs multiply the (Aside: Sometimes this coefficient is called .) first ()by and divide the last ()bythesame (c) While we are here let me also define, factor.) 1 ( ) 4. We can rewrite the above cyclic relation in terms of ≡ = Coef. of Tension. the coefficients defined above.
16 1 1=( )()( )or 2. Imagine a function (). The Legendre transforma- − − = tion changes from one function with an indepen- dent variable to another function with the inde- 5. To see the power of this math, pendent variable being the derivative of the old fxn wrt . Calculate the Coef. of Tension () for an IG. The function () has slopes at each point ()and intercept values of the slopes on the y axis that are 4.4 Integrating factors functions of both x and m (the slope).
An inexact differential can be converted into an exact 3. The set of tangent (slope) lines are defined by = differential by multiplication by an “integrating factor”. + . Therefore we can define a new function, an intercept function, as 1. First consider: + = ()= Which can be represented as: − ( ) − 1= = − = 1 new fxn = old fxn - (slope of old fxn) * (old inde- 6 − Therefore this differential is inexact. That is, the pendent variable). valueofanintegraldependsonthepathchosen. 4. Let us apply this to the case ( )inaneffort to 2 1 find the natural function of [= ( )]and Call 2. Now multiply by − to get: 2 ( ) − the new function ( ) 2 1 2 (− ) ( − ) 2 Now − = = − = − To start, we need a functional definition of
2 ( ) (A totally general definition We call the − an integrating factor for the differ- would require≡ additional− terms. However, for the ential in question. Integration of the factor times the present purposes, this will suffice.) (inexact) differential is path independent. Using the fact that is the slope of the old function (U) w.r.t. S, =( ) a new function can be 4.5 Legendre Transformations defined, ( ) Legendre Transformations allow us to change from one ≡ − variable to another, however we will also change the func- The differential of which (using the chain rule) is, tion. Lets see how this works. =() − − =1 =( ) 1. = (one component with only − − − − = “” work). This means ( ). While you can − − express in terms of ( ) (), or ( )itis 5. Note the old variable () is out and the new one the ( ) variables that has a simple expression. ( ) is in, which is the slope of the old function () wrt the old variable (). This can be done again to (in the other sector) to swap out the other old variable and replace it with . This, of course, would generate yet another function. This one we call 6. The entire set of energy functions and their interre- lationships in terms of Legendre transformations is given below. ( ) ======( ) ...... L. trans. on Newtonian sector ...... ⇓ ⇓ ...... ⇓ ⇓ .... L. Trans. . ⇓ ⇓ .... Thermal sector ...... ⇓ ⇓ Legendre Transformation ...... ⇓ ⇓ ( ) ======()
17 4.6 Euler’s theorem for homoge- (c) As i) and ii) are equal (if homo. fxn.), 1 ∗ ∗ ( )= 1 ( ) +( ) neous functions. − { ∗ ∗ ∗ ∗ } This last expression must be valid for all in- 1. Functions which have the property cluding =1 As 1 = 1 and the *’s
go away (when = 1), we have the previously (12) = (12) stated consequence: where is a constant, are called homogeneous ( )=( ) + ( ) functions of degree . 4. Lets apply this theorem to the internal energy. (a) Functions of degree 0 are intensive functions For a multicomponent system, while = + (b) functions of degree 1 (in at least one variable) − are extensive. Functions of state are exten- with ( 1P2) sive. Euler’s theorem gives, =( ) +( ) + 2. The consequence of a function being an degree homogeneous function is, =() []+ − { } ( )= ( ) Therefore the total derivative of also is 1 2 P =( + ) [ ] Let us considerP functions of two variables ( ). − − =1 + + ( ) = ( ) + ( ) = ( ) { } Comparing the total derivatives yields1, =1 P P = + = ( ) + =0 − At constant and , Remark 5 Note the use of the standard short-hand P notation for partials in the second line. Now the subscript means this is the variable in question and all others are fixed. (While I will avoid this nota- 0 = = (GD) tion, I wanted to acquaint you with it.) X X where the X’s are the mole fractions. This is the Remark 6 Our Thermodynamic energy functions Gibbs-Duhem equation. The importance of this are homogeneous of degree 1 in at least one vari- result will become clear later. For now appreciate able. The internal energy U is homogeneous in all. that the changes in one chemical potential, with a (Mathematicians would only accept U as a true “ho- change in composition, must be mated to changes mogeneous function” of degree 1.) in the chemical potentials the other components.
5. Consider a mixture of b and g where = 3. The proof of Euler’s theorem follows. ( ) (a) Let ∗ = and ∗ = If is homogeneous, ∗ (∗∗)=( ) = ( ) ≡ i) ii) (b)Takederivativeofi)andii)
∗ ∗ i) ∗ =( ) ∗ +( ) ∗ ∗ ∗ ∗ ∗ Now divide by 1 ∗ ∗ ∗ ∗ ∗ Note that ALL of the independent variables are intensive. =( ) +( ) ∗ ∗ ∗ ∗ What else could this differential expression be equal to? If none ∗ of the independent variables carry extensivity, increasing the size However = so, of the system by any factor can not change anything. 0 is the ∗ ∗ =( ) +( ) only number when multiplied by anything does not change. This ∗ ∗ ∗ ∗ expression can be used to show (for a one component system) that ∗ 1 ii) = [ ( )] = − ( ) 2 1 ( ) =( )− In about a month you can try to derive as ( ) = () this. 6
18 Part II
Lecture Notes
19