1 The Basic Structure of Thermodynamics

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1 1 The Basic Structure of Thermodynamics 1.1 Problems and Postulates 1. Many DOFs few thermodynamic variables 2. Extensive variables: V, N, U 3. Three overarching questions: (a) Characterize thermodynamic equilibrium state of a system (b) Determine differences in thermodynamic quantities between two states, and corresponding effect on environment (c) Determine equilibrium state when a constraint is removed (d) Bonus problem: Relate macroscopic to microscopic behavior 4. Conservation of energy 5. Postulate 1: Equilibrium states exist (a) Examples 6. Definitions (a) System and environment (b) Walls - permeable and impermeable; adiabatic and diathermal (c) Processes - Spontaneous, quasi-static, reversible 7. Work and heat - inexact differentials, path dependent 8. Internal energy du = dq + dw 9. Mechanical measurement of energy, work W = P ext dv 10. Postulate 2: A state function entropy S exists and is maximized at equilibrium (a) The fundamental equation 11. Postulate 3: S is a nice, extensive, homogeneous function, monotonically increasing in U, S/ U Postulate 4: Entropy vanishes at T = 0, third law of thermodynamics 13. Impossible, reversible, quasi-static, and real processes 1.2 Interlude: Multivariate calculus 1. Partial derivatives 2. State functions and exact differentials: df = ( ) ( ) f x dx + f y y dy x 3. Fun with partial differentials ( ) f x y ( ) x y f ( ) y = 1 f x ( ) f = x y ( ) x 1 f y ( ) f = x y ( ) f / t y ( ) x t y November 26, W. F. Schneider 1

2 4. Homogeneous functions of order n: f(λx, λy) = λ n f(x, y) (a) Thermodynamic state functions (extensive quantities) are homogeneous order 1 (b) First derivatives (intensive quantities) are homogeneous order 0 ( ) ( ) 5. Euler relation: = f x x + f y y y x 1.3 Calculus of energy, entropy, and equilibrium 1. U = U(S, V, N) state function (Table~1) (a) Molar state function u(s, v) = U(S/N, V/N, 1)/N 2. Partial derivatives of U: intensive variables temperature, pressure, and chemical potential 3. Equations of state relate intensive variables 4. Euler relation 5. Entropy representation S = S(U, V, N), s = s(u, v) 6. Thermal, mechanical, chemical equilibrium determined by equality of T, P, µ i 7. Gibbs-Duhem relationship between intensive variables (a) SdT V dp + i N idµ i = Ideal and van der Waals gases 1. Ideal gas EOS P v = RT, u = crt (a) Fundamental equation s(u, v) = cr ln(u/u 0 ) + R ln(v/v 0 ) + s 0 (u 0, v 0 ) 2. Generalized ideal gas (a) P v = RT and u(t ) = u 0 + T T 0 c v(t )dt (b) κ T = 1/P, α = 1/T, c p = c v + R 3. Isothermal process ( T = 0) (a) Ideal gas: u = 0, s = R ln(v/v 0 ), q QS = RT ln(v/v 0 ) = w QS 4. Adiabatic ( s = 0) processes (a) Ideal gas adiabats (γ = (c + 1)/c): P v γ = c, T v γ 1 = c, P (1 γ)/γ T = c (b) w QS = RT 0 [ 1 (v/v0 ) 1 γ] /(1 γ), q QS = 0 5. Free expansion: heat but no light! 6. van der Waals gas (a) P vdw = RT/(v b) a/v 2 and u = crt a/v November 26, W. F. Schneider 2

3 U = U(S, V, N) du = Table 1: Thermodynamic Potentials ( ) U ds + S V,N ( ) U dv + ( ) U dn i V S,N N i S,V S = S(U, V, N) ds = du = T ds P dv + µ i dn i U = T S P V + µn ( ) ( ) S S du + dv + ( ) S dn i U V,N V U,N N i U,V ds = 1 T du + P T dv µ i T dn i S = U/T + P V/T + µ i N i /T H = H(S, P, N) H = U + P V dh = ( ) H ds + S P,N ( ) H dp + ( ) H dn i P S,N N i S,P dh = T ds + V dp + µ i dn i H = T S + µ i N i F = F (T, V, N) F = U T S df = ( ) ( ) F F dt + dv + ( ) F dn i T V,N V T,N N i T,V df = SdT P dv + µ i dn i F = P V + µ i N i G = G(T, P, N) G = U T S + P V dg = ( ) G dt + T P,N ( ) G dp + ( ) G dn i P T,N N i T,P dg = SdT + V dp + µ i dn i G = µ i N i November 26, W. F. Schneider 3

4 (b) Fundamental equation s(u, v) = cr ln((u + a/v)/(u 0 + a/v 0 )) + R ln((v b)/(v 0 b)) + s 0 (u 0, v 0 ) (c) Simplest cubic EOS that gives qualitatively correct fluid properties i. Coexistent of two phases ii. Critical point (T c, P c, v c ) where two phases coalesce into one 1.5 Work and efficiency 1. Maximum work theorem: maximum work is delivered by a process that overall produces zero entropy (a) du sys + dq rhs + dw rws = 0, ds sys + ds rhs = ds sys + dq rhs /T rhs = 0 (b) Tells us what is possible, not how to achieve it! 2. Examples: expansion with a low T reservoir, separation, cooling water 3. Thermodynamic engines operate cyclically to convert heat to work or use work to move heat 4. Carnot efficiency and Carnot cycle, η = 1 T c /T h 1.6 Other thermodynamic potentials 1. Energy minimum principle minimum at constant entropy 2. Legendre transforms (a) Y = Y (X) ψ(p ) = Y (P ) P X(P ) P = Y/ X (b) P, ψ(p ) give intercept and slope of tangents of Y 3. Enthalpy H(S, P, N) = U + P V (a) Minimized at constant S, P, and N (b) Heat flow when only P V work done 4. Helmholtz A(T, V, N) = U T S (a) Minimized at constant T, V, and N (b) Maximum useful work from a process at temperature T 5. Gibbs G(T, P, N) = U + P V T S (a) Minimized at constant T, P, and N (b) Most useful for chemical problems ( ) (G/T ) (c) Gibbs-Helmholtz = H T T 2 6. Alles potential (a) Gibbs-Duhem redux 7. Maxwell relations, see Table 2. P,N November 26, W. F. Schneider 4

5 Table 2: Useful Maxwell Relationships Enthalpy Helmholtz Gibbs ( ) ( ) T V = P S S P ( ) S V T = ( ) P T S ( ) S P T ( ) V = T P Table 3: Susceptibilities ( ) 1 v Coefficient of thermal expansion α v T P Isothermal compressibility κ T 1 ( ) v v P T ( ) s Constant pressure heat capacity C p T Constant volume heat capacity C v T T ) ( s T P v ( 1 2 ) g v T P ( 2 ) g 1 v P 2 ( 2 ) g T T 2 N T,N P,N 8. Three unique susceptibilities of a one-component material (Table 3) (a) All thermodynamic properties can be described in terms of the susceptibilities (b) Integrating susceptibilities (c) Heat capacity and departure functions 9. Using thermodynamic relations (a) Joule-Thompson effect 1.7 Stability and phase equilibria 1. Local stability condition (a) (Free) energy minimized du = 0 d 2 U 0 (b) Entropy maximized ds = 0 d 2 S 0 (c) Implies c p c v 0, κ T κ s 0 (d) Microscopic fluctuations and Le Chatlier s principle 2. Global stability conditions (a) Common tangents and convex hull (b) Lever rule (c) Phase separation two phases have lower free energy than one. Balance of energetic attractions and entropic repulsion (d) Critical points (d 3 u = 0) attraction and repulsion exactly in balance (e) Stable, metastable (spinodal), and unstable regions November 26, W. F. Schneider 5

6 i. Extensive quantities discontinuous between phases ( latent quantities) ii. Intensive quantities equal between phases iii. Susceptibilities discontinuous between phases 3. Gibbs-Duhem integrations 4. Equal area construction, dµ = vdp along an isotherm 5. dµ = sdt, chemical potential of each phase decreases with T 6. Phase diagrams lines of equal chemical potential, µ(l) = µ(v) 7. Clausius equation (a) Along coexistence line dp/dt = s/ v = h/t v in general (b) Clausius-Clapeyron for ideal vapor d ln P/d(1/T ) = h/r 8. Gibb s phase rule and triple point (a) DOF = c π R The Microscopic View 2.1 Micro-canonical ensemble Energy is quantized at microscopic level 1. Consequence of quantum mechanics 2. electronic, vibrational, rotational, translational 3. Need machinary to average QM information over macroscopic systems 4. Equal a priori probabilities Two-state model 1. Box of particles, each of which can have energy 0 or ɛ 2. Thermodynamic state defined by number of elements N, and number of quanta q, U = qɛ 3. Degeneracy of given N and q given by binomial distribution: Ω = 4. Allow energy to flow between two such systems N! q!(n q)! (a) Energy of a closed system is conserved (first law!) (b) Degeneracy of total system is always degeneracy of the starting parts! (c) Boltzmann s tombstone, S = k B ln Ω (d) Clausius: entropy of the universe seeks a maximum! Second Law... November 26, W. F. Schneider 6

7 5. Energy flow/thermal equilibrium between two large systems (a) Each subsystem has energy U i and degeneracy Ω i (U i ) (b) Bring in thermal contact, U = U 1 + U 2, Ω = Ω 1 (U 1 )Ω 2 (U 2 ) (c) If systems are very large, one combination of U 1, U 2 and Ω will be much more probably than all others (d) What value of U 1 and U 2 = U U 1 maximizes Ω? ( ) ln Ω1 U 1 ( ) S1 U 1 N N ( ) ln Ω2 = U 2 ( ) S2 = U 2 1. Thermal equilibrium is determined by equal temperature! N ( ) 1 S T = U N (a) When the temperatures of the two subsystems are equal, the entropy of the combined system is maximized! (b) (Same arguments lead to requirement that equal pressures (P i ) and equal chemical potentials (µ i ) maximize entropy when volumes or particles are exchanged) Two-state model in limit of large N 1. Large N and Stirling s approximation 2. Fundamental thermodynamic equation of two-state system: S(U) = k B (x ln x + (1 x) ln(1 x)), where x = q/n = U/Nɛ 3. Temperature is derivative of entropy wrt energy yields U(T ) = (a) T 0, U 0, S 0, minimum disorder Nɛ 1 + e ɛ/k BT (b) T, U Nɛ/2, S k B ln 2, maximum disorder 4. Differentiate again to get heat capacity 2.2 Canonical ensemble Partition function 1. Where do fundamental equations come from? 2. Direct construction of S(U) is generally intractable, so seek simpler approach 3. Imagine a system brought into thermal equilibrium with a much larger reservoir of constant T, such that the aggregate has a total energy U N November 26, W. F. Schneider 7

8 4. Degeneracy of a given system microstate j with energy U j is Ω res (U U j ) du res T = k B d ln Ω res Ω res (U U j ) e U j/k B T 5. Probability for system to be in a microstate with energy U j given by Boltzmann distribution! P (U j ) e U j/k B T = e U j 6. Partition function normalizes distribution, Q(T ) = j e U j 7. For system of identical (distinguishable) elements with energy states ɛ i, can factor probability to show P (ɛ i ) e ɛ i/k B T = e ɛ i, = 1/k B T Energy factoring 1. If system is large, how to determine it s energy states U j? There would be many, many of them! 2. One simplification is if we can write energy as sum of energies of individual elements (atoms, molecules) of system: Q(N, V, T ) = j U j = ɛ j (1) + ɛ j (2) ɛ j (N) (1) e U j (2) = j e (ɛ j(1)+ɛ j (2)+...+ɛ j (N)) (3) 3. If molecules/elements of system can be distinguished from each other (like atoms in a fixed lattice), expression can be factored: Q(N, V, T ) = j e ɛ j(1) j e ɛ j(n) (4) = q(1) q(n) (5) Assuming all the elements are the same: (6) = q N (7) q = e ɛj : molecular partition function j (8) 4. If not distinguishable (like molecules in a liquid or gas, or electrons in a solid), problem is difficult, because identical arrangements of energy amongst elements should only be counted once. Approximate solution, good almost all the time:} Q(N, V, T ) = q N /N! (9) 5. Sidebar: Correct factoring depends on whether individual elements are fermions or bosons, leads to funny things like superconductivity and superfluidity. November 26, W. F. Schneider 8

9 2.2.3 Two-state system again 1. Partition function, q(t ) = 1 + e ɛ 2. State probabilities 3. Internal energy U(T ) U(T ) = N ( ln(1 + e ɛ ) ) = Nɛe ɛ 1 + e ɛ (10) 4. Heat capacity C v (a) Minimum when change in states with T is small (b) Maximize when chagne in states with T is large 5. Helmholtz energy, A = ln q/, decreasing function of T 6. Entropy 7. Distinguishable vs.\ indistinguishable particles (a) Distinguishable (e.g., in a lattice): Q(N, V, T ) = q(v, T ) N (b) Indistinguishable (e.g., a gas): Q(N, V, T ) q(v, T ) N /N! 8. Thermodynamic functions in canonical ensemble 2.3 Ideal gases redux Separability Q ig (N, V, T ) = (q transq rot q vib ) N N! Particle-in-a-box (translational states of a gas) 1. Energy states ɛ n = n 2 ɛ 0, n = 1, 2,..., ɛ 0 tiny for macroscopic V 2. Θ trans = ɛ 0 /k B translational temperature 3. Θ trans << T many states contribute to q trans integral approximation 4. Internal energy 5. Heat capacity 6. Equation of state (!) Λ = 7. Entropy: Sackur-Tetrode equation q trans,1d = 0 e x2 ɛ 0 dx = L/Λ ( h 2 ) 1/2 thermal wavelength 2πm q trans,3d = V/Λ 3 November 26, W. F. Schneider 9

10 Table 4: Equations of the Canoncial (NV T ) Ensemble = 1/k B T Full Ensemble Distinguishable particles Indistinguishable particles (e.g. atoms in a lattice) (e.g. molecules in a fluid) Single particle partition function q(v, T ) = e ɛi i q(v, T ) = e ɛi i Full partition function Q(N, V, T ) = j e Uj Q = q(v, T ) N Q = q(v, T ) N /N! Log partition ln Q N log q N ln q ln N! function N(ln Q ln N + 1) Helmholtz energy (A = U T S) Internal energy (U) Pressure (P ) ln Q ( ) ln Q 1 ( ) ln Q V NV N N N N ln q ( ) ln q V ( ) ln q V N N N (ln q N + 1 ) ( ) ln q V ( ) ln q Entropy (S/k B ) U + ln Q U + N ln q U + N (ln(q/n) + 1) Chemical potential (µ) 1 ( ) ln Q N V T ln q V ln(q/n) NOTE! All energies are referenced to their values at 0 K. Enthalpy H = U + P V, Gibb s Energy G = A + P V. November 26, W. F. Schneider 10

11 2.3.3 Rigid rotor (rotational states of a gas) 1. energy states and degeneracies 2. Θ rot = 2 /2Ik B 3. High T q rot (T ) σθ rot /T Harmonic oscillator (vibrational states of a gas) 1. Θ vib = hν/k B Electronic partition functions spin multiplicity Solids 1. Equipartition 2. Law of Dulong and Petitt 3. Einstein crystal and heat capacity 4. Debye crysal Other ensembles 1. Isothermal/isobaric (a) (T, P, N) = j e Uj e P V j (b) G(T, P, N) = k B T ln (T, P, N) 2. Grand canonical (a) Ξ(T, V, µ) = j e Uj e P Vj e µn j (b) Ψ(T, V, µ) = k B T ln Ξ(T, V, µ) (c) Langmuir isotherm example 3 Thermodynamics of Stuff 3.1 Theory of non-ideal fluids Non-ideality 1. Real molecules interact through vdw interactions (a) dipole-dipole, dipole-induced dipole, induced dipole-induced dipole (London dispersion) (b) scale with dipole moments( µ and polarizability volumes (α) of molecules (c) U(r) c/r 6 2. Particle-in-a-box model breaks down, have to work harder but can still get at same ideas 3. Configurational integral Q config =... e U(r) dr 1... dr n November 26, W. F. Schneider 11

12 Table 5: Statistical Thermodynamics of an Ideal Gas Translational DOFs 3-D particle in a box model θ trans = π2 2 ( ) 1/2 2mL 2, Λ = h k B 2πm For T >> Θ trans, Λ << L, q trans = V/Λ 3 (essentially always true) U trans = 3 2 RT C v,trans = 3 ( e 5/2 2 R V ) ( e 5/2 ) k B T S trans = R ln N Λ 3 = R ln P Λ 3 Rotational DOFs Rigid rotor model Linear molecule θ rot = hcb/k B q rot = 1 (2l + 1)e l(l+1)θrot/t, 1 { T 1, unsymmetric, T >> θ rot σ = σ σ θ l=0 rot 2, symmetric U rot = RT C v,rot = R S rot = R(1 ln(σθ rot /T )) Non-linear molecule θ rot,α = hcb α /k B q rot 1 σ ( πt 3 θ rot,α θ rot, θ rot,γ U rot = 3 2 RT C v,rot = 3 2 R S rot = R 2 Vibrational DOFs Harmonic oscillator model Single harmonic mode θ vib = hν/k B ) 1/2, T >> θ rot,α,,γ σ = rotational symmetry number ( 3 ln σθ rot,αθ rot, θ rot,γ πt 3 ) q vib = 1 1 e θ vib/t T θ vib, T >> θ vib U vib = C v,vib = Svib,i ( = θ vib θvib e θ ) 2 ( ) vib/2t θvib /T R e θvib/t R 1 T e θvib/t R 1 e θvib/t 1 ln(1 e θvib/t ) Multiple harmonic modes θ vib,i = hν i /k B q vib = i 1 1 e θ vib,i/t R i U vib = C v,vib = Svib,i = θ vib,i e θvib,i/t R ( θvib,i e θ ) 2 ( ) vib,i/2t θvib,i /T 1 T i e θvib,i/t R 1 e θvib,i/t 1 ln(1 e θvib,i/t ) Electronic DOFs q elec = spin multiplicity November 26, W. F. Schneider 12

13 Table 6: Contributions of Molecular Degrees of Freedom to Gas Thermodynamics DOF Characteristic Characteristic #states at Internal energy temperature 300 K energy Translational ɛ trans = 2 2mL cm 1 θ trans K U = 3 2 RT Rotational ɛ rot 1 cm 1 θ rot 1 K 100s #DOF RT Vibrational ɛ vib 1000 cm 1 θ vib 1000 K 1 non-classical, 0 RT Electronic ɛ elec cm 1 θ elec K 1 0 Q = (q trans q rot q vib q elec ) N /N! U = U trans + U rot + U vib + U elec, van der Waals gas 1. Hard sphere + 1/r 6 potential + mean-field approximation (g(r) = 1) 2. Q config = ((V Nb) exp( (φ/2))) N vdw EOS 3. See Hill, J. Chem. Ed. 1948, 25, 347, 4. Free energy has two competing contributions 5. f vdw = k B T ln{(v b)(k B T ) 3/2 } a/v k B T 6. P vdw = RT/(v b) a/v Radial distribution functions, g(r), for gases, liquids, solids Virial expansion 1. Configurational integral can be expanded in powers of 1/v times virial coefficients B j (T ) 2. f(t, P ) = f ig (T, P ) RT {B 2 (T )/v + B 3 (T )/v } 3. Z = 1 + B 2 (T )/v + B 3 (T )/v Second virial coefficient B 2 (T ) limiting low density departure of volume from that of an ideal gas, v res (T, v) = v v ig = (RT/P )(Z 1) B 2 (T ) = lim v v res, Virial coefficients integrate clusters of intermolecular interactions, 2-body, 3-body, B 2 (T ) = 2πN A 0 {e u(r)/k BT 1}r 2 dr 6. B vdw 2 (T ) = b a/rt 7. Lennard-Jones potential November 26, W. F. Schneider 13

14 3.1.5 Modern approach is to use numerical methods 1. forcefield to represent intra- and intermolecular properties 2. molecular dynamics or Monte Carlo to sample interactions 3. Fluid property challenge at AIChE 3.2 Engineering representations of fluids How to represent properties of real fluids? 1. Mechanical equations of state (empirical) 2. Thermodynamic tabulations (JANAF, steam tables) 3. Theoretical models (virial expansion) 4. Computer simulations (molecular interactions) Thermodynamic integrations 1. Integrate susceptibilities 2. Integrate P vt relationship plus C ig p (T ) Compressibility 1. Fluids deviate from ideality because they have finite size and interact over distances, origin of two-parameter EOS 2. Compressibility Z(T, P ) = P/P ig = P v/rt measures deviation from ideality 3. Critical compressibility Z c 0.27 for all normal fluids 4. Reduced variable T R = T/T c, P R = P/P c, v R = v/v c 5. Law of corresponding states all normal fluids have the same P vt behavior in reduced variables 6. Allows fluids to be described on generalized compressibility chart Cubic mechanical equations of state 1. van der Waals, Z c = Redlich-Kwong, Z c = Peng-Robinson, Z c = Empirically parameterized, all obey law of corresponding states, none perfect November 26, W. F. Schneider 14

15 3.2.5 Departure functions measure thermodynamic potential difference between real state and hypothetical ideal gas state 1. whatever = ideal + departure 2. From generalized compressibility or accurate EOS integration 3. Entropy departure s(t, P ) s ig (T, P ) = R ln Z(T, P ) + v(t,p ) 4. Enthalpy departure h(t, P ) h ig (T, P ) = RT (Z 1) + [ v(t,p ) T Fugacity measures departure of free energy ideality 1. Define µ(t, P ) = µ (T ) + RT ln f(t, P )/f (T ) [( ) P T ( P T v R v ] dv )v P ] dv lim µ = P 0 ln f/p = RT P lim f = P ig = 0 P 0 P 0 (Z 1)dP 2. µ α = µ implies f α = f. 3. Fugacity coefficient γ(t, P ) = f(t, P )/P ln γ = P o Z 1 P dp 3.3 Single-phase mixtures Ideal gas mixtures 1. Statistical mechanical approach 2. Properties of ideal mixture u mix = v mix = 0 s mix = k B y i ln y i i 3. Partial pressure P i = y i P 4. Chemical potential µ i (y i, T ) = µ i (T ) + k BT ln (y i P/P ) 5. Work of separation and Gibbs paradox Non-ideal gas mixtures 1. Inconvenient that lim yi 0 µ i (y i, T, P ) = 2. Construct fugacity to obey same equilibrium conditions as chemical potential but to tend to y i P in infinite dilution f i (y i, T, P ) = P exp [(µ i (y i, T, P ) µ i (T ))/k B T ] November 26, W. F. Schneider 15

16 3. Chemical potential µ i (y i, T, P ) = µ i (T ) + k BT ln (/P ) 4. Fugacity coefficient φ i (y i,t,p ) = f i (y i, T, P )/(y i P ) 5. Fugacity can be computed from a mixture-explicit EOS ( ) P ( RT ln fi /y i P = v i RT ) dp = 1 ( (F F ig ) ) ln Z 0 P RT N i T,V,N j i 6. Virial mixture equation, vdw equation, Lorentz-Berthelot mixing b = i b i a = i y i y j a ij j a ij = a i a j 8. Lewis (ideal) fugacity rule f i y i f i generalization of ideal gas Ideal liquid mixtures 1. Lattice model with random distribution of molecules u mix = v mix = 0 s mix = k B x i ln x i µ i (x i, T ) = µ i (T ) + k B T ln (x i ) 2. True for liquids of comparable molecular dimensions and interactions Partial molar quantities i 1. For any extensive quantity or susceptibility, J: ( ) J J i = J = N i N j i i J i N i j = Ji x i j mix = i J i x i i j i x i 2. Gibbs-Duhem says partial molar properties are not independent: i N id J i = Non-ideal liquid mixtures 1. Bragg-Williams/mean field approximation 2. Differential exchange parameter χ AB χ 12 = z k B T ( u 12 u 11 + u Hildebrand regular solution and excess mixing quantities g xs /k B T = χ 12 x 1 (1 x 2 ) µ 1 (T, x 1 ) = µ ideal 1 (T, x 1 ) + k B T χ 12 (1 x 1 ) 2 November 26, W. F. Schneider 16 )

17 4. Activity and activity coefficient (a) Solvent convention (b) Solute convention 5. Liquid-liquid phase diagrams (a) Phase separation, critical point 6. Freezing point depression/boiling point elevation Ionic mixtures 1. Debye-Huckel 3.4 Gas-liquid equilibria Ideal-ideal liquid-vapor mixtures 1. Equal chemical potentials in each phase 2. Equilibrium cycle (relate to single component phase equilibrium): start from single component l-v equilibrium, compress each component to desired pressure, then mix. 3. Gas compression important, liquid compression (Poynting correction) less so. 4. Ideal vapor-liquid equilibrium Raoult s Law: 5. Pressure-composition diagram 6. Temperature-composition diagram Non-ideal liquid/ideal vapor µ gas i = µ liq i y i P = x i P i 1. Regular solution χ 12 > 0 positive deviation from Raoult s Law, = 0 Raoult s Law, < 0 negative deviation from Raoult s Law 2. Temperature-composition diagrams (a) Liquid-vapor (b) Eutectics, Henry s Law (dilute) limit 4. Gibbs-Duhem consequences November 26, W. F. Schneider 17

18 4 Thermodynamics of Change 4.1 Chemical thermodynamics and equilibria 1. Chemical reactions 2. Thermodynamic potential differences (a) Standard states (b) Reaction entropy S (T ) = S B (T ) S A (T ) (c) Reaction energy U (T ) = UB (T ) U A (T ) + E(0) (d) Gibbs-Helmholtz 3. Equilibrium-closed system (a) Equilibrium constants and algebraic solutions (b) Free energy minimization (c) Parallel reactions 4. Equilibrium-open system (a) Reaction phase diagrams 5. Partition functions and K eq 6. Non-ideal activities 7. Electrochemical reactions 4.2 Non-equilibrium thermodynamics November 26, W. F. Schneider 18

19 Table 7: Equilibrium and Rate Constants Equilibrium Constants a A + b B c C + d D K eq (T ) = e S (T,V )/k B e H (T,V )/k B T K c (T ) = K p (T ) = ( 1 c ) νc+ν d ν a ν b (q c /V ) νc (q d /V ) ν d ( kb T P (q a /V ) νa (q b /V ) ν b e E(0) ) νc+ν d ν a ν b (q c /V ) νc (q d /V ) ν d (q a /V ) νa (q b /V ) ν b e E(0) Unimolecular Reaction [A] [A] C k(t ) = ν K = k BT h q (T )/V q A (T )/V e E (0) E a = H + k B T A = e 1 k BT h e S Bimolecular Reaction A + B [AB] C k(t ) = ν K = k BT h ( ) q (T )/V 1 1 (q A (T )/V )(q B (T )/V ) c e E (0) E a = H + 2k B T A = e 2 k BT h e S Table 8: Physical units N Av : mol 1 1 amu: kg k B : J K ev K 1 R: J K 1 mol l atm mol 1 K 1 σ SB : J s 1 m 2 K 4 c: m s 1 h: J s ev s : J s ev s hc: ev nm e: C m e : kg MeV c 2 ɛ 0 : C 2 J 1 m e 2 Å 1 ev 1 e 2 /4πɛ 0 : J m ev Å a 0 : m Å E H : 1 Ha ev November 26, W. F. Schneider 19

Introduction Statistical Thermodynamics. Monday, January 6, 14

Introduction Statistical Thermodynamics 1 Molecular Simulations Molecular dynamics: solve equations of motion Monte Carlo: importance sampling r 1 r 2 r n MD MC r 1 r 2 2 r n 2 3 3 4 4 Questions How can