MSE265 Equations

Fundamental Thermodynamic Equation

dE = TdS - PdV +udN
E(S,V,N)

Enthalpy H: definition, differential, natural variables

H = E + PV
dH = TdS + VdP + udN
H(S,P,N)

Gibbs free energy G: definition, differential, natural variables

G = E + PV - TS = H - TS
dG = -SdT + VdP + udN
G(T,P,N)

Helmholtz free energy A: definition, differential, natural variables

A = E - TS
dA = -SdT - PdV + udN
A(T,V,N)

Which free energy is minimized at equilibrium

T, V, N fixed = A minimized
T, P, N fixed = G minimized

How do I derive a Maxwell relation

1. Write thermodynamic potential

2. Read off first derivatives from coefficients

3. Cross-differentiate with respect to the other natural variable

4. Set mixed second derivatives equal

Maxwell relation from E

Maxwell relation from H

Maxwell relation from A

Maxwell relation from G

Definitions of C_v and C_P

Ideal-gas entropy change using T,V

I

Ideal-gas entropy change using T, P

First law sign convention

Q > 0: heat enters system. W > 0: system does work on surroundings

Expansion vs compression work signs

Expansion: W > 0 | Compression: W < 0

Reversible vs irreversible etnropy

Microcanonical vs canonical ensemble

Micro: E,V,N isolated | Canonical: T,V, N energy exchange with reservoir

Boltzmann factor

Lower E → greater statistical weight

Canonical partition function

Probability of state i

Average Energy

Chemical potential equilibrium condition

Compare the same component across phases

Gibbs Phase Rule

Clapeyron equation

Phase transition at equilibrium

Delta G = 0 → Delta H = TDeltaS

Ideal-solution assumption

DeltaH_mix = 0

Ideal entropy of mixing

Ideal Gibbs free energy of mixing

Regular-solution enthalpy of mixing

Regular-solution Gibbs free energy of mixing

Meaning of Omega

Omega > 0 = unfavorable A-B interactions → phase separation favored | Omega < 0 = favorable A-B interactions → mixing favored | Omega = 0 →ideal solution limit

General chemical potential/activity equation

Ideal solution activity coefficient

E

Effect of gamma_i on mu_i at fixed x_i

Activity vs standard-state chemical potential

Stability from curvature of G(x)

Spinodal vs binodal

Inside spinodal → unstable, no nucleation barrier | Between spinodal and binodal →metastable, nucleation required | Outside binoday → stable

Multiplicity for binary lattice

Boltzmann Entropy

S = k_b ln Omega