// 05 · Fugacity · EOS · Activity coefficients · Chemical equilibrium

Chemical Thermodynamics

Advanced phase and chemical equilibrium. Covers fugacity, virial and Peng-Robinson EOS, excess functions, activity coefficients (Margules, Wilson, NRTL), and chemical equilibrium constant K. Based on Prausnitz, Smith & Van Ness.

Ref: Prausnitz · Smith & Van Ness · Sandler | Termodinámica Química 1426 — Facultad de Química UNAM
P Vm Gas ideal T > Tc T ≈ Tc T < Tc Maxwell Cúpula bifásica Pc,Vc,Tc Líquido Vapor L + V Z = PVm/RT 0.89 Peng-Robinson Tr=0.85 Pr=0.60 φ=0.82 f=0.49MPa Virial Z=1+B/Vm B=-120 cm³/mol Z_vir=0.91 ω=0.344
Peng-Robinson: P=RT/(Vm-b) - a(T)/[Vm(Vm+b)+b(Vm-b)] · a(Tc)=0.45724R²Tc²/Pc
Fugacidad: ln φ = Z-1-ln(Z-B')-A'/(2√2B')·ln[(Z+(1+√2)B')/(Z+(1-√2)B')]
Virial (Pitzer): B·Pc/(RTc) = B⁰(Tr)+ω·B¹(Tr) · B⁰=0.083-0.422/Tr^1.6
Diagrama T-x-y binario T x,y 0 0.5 1 Burbuja (x) Rocío (y) L + V Tb,B Tb,A x=0.32 y=0.65 Diagrama ternario A B C Inmiscible Punto plait Feed Palanca: V/F=(z-x)/(y-x) VF=0.40 K=y/x=2.03 Gibbs: F = C - π + 2 Binario bifásico: F=1 Ternario 3 fases: F=0
Raoult modificada (VLE): yᵢ·φᵢᵛ·P = xᵢ·γᵢ·φᵢˢᵃᵗ·Pᵢˢᵃᵗ  →  K-value: Kᵢ=yᵢ/xᵢ
Criterio: fᵢᴸ = fᵢᵛ ∀i  |  Gibbs: F = C - π + 2
Flash isotérmico: Σ[zᵢ(Kᵢ-1)/(1+β(Kᵢ-1))] = 0 → β=V/F (Rachford-Rice)
GE/RT vs x₁ GE/RT x₁ 0 Margules Wilson NRTL 0 0.5 1 GE/RT 0.18 γ₁∞ 2.23 ln(γᵢ) vs x₁ ln γ 0 ln γ₁ ln γ₂ ln γ₁∞ G-D: x₁d(lnγ₁)+x₂d(lnγ₂)=0 Margules 2-suf. A₁₂=0.80, A₂₁=0.60
Margules 2-suf.: GE/RT=x₁x₂(A₂₁x₁+A₁₂x₂) · ln γ₁=x₂²[A₁₂+2(A₂₁-A₁₂)x₁]
Wilson: GE/RT=-x₁ln(x₁+Λ₁₂x₂)-x₂ln(x₂+Λ₂₁x₁) · Λ=Vⱼ/Vᵢ·exp(-aᵢⱼ/RT)
NRTL: GE/RT=x₁x₂[τ₂₁G₂₁/(x₁+x₂G₂₁)+τ₁₂G₁₂/(x₂+x₁G₁₂)] · Gᵢⱼ=exp(-ατᵢⱼ)
A + B ⇌ C + D | ΔG° = -RT·ln(K) Diagrama de energía G ξ ξ_eq Reactivos Productos dG/dξ=0 ΔG°rxn 0 1 ln K vs 1/T (Van't Hoff) ln K 1/T Exotérmico (ΔH<0) Endotérmico (ΔH>0) K(T) Pend = -ΔH°/R K(T) 4.82 X_eq 69% ΔG°(T) kJ/mol -3.89
K(T): ΔG° = -RT·ln K = Σνᵢ·μᵢ° · K = Π(âᵢ^νᵢ) = Π(γᵢ·xᵢ)^νᵢ · (P/P°)^Δν
Van't Hoff: d(ln K)/dT = ΔH°/RT² → K(T₂)=K(T₁)·exp[-ΔH°/R·(1/T₂-1/T₁)]
A+B⇌C+D equimolar: K = X²/(1-X)² → X=√K/(1+√K) · A⇌2B: K=4X²P/(1-X²)
// Equation of state
// Critical properties
Tc (K)647
Pc (MPa)22.1
ω (acentric)0.344
// Reduced conditions
Tr = T/Tc0.85
Pr = P/Pc0.60
// Results
Z
0.89
Phase
Vapor
φ
0.82
f (MPa)
0.49
Z_virial
0.91
B (cm³/mol)
-120
Graphical analysis — Chemical thermodynamics
Generalized Z correlation · Activity coefficients γ vs x₁
Z vs Pr — Peng-Robinson (multiple Tr)
Activity coefficients γᵢ vs x₁