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Cheatsheet: Phase Equilibria

Table of Contents
1. Fundamental Concepts
2. Vapor-Liquid Equilibrium (VLE)
3. Liquid-Liquid Equilibrium (LLE)
4. Vapor-Liquid-Liquid Equilibrium (VLLE)
5. Solid-Liquid Equilibrium (SLE)
6. High-Pressure VLE
7. Fugacity and Fugacity Coefficient
8. Flash Calculations
9. Henry's Law
10. Excess Properties
11. Critical Properties and Corresponding States
12. Multi-Component VLE
13. Phase Diagrams
14. Thermodynamic Consistency Tests
15. Special Topics
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1. Fundamental Concepts

1.1 Phase Rule

Parameter	Definition
Gibbs Phase Rule	F = C - P + 2, where F = degrees of freedom, C = number of components, P = number of phases
Degrees of Freedom	Number of intensive variables that can be independently varied without changing the number of phases
Phase	Homogeneous, physically distinct portion of a system separated by definite boundaries
Component	Chemically independent constituents of a system

1.2 Equilibrium Criteria

Criterion	Mathematical Expression
Temperature Equality	T^α = T^β = T^γ = ... for all phases
Pressure Equality	P^α = P^β = P^γ = ... for all phases (no curved interfaces)
Chemical Potential Equality	μ_i^α = μ_i^β = μ_i^γ = ... for each component i in all phases
Fugacity Equality	f_i^α = f_i^β = f_i^γ = ... for each component i in all phases

2. Vapor-Liquid Equilibrium (VLE)

2.1 Ideal Systems - Raoult's Law

Equation	Expression
Raoult's Law	y_iP = x_iP_i^sat, where y_i = vapor mole fraction, x_i = liquid mole fraction
K-value (Ideal)	K_i = y_i/x_i = P_i^sat/P
Relative Volatility	α_ij = K_i/K_j = (y_i/x_i)/(y_j/x_j) = P_i^sat/P_j^sat
Bubble Point (T)	Σx_iP_i^sat(T) = P
Dew Point (T)	Σy_i/K_i = Σy_iP/P_i^sat(T) = 1

2.2 Non-Ideal Systems - Modified Raoult's Law

Equation	Expression
Modified Raoult's Law	y_iφ_iP = x_iγ_iP_i^sat, where γ_i = activity coefficient, φ_i = fugacity coefficient
Low Pressure Form	y_iP = x_iγ_iP_i^sat (assumes φ_i ≈ 1)
Activity	a_i = γ_ix_i
K-value (Non-Ideal)	K_i = (γ_iP_i^sat)/(φ_iP)

2.3 Vapor Pressure Correlations

Correlation	Equation
Antoine Equation	log₁₀P^sat = A - B/(C + T), where T in °C or K, P^sat in mmHg or kPa
Clausius-Clapeyron	d(ln P^sat)/dT = ΔH_vap/(RT²)
Integrated C-C	ln(P₂^sat/P₁^sat) = -(ΔH_vap/R)(1/T₂ - 1/T₁)

2.4 Activity Coefficient Models

2.4.1 Margules Equations

Model	Equation
Two-Suffix Margules	ln γ₁ = Ax₂²; ln γ₂ = Ax₁²
Three-Suffix Margules	ln γ₁ = x₂²[A₁₂ + 2(A₂₁ - A₁₂)x₁]; ln γ₂ = x₁²[A₂₁ + 2(A₁₂ - A₂₁)x₂]

2.4.2 Van Laar Equation

Component	Equation
Component 1	ln γ₁ = A₁₂/[1 + (A₁₂x₁)/(A₂₁x₂)]²
Component 2	ln γ₂ = A₂₁/[1 + (A₂₁x₂)/(A₁₂x₁)]²

2.4.3 Wilson Equation

Expression	Equation
Binary System	ln γ₁ = -ln(x₁ + Λ₁₂x₂) + x₂[Λ₁₂/(x₁ + Λ₁₂x₂) - Λ₂₁/(x₂ + Λ₂₁x₁)]
Wilson Parameter	Λ_ij = (V_j^L/V_i^L)exp[-(λ_ij - λ_ii)/(RT)]

2.4.4 NRTL Equation

Parameter	Definition
Binary Activity Coefficient	ln γ₁ = x₂²[τ₂₁(G₂₁/(x₁ + x₂G₂₁))² + (τ₁₂G₁₂)/(x₂ + x₁G₁₂)²]
G Parameters	G₁₂ = exp(-α₁₂τ₁₂); G₂₁ = exp(-α₂₁τ₂₁)
τ Parameters	τ₁₂ = (g₁₂ - g₂₂)/(RT); τ₂₁ = (g₂₁ - g₁₁)/(RT)
α Parameter	Non-randomness parameter, 0.2 ≤ α ≤ 0.47 for most systems

2.4.5 UNIQUAC Equation

Uses molecular size (r) and surface area (q) parameters
Two contributions: combinatorial (entropic) and residual (enthalpic)
ln γ_i = ln γ_i^C + ln γ_i^R
Combinatorial: ln γ_i^C = ln(Φ_i/x_i) + (z/2)q_iln(θ_i/Φ_i) + l_i - (Φ_i/x_i)Σx_jl_j
z = 10 (coordination number)

2.5 Azeotropes

Type	Characteristics
Minimum Boiling	x_i = y_i for all i; positive deviation from Raoult's Law; γ_i > 1
Maximum Boiling	x_i = y_i for all i; negative deviation from Raoult's Law; γ_i <>
Azeotrope Condition	γ₁P₁^sat = γ₂P₂^sat at azeotropic composition

3. Liquid-Liquid Equilibrium (LLE)

3.1 Fundamentals

Concept	Description
Equilibrium Condition	x_i^αγ_i^α = x_i^βγ_i^β for each component in phases α and β
Mutual Solubility	Concentrations of components in each phase at equilibrium
Distribution Coefficient	K_D,i = x_i^β/x_i^α
Plait Point	Point on phase diagram where two liquid phases become identical in composition

3.2 Ternary LLE Diagrams

Triangular diagram: three components at vertices
Binodal curve: boundary between one-phase and two-phase regions
Tie lines: connect equilibrium compositions of two phases
All tie lines meet at plait point
Lever rule applies for calculating phase amounts

3.3 Extraction Calculations

Parameter	Definition
Distribution Ratio	K_D = concentration in extract/concentration in raffinate
Selectivity	β = (K_D,solute)/(K_D,carrier)
Extraction Factor	E = K_D(S/F), where S = solvent flow, F = feed flow

4. Vapor-Liquid-Liquid Equilibrium (VLLE)

4.1 Heterogeneous Azeotropes

Property	Description
Three Phases	Vapor phase in equilibrium with two immiscible liquid phases
Total Pressure	P = P₁^sat + P₂^sat for partially miscible binary at low mutual solubility
Vapor Composition	y₁ = P₁^sat/(P₁^sat + P₂^sat)
Degrees of Freedom	F = C - P + 2 = 2 - 3 + 2 = 1 for binary VLLE

5. Solid-Liquid Equilibrium (SLE)

5.1 Solubility

Equation	Expression
Ideal Solubility	ln x_i = -(ΔH_fus/R)(1/T - 1/T_m), where T_m = melting point
Non-Ideal Solubility	ln(γ_ix_i) = -(ΔH_fus/R)(1/T - 1/T_m)
Eutectic Point	Lowest temperature at which liquid phase exists for binary system

5.2 Crystallization

Supersaturation drives crystallization: S = C/C^sat
Yield = (m₀ - m_sat)/(m₀), where m₀ = initial mass, m_sat = mass in saturated solution
Material balance includes solid and liquid phases
Enthalpy balance includes heat of crystallization

6. High-Pressure VLE

6.1 Equations of State Approach

Method	Expression
Fugacity from EOS	ln(f_i/P) = ln φ_i = ∫_∞^V[(∂P/∂n_i)_{T,V,n_j} - RT/V]dV - ln Z
Equilibrium Condition	φ_i^Vy_iP = φ_i^Lx_iP
K-value from EOS	K_i = y_i/x_i = φ_i^L/φ_i^V

6.2 Cubic Equations of State

6.2.1 van der Waals EOS

Parameter	Expression
Equation	(P + a/V²)(V - b) = RT
Parameter a	a = 27R²T_c²/(64P_c)
Parameter b	b = RT_c/(8P_c)

6.2.2 Redlich-Kwong EOS

Parameter	Expression
Equation	P = RT/(V - b) - a/(T^0.5V(V + b))
Parameter a	a = 0.42748R²T_c^2.5/P_c
Parameter b	b = 0.08664RT_c/P_c

6.2.3 Soave-Redlich-Kwong (SRK) EOS

Parameter	Expression
Equation	P = RT/(V - b) - aα/(V(V + b))
Parameter a	a = 0.42748R²T_c²/P_c
Parameter b	b = 0.08664RT_c/P_c
α Function	α = [1 + m(1 - T_r^0.5)]², where T_r = T/T_c
Parameter m	m = 0.480 + 1.574ω - 0.176ω², where ω = acentric factor

6.2.4 Peng-Robinson (PR) EOS

Parameter	Expression
Equation	P = RT/(V - b) - aα/(V(V + b) + b(V - b))
Parameter a	a = 0.45724R²T_c²/P_c
Parameter b	b = 0.07780RT_c/P_c
α Function	α = [1 + κ(1 - T_r^0.5)]²
Parameter κ	κ = 0.37464 + 1.54226ω - 0.26992ω²

6.3 Mixing Rules

Rule	Expression
van der Waals One-Fluid	a_m = ΣΣy_iy_ja_ij; b_m = Σy_ib_i
Combining Rule	a_ij = (a_ia_j)^0.5(1 - k_ij), where k_ij = binary interaction parameter
Geometric Mean	b_ij = (b_i + b_j)/2

6.4 Compressibility Factor

Definition	Expression
Z Factor	Z = PV/(RT)
Cubic Form (SRK/PR)	Z³ - Z² + (A - B - B²)Z - AB = 0
Reduced Parameters	A = aαP/(R²T²); B = bP/(RT)

7. Fugacity and Fugacity Coefficient

7.1 Definitions

Property	Definition
Fugacity	Corrected pressure for non-ideal behavior: dG_i = RTd(ln f_i)
Fugacity Coefficient	φ_i = f_i/(y_iP) for vapor; φ_i = f_i/(x_iP) for liquid
Activity Coefficient	γ_i = f_i/(x_if_i⁰), where f_i⁰ = standard state fugacity

7.2 Pure Component Fugacity

Phase	Expression
Vapor	ln(f/P) = ∫₀^P[(Z - 1)/P]dP
Liquid at Saturation	f_i^sat = φ_i^satP_i^sat
Poynting Correction	f_i = φ_i^satP_i^satexp[V_i^L(P - P_i^sat)/(RT)]

7.3 Mixture Fugacity

ln φ_i = (1/RT)∫_∞^V[(∂P/∂n_i)_{T,V,n_j} - RT/V]dV - ln Z
For SRK: ln φ_i = (b_i/b_m)(Z - 1) - ln(Z - B) - (A/B)[2Σy_ja_ij/a_m - b_i/b_m]ln(1 + B/Z)
For PR: Similar form with modified ln term

8. Flash Calculations

8.1 Isothermal Flash

Equation	Expression
Material Balance	z_i = x_iL + y_iV, where z_i = feed composition, L + V = 1
Rachford-Rice	Σ[z_i(K_i - 1)/(1 + V(K_i - 1))] = 0
Liquid Composition	x_i = z_i/[1 + V(K_i - 1)]
Vapor Composition	y_i = K_iz_i/[1 + V(K_i - 1)]

8.2 Solution Procedure

Calculate K-values at given T and P
Check: if all K_i > 1, all vapor; if all K_i < 1,="" all="">
Solve Rachford-Rice equation for V (vapor fraction)
Calculate x_i and y_i from V and K_i
Update K-values if using non-ideal models and iterate

9. Henry's Law

9.1 Gas Solubility

Equation	Expression
Henry's Law	f_i = x_iH_i, where H_i = Henry's constant
Low Pressure Form	y_iP = x_iH_i
Temperature Dependence	d(ln H)/dT = -ΔH_sol/(RT²)
Combined Approach	For dilute solute: use Henry's Law; for solvent: use Raoult's Law

9.2 Applications

Gas absorption: CO₂, O₂, N₂ in water
Sparingly soluble gases in liquids
Valid for dilute solutions (x_i → 0)
H_i units must be consistent with P and x

10. Excess Properties

10.1 Definitions

Property	Expression
Excess Gibbs Energy	G^E = G - G^id = RT Σx_iln γ_i
Excess Enthalpy	H^E = H - H^id
Excess Entropy	S^E = S - S^id
Excess Volume	V^E = V - V^id

10.2 Gibbs-Duhem Equation

Form	Expression
General	Σx_id(ln γ_i) = 0 at constant T and P
Binary	x₁d(ln γ₁) + x₂d(ln γ₂) = 0
Consistency Test	∫₀¹ln(γ₁/γ₂)dx₁ = 0 for thermodynamically consistent data

10.3 Activity Coefficient Relationships

Relationship	Expression
From G^E	ln γ_i = [∂(nG^E/RT)/∂n_i]_{T,P,n_j}
Temperature Dependence	[∂ln γ_i/∂T]_P,x = -H_i^E/(RT²)
Infinite Dilution	γ_i^∞ = limit as x_i → 0

11. Critical Properties and Corresponding States

11.1 Critical Point

Property	Definition
Critical Temperature	Highest temperature at which liquid and vapor can coexist
Critical Pressure	Vapor pressure at critical temperature
Critical Volume	Molar volume at critical point
Critical Criteria	(∂P/∂V)_T = 0 and (∂²P/∂V²)_T = 0

11.2 Reduced Properties

Property	Definition
Reduced Temperature	T_r = T/T_c
Reduced Pressure	P_r = P/P_c
Reduced Volume	V_r = V/V_c
Acentric Factor	ω = -log₁₀(P_r^sat)_Tr=0.7 - 1.000

11.3 Generalized Correlations

Correlation	Expression
Compressibility	Z = Z⁽⁰⁾ + ωZ⁽¹⁾, where Z⁽⁰⁾ and Z⁽¹⁾ are functions of T_r and P_r
Fugacity Coefficient	ln φ = ln φ⁽⁰⁾ + ω ln φ⁽¹⁾
Lee-Kesler	Three-parameter corresponding states using ω

12. Multi-Component VLE

12.1 Bubble Point Calculation

Type	Equation
Bubble Point T	ΣK_i(T)x_i = 1 at specified P
Bubble Point P	P = Σx_iK_iP at specified T (rearranged form)
Vapor Composition	y_i = K_ix_i

12.2 Dew Point Calculation

Type	Equation
Dew Point T	Σy_i/K_i(T) = 1 at specified P
Dew Point P	Σy_i/K_i = 1 at specified T
Liquid Composition	x_i = y_i/K_i

12.3 K-value Estimation

Method	Expression
Wilson Approximation	K_i = (P_c,i/P)exp[5.373(1 + ω_i)(1 - T_c,i/T)]
DePriester Charts	Graphical correlation of K vs T and P for light hydrocarbons

13. Phase Diagrams

13.1 Binary VLE Diagrams

Type	Characteristics
T-x-y (Isobaric)	Temperature vs composition at constant pressure; upper curve = bubble point, lower curve = dew point
P-x-y (Isothermal)	Pressure vs composition at constant temperature; upper curve = dew point, lower curve = bubble point
x-y Diagram	y vs x at constant T or P; 45° line = x = y; equilibrium curve above = more volatile in vapor

13.2 Ternary Phase Diagrams

Triangular coordinates: each vertex = pure component (100%)
Distance from side = concentration of opposite component
Constant composition lines: parallel to opposite side
Sum of perpendicular distances = constant (height of triangle)

14. Thermodynamic Consistency Tests

14.1 Area Test

Test	Criterion
Redlich-Kister	\|∫₀¹ln(γ₁/γ₂)dx₁\| < 0.01="" for="" consistent="">
Herington Test	D - J < 10,="" where="" d="">₊ - A_-\|/(A₊ + A_-), J = 150\|T_max - T_min\|/T_min

14.2 Point Test

Check Gibbs-Duhem at individual data points
Slope test: d(ln γ₁)/dx₁ vs d(ln γ₂)/dx₁ must satisfy Gibbs-Duhem
Infinite dilution: γ₁^∞ and γ₂^∞ must be consistent with G^E model

15. Special Topics

15.1 Supercritical Extraction

Parameter	Description
Supercritical Fluid	T > T_c and P > P_c; properties between liquid and gas
Solvent Power	Increases with density (pressure); tuneable by adjusting P and T
Common Solvents	CO₂ (T_c = 31.1°C, P_c = 73.8 bar); ethane; propane

15.2 Polymer Solutions

Model	Application
Flory-Huggins	Activity coefficients for polymer-solvent systems
UNIFAC-FV	Free-volume modification for polymers
SAFT	Statistical Associating Fluid Theory for complex molecules

15.3 Electrolyte Solutions

Model	Description
Debye-Hückel	Activity coefficients for dilute electrolytes based on ionic strength
Pitzer Equations	Virial expansion for concentrated electrolytes
NRTL-Electrolyte	Extension of NRTL for systems with ions

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