Formula Sheet: Phase Equilibria

Fundamental Phase Equilibria Concepts

Gibbs Phase Rule

Degrees of Freedom:

\[F = C - P + 2\]

• F = degrees of freedom (number of intensive variables that can be varied independently)
• C = number of components
• P = number of phases in equilibrium
• The "2" represents temperature and pressure
• For systems at constant temperature or pressure, use: \(F = C - P + 1\)

Chemical Potential Equilibrium Criterion

Phase Equilibrium Condition:

\[\mu_i^{\alpha} = \mu_i^{\beta} = \mu_i^{\gamma} = \cdots\]

• μ_i = chemical potential of component i
• Superscripts α, β, γ denote different phases
• At equilibrium, chemical potential of each component is equal in all phases

Fugacity and Fugacity Coefficient

Pure Component Fugacity

Definition of Fugacity:

\[d\mu_i = RT \, d(\ln f_i)\]

• f_i = fugacity of component i (same units as pressure)
• μ_i = chemical potential of component i
• R = universal gas constant
• T = absolute temperature (K or °R)

Fugacity Coefficient:

\[\phi_i = \frac{f_i}{P}\]

• φ_i = fugacity coefficient (dimensionless)
• P = total pressure
• For ideal gas: φ_i = 1

Fugacity Coefficient from Equation of State:

\[\ln \phi = \int_0^P \left(\frac{Z-1}{P}\right) dP = \int_0^P \left[\left(\frac{\partial Z}{\partial n_i}\right)_{T,P,n_{j \neq i}} - \frac{1}{P}\right] dP\]

Or in terms of compressibility factor:

\[\ln \phi = \int_{\infty}^{V} \left(\frac{P}{RT} - \frac{1}{V}\right) dV - \ln Z\]

• Z = compressibility factor
• V = molar volume

Mixture Fugacity

Component Fugacity in Mixture:

\[\hat{f}_i = y_i \hat{\phi}_i P\]

• ŷ_i = fugacity of component i in mixture
• y_i = mole fraction of component i in vapor phase
• φ̂_i = fugacity coefficient of component i in mixture

Fugacity Coefficient in Mixture from EOS:

\[\ln \hat{\phi}_i = \int_0^P \left[\frac{1}{RT}\left(\frac{\partial n G}{\partial n_i}\right)_{T,P,n_{j \neq i}} - \frac{1}{P}\right] dP\]

Vapor-Liquid Equilibrium (VLE)

General VLE Criteria

Equilibrium Condition:

\[\hat{f}_i^V = \hat{f}_i^L\]

• Superscript V = vapor phase
• Superscript L = liquid phase
• Must be satisfied for all components at equilibrium

Raoult's Law (Ideal System)

Raoult's Law:

\[y_i P = x_i P_i^{sat}\]

• x_i = mole fraction of component i in liquid phase
• P_i^sat = vapor pressure of pure component i at system temperature
• Valid for ideal solutions (similar molecular size and intermolecular forces)
• Valid at low to moderate pressures

Total Pressure for Binary System:

\[P = x_1 P_1^{sat} + x_2 P_2^{sat} = P_2^{sat} + x_1(P_1^{sat} - P_2^{sat})\]

Modified Raoult's Law (Non-Ideal Liquid, Ideal Vapor)

Modified Raoult's Law:

\[y_i P = x_i \gamma_i P_i^{sat}\]

• γ_i = activity coefficient of component i in liquid phase (dimensionless)
• Valid at low to moderate pressures where vapor phase is ideal
• Accounts for non-ideality in liquid phase

Equilibrium Ratio (K-value):

\[K_i = \frac{y_i}{x_i} = \frac{\gamma_i P_i^{sat}}{P}\]

• K_i = equilibrium ratio (distribution coefficient)
• K_i > 1: component tends toward vapor phase
• K_i < 1:="" component="" tends="" toward="" liquid="">

General VLE Using Fugacity Coefficients

Phi-Phi Approach (for high pressures):

\[y_i \hat{\phi}_i^V P = x_i \hat{\phi}_i^L P\]

Simplified:

\[y_i \hat{\phi}_i^V = x_i \hat{\phi}_i^L\]

• Both liquid and vapor non-idealities accounted for
• Required at high pressures
• Fugacity coefficients calculated from equation of state

Gamma-Phi Approach (most common):

\[y_i \hat{\phi}_i^V P = x_i \gamma_i f_i^L\]

Where liquid fugacity:

\[f_i^L = \phi_i^{sat} P_i^{sat} \exp\left[\frac{V_i^L(P - P_i^{sat})}{RT}\right]\]

• V_i^L = molar volume of pure liquid component i
• φ_i^sat = fugacity coefficient of pure i at saturation
• The exponential term is the Poynting correction factor
• At low pressure, φ_i^sat ≈ 1 and Poynting factor ≈ 1, reducing to modified Raoult's law

Simplified Gamma-Phi Form:

\[y_i \hat{\phi}_i P = x_i \gamma_i \phi_i^{sat} P_i^{sat} \exp\left[\frac{V_i^L(P - P_i^{sat})}{RT}\right]\]

Henry's Law (Dilute Solutions)

Henry's Law:

\[y_i P = x_i H_{i,j}\]

• H_i,j = Henry's constant for component i in solvent j (units of pressure)
• Valid for sparingly soluble gases (x_i → 0)
• Valid for solute at infinite dilution
• Temperature dependent

Alternative Henry's Law Form:

\[P_i = H_{i,j} x_i\]

Relationship to Activity Coefficient:

\[H_{i,j} = \lim_{x_i \to 0} \gamma_i P_i^{sat}\]

Activity Coefficients

Definition and Properties

Activity Coefficient:

\[\gamma_i = \frac{a_i}{x_i}\]

• a_i = activity of component i
• γ_i = 1 for ideal solution
• γ_i > 1 indicates positive deviation from ideality
• γ_i < 1="" indicates="" negative="" deviation="" from="">

Infinite Dilution Activity Coefficient:

\[\gamma_i^{\infty} = \lim_{x_i \to 0} \gamma_i\]

• Represents maximum deviation from ideality
• Important for highly non-ideal systems

Gibbs-Duhem Equation

Gibbs-Duhem Equation (isothermal, isobaric):

\[\sum_{i=1}^{n} x_i d\ln \gamma_i = 0\]

For binary system:

\[x_1 d\ln \gamma_1 + x_2 d\ln \gamma_2 = 0\]

• Provides thermodynamic consistency check for activity coefficient data
• Activity coefficients of components in a mixture are not independent

Thermodynamic Consistency Test (binary):

\[\int_0^1 \ln\left(\frac{\gamma_1}{\gamma_2}\right) dx_1 = 0\]

Excess Gibbs Free Energy

Excess Gibbs Energy:

\[G^E = G - G^{ideal} = RT \sum_{i} x_i \ln \gamma_i\]

• G^E = excess Gibbs free energy
• Measure of deviation from ideal solution behavior

Activity Coefficient from Excess Gibbs Energy:

\[\ln \gamma_i = \frac{1}{RT}\left[\frac{\partial (nG^E)}{\partial n_i}\right]_{T,P,n_{j \neq i}}\]

For constant T and P:

\[\ln \gamma_i = \frac{\partial (G^E/RT)}{\partial x_i} + \frac{G^E}{RT} - \sum_j x_j \frac{\partial (G^E/RT)}{\partial x_j}\]

Activity Coefficient Models

Margules Equations

Two-Suffix (One-Parameter) Margules (symmetric):

\[\ln \gamma_1 = A x_2^2\] \[\ln \gamma_2 = A x_1^2\] \[\frac{G^E}{RT} = A x_1 x_2\]

• A = Margules parameter (dimensionless)
• Valid for symmetric systems

Three-Suffix (Two-Parameter) Margules:

\[\ln \gamma_1 = x_2^2[A_{12} + 2(A_{21} - A_{12})x_1]\] \[\ln \gamma_2 = x_1^2[A_{21} + 2(A_{12} - A_{21})x_2]\] \[\frac{G^E}{RT} = x_1 x_2 (A_{12} x_2 + A_{21} x_1)\]

• A₁₂, A₂₁ = Margules parameters (dimensionless)
• More flexible than two-suffix form

Van Laar Equations

Van Laar Equations (binary):

\[\ln \gamma_1 = \frac{A}{\left(1 + \frac{A x_1}{B x_2}\right)^2}\] \[\ln \gamma_2 = \frac{B}{\left(1 + \frac{B x_2}{A x_1}\right)^2}\] \[\frac{G^E}{RT} = \frac{A B x_1 x_2}{A x_1 + B x_2}\]

• A, B = Van Laar parameters (dimensionless)
• At infinite dilution: A = ln γ₁^∞ and B = ln γ₂^∞
• Good for systems with large deviations from ideality

Wilson Equation

Wilson Equation (binary):

\[\ln \gamma_1 = -\ln(x_1 + \Lambda_{12} x_2) + x_2 \left(\frac{\Lambda_{12}}{x_1 + \Lambda_{12} x_2} - \frac{\Lambda_{21}}{x_2 + \Lambda_{21} x_1}\right)\] \[\ln \gamma_2 = -\ln(x_2 + \Lambda_{21} x_1) - x_1 \left(\frac{\Lambda_{12}}{x_1 + \Lambda_{12} x_2} - \frac{\Lambda_{21}}{x_2 + \Lambda_{21} x_1}\right)\]

Wilson Parameters:

\[\Lambda_{12} = \frac{V_2^L}{V_1^L} \exp\left(-\frac{\lambda_{12} - \lambda_{11}}{RT}\right)\] \[\Lambda_{21} = \frac{V_1^L}{V_2^L} \exp\left(-\frac{\lambda_{21} - \lambda_{22}}{RT}\right)\]

• Λ₁₂, Λ₂₁ = Wilson parameters (dimensionless)
• λ_ij = interaction energy parameters
• V_i^L = liquid molar volume of pure component i
• Cannot predict liquid-liquid equilibria
• Good for highly non-ideal vapor-liquid systems

NRTL (Non-Random Two-Liquid) Equation

NRTL Equation (binary):

\[\ln \gamma_1 = x_2^2 \left[\tau_{21}\left(\frac{G_{21}}{x_1 + x_2 G_{21}}\right)^2 + \frac{\tau_{12} G_{12}}{(x_2 + x_1 G_{12})^2}\right]\] \[\ln \gamma_2 = x_1^2 \left[\tau_{12}\left(\frac{G_{12}}{x_2 + x_1 G_{12}}\right)^2 + \frac{\tau_{21} G_{21}}{(x_1 + x_2 G_{21})^2}\right]\]

NRTL Parameters:

\[G_{12} = \exp(-\alpha_{12} \tau_{12})\] \[G_{21} = \exp(-\alpha_{12} \tau_{21})\] \[\tau_{12} = \frac{g_{12} - g_{22}}{RT}\] \[\tau_{21} = \frac{g_{21} - g_{11}}{RT}\]

• τ₁₂, τ₂₁ = NRTL energy parameters (dimensionless)
• α₁₂ = non-randomness parameter (typically 0.2 to 0.47)
• g_ij = interaction energy parameters
• Can predict both VLE and LLE
• Three adjustable parameters per binary pair

UNIQUAC (Universal Quasi-Chemical) Equation

UNIQUAC Equation:

\[\ln \gamma_i = \ln \gamma_i^{combinatorial} + \ln \gamma_i^{residual}\]

Combinatorial Part:

\[\ln \gamma_i^{combinatorial} = \ln\frac{\Phi_i}{x_i} + \frac{z}{2}q_i \ln\frac{\theta_i}{\Phi_i} + l_i - \frac{\Phi_i}{x_i}\sum_j x_j l_j\]

Where:

\[\Phi_i = \frac{x_i r_i}{\sum_j x_j r_j}\] \[\theta_i = \frac{x_i q_i}{\sum_j x_j q_j}\] \[l_i = \frac{z}{2}(r_i - q_i) - (r_i - 1)\]

Residual Part (binary):

\[\ln \gamma_i^{residual} = q_i \left[1 - \ln(\theta_1 \tau_{i1} + \theta_2 \tau_{i2}) - \frac{\theta_1 \tau_{1i}}{\theta_1 \tau_{1i} + \theta_2 \tau_{2i}} - \frac{\theta_2 \tau_{2i}}{\theta_1 \tau_{1i} + \theta_2 \tau_{2i}}\right]\]

UNIQUAC Parameters:

\[\tau_{ij} = \exp\left(-\frac{u_{ij} - u_{jj}}{RT}\right)\]

• r_i = volume parameter for component i
• q_i = surface area parameter for component i
• z = coordination number (typically 10)
• τ_ij = UNIQUAC interaction parameters
• u_ij = interaction energy parameters
• r and q obtained from group contribution methods
• Can predict VLE and LLE

Vapor Pressure Correlations

Antoine Equation

Antoine Equation:

\[\log_{10} P^{sat} = A - \frac{B}{C + T}\]

Or in natural logarithm form:

\[\ln P^{sat} = A - \frac{B}{C + T}\]

• A, B, C = Antoine constants (component-specific)
• T = temperature (°C or K, depending on constant set)
• P^sat = vapor pressure (units depend on constants, often mmHg, bar, or kPa)
• Valid over limited temperature ranges
• Most common vapor pressure correlation

Clausius-Clapeyron Equation

Clausius-Clapeyron Equation:

\[\frac{dP^{sat}}{dT} = \frac{\Delta H_{vap}}{T \Delta V_{vap}}\]

For ideal gas vapor phase:

\[\frac{dP^{sat}}{dT} = \frac{\Delta H_{vap} P^{sat}}{RT^2}\]

• ΔH_vap = enthalpy of vaporization
• ΔV_vap = volume change upon vaporization

Integrated Form (constant ΔH_vap):

\[\ln P^{sat} = -\frac{\Delta H_{vap}}{RT} + C\]

Or between two states:

\[\ln\left(\frac{P_2^{sat}}{P_1^{sat}}\right) = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\]

• Valid when ΔH_vap is approximately constant
• Useful for estimating vapor pressure at different temperatures

Cox Chart Method

Cox Chart Relationship:

\[\log P^{sat} = a + b \cdot f(T)\]

• Plots log P^sat vs. special temperature function
• Produces nearly straight lines for many substances
• Useful for graphical interpolation and extrapolation

Flash Calculations

Single-Stage Flash

Overall Material Balance:

\[F = L + V\]

• F = feed molar flow rate
• L = liquid product molar flow rate
• V = vapor product molar flow rate

Component Material Balance:

\[F z_i = L x_i + V y_i\]

• z_i = mole fraction of component i in feed

Vapor Fraction:

\[\psi = \frac{V}{F}\]

• ψ = vapor fraction (0 ≤ ψ ≤ 1)
• Liquid fraction: (1 - ψ) = L/F

Rachford-Rice Equation:

\[\sum_{i=1}^{n} \frac{z_i(K_i - 1)}{1 + \psi(K_i - 1)} = 0\]

• Solved iteratively for ψ given K_i values
• Valid for 0 < ψ=""><>

Component Distribution:

\[x_i = \frac{z_i}{1 + \psi(K_i - 1)}\] \[y_i = \frac{K_i z_i}{1 + \psi(K_i - 1)}\]

Or equivalently:

\[x_i = \frac{z_i}{(1 - \psi) + \psi K_i}\] \[y_i = \frac{K_i z_i}{(1 - \psi) + \psi K_i}\]

Summation Checks:

\[\sum x_i = 1\] \[\sum y_i = 1\]

Bubble Point and Dew Point

Bubble Point (incipient vaporization):

\[\sum_{i=1}^{n} y_i = \sum_{i=1}^{n} K_i x_i = 1\]

• System is all liquid with first bubble of vapor
• Given liquid composition, solve for T (at fixed P) or P (at fixed T)
• ψ = 0

Bubble Point Temperature Calculation:
At given P and liquid composition x_i:

\[\sum_{i=1}^{n} K_i(T) x_i = 1\]

Bubble Point Pressure Calculation:
At given T and liquid composition x_i:

\[P_{bubble} = \sum_{i=1}^{n} x_i P_i^{sat}\]

Dew Point (incipient condensation):

\[\sum_{i=1}^{n} x_i = \sum_{i=1}^{n} \frac{y_i}{K_i} = 1\]

• System is all vapor with first droplet of liquid
• Given vapor composition, solve for T (at fixed P) or P (at fixed T)
• ψ = 1

Dew Point Temperature Calculation:
At given P and vapor composition y_i:

\[\sum_{i=1}^{n} \frac{y_i}{K_i(T)} = 1\]

Dew Point Pressure Calculation:
At given T and vapor composition y_i:

\[\frac{1}{P_{dew}} = \sum_{i=1}^{n} \frac{y_i}{P_i^{sat}}\]

Liquid-Liquid Equilibrium (LLE)

LLE Criteria

Phase Equilibrium for LLE:

\[\hat{f}_i^{\alpha} = \hat{f}_i^{\beta}\]

In terms of activity coefficients:

\[x_i^{\alpha} \gamma_i^{\alpha} = x_i^{\beta} \gamma_i^{\beta}\]

• Superscripts α and β denote two liquid phases
• Must be satisfied for all components

Distribution Coefficient

Distribution Coefficient (Partition Coefficient):

\[K_{D,i} = \frac{x_i^{\beta}}{x_i^{\alpha}}\]

• K_D,i = distribution coefficient for component i
• At equilibrium: \(K_{D,i} = \gamma_i^{\alpha} / \gamma_i^{\beta}\)

Mutual Solubility

For binary systems:

\[x_1^{\alpha} \gamma_1^{\alpha} = x_1^{\beta} \gamma_1^{\beta}\] \[x_2^{\alpha} \gamma_2^{\alpha} = x_2^{\beta} \gamma_2^{\beta}\]

• Solved simultaneously with activity coefficient model
• Produces binodal curve (solubility limits)

Plait Point

Plait Point Condition:

\[x_i^{\alpha} = x_i^{\beta}\] \[\gamma_i^{\alpha} = \gamma_i^{\beta}\]

• Point where two liquid phases become identical
• Upper or lower critical solution temperature

Solid-Liquid Equilibrium (SLE)

Ideal Solubility

Ideal Solubility Equation:

\[\ln x_i = -\frac{\Delta H_{fus}}{R}\left(\frac{1}{T} - \frac{1}{T_{fus}}\right)\]

• ΔH_fus = enthalpy of fusion (melting)
• T_fus = normal melting point (fusion temperature)
• Valid for ideal solutions
• Assumes no solid-solid transitions

Including Heat Capacity Change:

\[\ln x_i = -\frac{\Delta H_{fus}}{R}\left(\frac{1}{T} - \frac{1}{T_{fus}}\right) + \frac{\Delta C_p}{R}\left[\ln\left(\frac{T_{fus}}{T}\right) + \frac{T}{T_{fus}} - 1\right]\]

• ΔC_p = heat capacity change upon melting
• More accurate for wide temperature ranges

Non-Ideal Solubility

Real Solubility with Activity Coefficient:

\[\ln(x_i \gamma_i) = -\frac{\Delta H_{fus}}{R}\left(\frac{1}{T} - \frac{1}{T_{fus}}\right)\]

• Activity coefficient accounts for solution non-ideality
• Requires activity coefficient model (NRTL, UNIQUAC, etc.)

Eutectic Systems

Eutectic Point:

• Lowest temperature at which liquid phase exists
• Unique composition where multiple solid phases coexist with liquid
• For binary: two solid phases and one liquid phase at eutectic

Azeotropes

Azeotrope Conditions

Azeotropic Condition:

\[x_i = y_i \quad \text{for all components}\]

• Vapor and liquid compositions are identical
• Cannot be separated by simple distillation
• Occurs at extrema (maximum or minimum) in T-x-y or P-x-y diagrams

Types of Azeotropes

Minimum Boiling Azeotrope:

\[\left(\frac{\partial P}{\partial x_i}\right)_{T} = 0 \quad \text{and} \quad P > P_i^{sat}\]

• Positive deviation from Raoult's law (γ_i > 1)
• Boils at lower temperature than pure components at constant P

Maximum Boiling Azeotrope:

\[\left(\frac{\partial P}{\partial x_i}\right)_{T} = 0 \quad \text{and} \quad P < p_i^{sat}\]="">

• Negative deviation from Raoult's law (γ_i <>
• Boils at higher temperature than pure components at constant P

Azeotrope Prediction

For Binary System at Azeotrope:

\[K_1 = K_2 = 1\] \[\gamma_1 P_1^{sat} = \gamma_2 P_2^{sat}\]

• Requires activity coefficient model to predict composition
• Azeotrope composition is temperature and pressure dependent

Relative Volatility

Definition

Relative Volatility:

\[\alpha_{ij} = \frac{K_i}{K_j} = \frac{y_i/x_i}{y_j/x_j}\]

• α_ij = relative volatility of component i with respect to component j
• α_ij > 1: component i is more volatile than j
• α_ij = 1: no separation possible (azeotrope)

For Binary System:

\[\alpha = \frac{y_1/x_1}{y_2/x_2} = \frac{y_1 x_2}{x_1 y_2}\]

Using modified Raoult's law:

\[\alpha = \frac{\gamma_1 P_1^{sat}}{\gamma_2 P_2^{sat}}\]

Simplified VLE Relations

When Relative Volatility is Constant:

\[y_1 = \frac{\alpha x_1}{1 + (\alpha - 1)x_1}\]

• Simplifies distillation calculations
• Valid when α is approximately constant over operating range

Rearranged for x₁:

\[x_1 = \frac{y_1}{\alpha - (\alpha - 1)y_1}\]

Equations of State for Phase Equilibria

Van der Waals Equation

Van der Waals EOS:

\[\left(P + \frac{a}{V^2}\right)(V - b) = RT\]

• a = attraction parameter
• b = covolume (excluded volume parameter)

Parameters at Critical Point:

\[a = \frac{27 R^2 T_c^2}{64 P_c}\] \[b = \frac{RT_c}{8P_c}\]

• T_c = critical temperature
• P_c = critical pressure

Redlich-Kwong Equation

Redlich-Kwong EOS:

\[P = \frac{RT}{V - b} - \frac{a}{T^{0.5} V(V + b)}\]

Parameters:

\[a = \frac{0.42748 R^2 T_c^{2.5}}{P_c}\] \[b = \frac{0.08664 RT_c}{P_c}\]

Soave-Redlich-Kwong (SRK) Equation

SRK EOS:

\[P = \frac{RT}{V - b} - \frac{a(T)}{V(V + b)}\]

Temperature-Dependent Parameter:

\[a(T) = 0.42748 \frac{R^2 T_c^2}{P_c} \alpha(T)\] \[\alpha(T) = \left[1 + m\left(1 - \sqrt{T_r}\right)\right]^2\] \[m = 0.480 + 1.574\omega - 0.176\omega^2\]

• ω = acentric factor
• T_r = reduced temperature (T/T_c)

Covolume Parameter:

\[b = 0.08664 \frac{RT_c}{P_c}\]

Peng-Robinson Equation

Peng-Robinson EOS:

\[P = \frac{RT}{V - b} - \frac{a(T)}{V(V + b) + b(V - b)}\]

Parameters:

\[a(T) = 0.45724 \frac{R^2 T_c^2}{P_c} \alpha(T)\] \[\alpha(T) = \left[1 + \kappa\left(1 - \sqrt{T_r}\right)\right]^2\] \[\kappa = 0.37464 + 1.54226\omega - 0.26992\omega^2\] \[b = 0.07780 \frac{RT_c}{P_c}\]

• Widely used for hydrocarbon systems
• Good for both liquid and vapor phases

Mixing Rules for EOS

Van der Waals One-Fluid Mixing Rules:

\[a_m = \sum_i \sum_j x_i x_j a_{ij}\] \[b_m = \sum_i x_i b_i\]

Geometric Mean Combining Rule:

\[a_{ij} = \sqrt{a_i a_j}(1 - k_{ij})\]

• k_ij = binary interaction parameter (typically 0 to 0.1)
• k_ii = 0 for pure components
• Fitted to experimental VLE data

Thermodynamic Relations

Maxwell Relations

From Gibbs Energy:

\[\left(\frac{\partial V}{\partial T}\right)_P = -\left(\frac{\partial S}{\partial P}\right)_T\]

From Helmholtz Energy:

\[\left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial S}{\partial V}\right)_T\]

Departure Functions

Enthalpy Departure:

\[\frac{H - H^{ig}}{RT} = Z - 1 + \int_{\infty}^{V} \left[\frac{T}{V}\left(\frac{\partial P}{\partial T}\right)_V - \frac{P}{V}\right] dV\]

Entropy Departure:

\[\frac{S - S^{ig}}{R} = \ln Z + \int_{\infty}^{V} \left[\frac{1}{V}\left(\frac{\partial P}{\partial T}\right)_V - \frac{R}{V}\right] dV\]

• Superscript "ig" denotes ideal gas state
• Used to calculate real fluid properties from EOS

Special Topics

Critical Point Criteria

Stability Criteria at Critical Point:

\[\left(\frac{\partial P}{\partial V}\right)_T = 0\] \[\left(\frac{\partial^2 P}{\partial V^2}\right)_T = 0\]

• Both conditions must be satisfied simultaneously
• Used to determine critical constants from EOS

Corresponding States Principle

Reduced Properties:

\[P_r = \frac{P}{P_c}, \quad T_r = \frac{T}{T_c}, \quad V_r = \frac{V}{V_c}\]

Compressibility Factor Correlation:

\[Z = Z^{(0)}(T_r, P_r) + \omega Z^{(1)}(T_r, P_r)\]

• Z⁽⁰⁾ = simple fluid compressibility factor
• Z⁽¹⁾ = correction for molecular complexity
• ω = acentric factor

Acentric Factor:

\[\omega = -\log_{10}(P_r^{sat}|_{T_r=0.7}) - 1.000\]

• Measure of molecular non-sphericity
• ω = 0 for simple fluids (Ar, Kr, Xe)

Lever Rule

Lever Rule (binary VLE):

\[\frac{L}{V} = \frac{y - z}{z - x}\]

• Determines phase amounts in two-phase region
• Derived from material balance
• Applies to any intensive property plotted on phase diagram

Ternary Systems

Material Balance for Ternary LLE:

\[F z_i = E x_i^E + R x_i^R\]

• E = extract phase flow rate
• R = raffinate phase flow rate
• Superscripts E and R denote extract and raffinate phases

Triangular Diagram Representation:

• Vertices represent pure components (100%)
• Sides represent binary mixtures
• Interior points represent ternary mixtures
• Tie lines connect equilibrium phases
• Binodal curve separates one-phase and two-phase regions

Pressure Swing and Temperature Swing

Effect of Pressure on VLE

Pressure Effect on K-value:

\[K_i = \frac{\gamma_i \phi_i^{sat} P_i^{sat}}{P \hat{\phi}_i^V} \exp\left[\frac{V_i^L(P - P_i^{sat})}{RT}\right]\]

• Increasing P generally decreases K_i
• Can shift azeotrope composition or eliminate azeotrope
• Pressure swing distillation exploits this behavior

Effect of Temperature on VLE

Temperature Effect via Vapor Pressure:

\[\frac{d \ln P_i^{sat}}{dT} = \frac{\Delta H_{vap,i}}{RT^2}\]

• Higher T increases vapor pressure exponentially
• Can change relative volatility
• May shift or break azeotropes

Multicomponent Systems

Generalized Flash Equations

Rachford-Rice for Multicomponent:

\[\sum_{i=1}^{n} \frac{z_i(K_i - 1)}{1 + \psi(K_i - 1)} = 0\]

Derivative for Newton-Raphson Solution:

\[\frac{d}{d\psi}\left[\sum_{i=1}^{n} \frac{z_i(K_i - 1)}{1 + \psi(K_i - 1)}\right] = -\sum_{i=1}^{n} \frac{z_i(K_i - 1)^2}{[1 + \psi(K_i - 1)]^2}\]

Fugacity in Multicomponent Mixtures

Fugacity from EOS:

\[\ln \hat{\phi}_i = \frac{1}{RT}\int_0^P \left[\left(\frac{\partial V}{\partial n_i}\right)_{T,P,n_{j \neq i}} - \frac{RT}{P}\right] dP\]

Or in compressibility form:

\[\ln \hat{\phi}_i = \int_0^P \left[\frac{1}{P}\left(\frac{\partial n Z}{\partial n_i}\right)_{T,P,n_{j \neq i}} - \frac{1}{P}\right] dP\]

K-value Correlations

Wilson Correlation for K-values:

\[K_i = \frac{P_{c,i}}{P} \exp\left[5.373(1 + \omega_i)\left(1 - \frac{T_{c,i}}{T}\right)\right]\]

• Quick estimate for hydrocarbon systems
• Used as initial guess for iterative calculations
• Accuracy decreases at high pressure

Colligative Properties

Boiling Point Elevation

Boiling Point Elevation:

\[\Delta T_b = K_b m\]

• ΔT_b = boiling point elevation
• K_b = ebullioscopic constant (solvent-specific)
• m = molality of solute

Ebullioscopic Constant:

\[K_b = \frac{RT_b^2 M_A}{\Delta H_{vap}}\]

• T_b = normal boiling point of pure solvent
• M_A = molar mass of solvent

Freezing Point Depression

Freezing Point Depression:

\[\Delta T_f = K_f m\]

• ΔT_f = freezing point depression
• K_f = cryoscopic constant (solvent-specific)

Cryoscopic Constant:

\[K_f = \frac{RT_f^2 M_A}{\Delta H_{fus}}\]

• T_f = normal freezing point of pure solvent

Osmotic Pressure

Van't Hoff Equation:

\[\Pi = MRT\]

• Π = osmotic pressure
• M = molar concentration of solute
• Valid for dilute solutions

General Form:

\[\Pi = -\frac{RT}{V_A} \ln a_A\]

• V_A = molar volume of pure solvent
• a_A = activity of solvent in solution