7.2 Vapor-Liquid Equilibrium (VLE), Raoult's Law, and Activity Coefficients

7.2 Vapor-Liquid Equilibrium (VLE), Raoult's Law, and Activity Coefficients

Vapor-liquid equilibrium (VLE) calculations dictate the stage requirements, reflux ratios, and separation feasibility of distillation columns, flash separators, and absorption towers throughout chemical process plants. On the NCEES PE Chemical Exam, VLE questions test your ability to navigate the continuum between ideal solution behavior (Raoult's Law) and highly non-ideal liquid mixtures requiring activity coefficient models and fugacity corrections.

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1. Rigorous Thermodynamic Framework: The Equal Fugacity Criterion

For a multicomponent mixture containing $C$ chemical species coexisting in vapor ($V$) and liquid ($L$) phases at uniform temperature ($T$) and pressure ($P$), phase equilibrium demands that the fugacity ($f_i$) of each component be identical in both phases:

$$f_i^V = f_i^L \quad \text{for } i = 1, 2, \dots, C$$

Fugacity (having units of pressure, such as $\text{bar}$ or $\text{psia}$) represents the "effective thermodynamic pressure" or escaping tendency of a chemical species from a given phase.

Vapor Phase Fugacity (Fugacity Coefficient Formulation)

In the vapor mixture, non-ideality arises from gas-phase intermolecular interactions and volumetric departures from the ideal gas law, quantified by the vapor fugacity coefficient ($\phi_i^V$):

$$f_i^V = y_i \phi_i^V P$$

Where:

• $y_i$ = mole fraction of component $i$ in the vapor phase.
• $P$ = total absolute system pressure.
• $\phi_i^V$ = partial fugacity coefficient of component $i$ in the gas mixture (evaluated from an equation of state such as Peng-Robinson or Soave-Redlich-Kwong; $\phi_i^V \to 1.0$ as $P \to 0$).

Liquid Phase Fugacity (Activity Coefficient Formulation)

In the liquid mixture, non-ideality arises from differences in molecular size, shape, polarity, and hydrogen bonding between dissimilar liquid molecules. This behavior is captured by the liquid activity coefficient ($\gamma_i$) relative to an ideal solution reference:

$$f_i^L = x_i \gamma_i f_i^\circ$$

Where:

• $x_i$ = mole fraction of component $i$ in the liquid phase.
• $\gamma_i$ = liquid-phase activity coefficient of component $i$ (dimensionless; $\gamma_i = 1.0$ for an ideal liquid solution).
• $f_i^\circ$ = standard state fugacity of pure liquid component $i$ at system temperature $T$ and pressure $P$.

Pure Liquid Standard State and the Poynting Correction Factor

The standard state fugacity $f_i^\circ$ is related to the pure component saturation pressure ($P_i^{sat}$) at temperature $T$ by integrating the fundamental property relation $(\partial \ln f_i / \partial P)_T = v_i / (R T)$ from saturation pressure to total system pressure $P$:

$$f_i^\circ = P_i^{sat} \phi_i^{sat} \exp\left[ \frac{v_i^L (P - P_i^{sat})}{R T} \right]$$

Where:

• $\phi_i^{sat}$ = fugacity coefficient of pure saturated vapor $i$ at $(T, P_i^{sat})$.
• $v_i^L$ = molar liquid volume of pure component $i$ (assumed incompressible over $P_i^{sat} \to P$).
• $\mathcal{P}_i = \exp\left[ \frac{v_i^L (P - P_i^{sat})}{R T} \right]$ is the Poynting Correction Factor.

The Complete (Gamma-Phi) VLE Equation

Equating vapor and liquid fugacities yields the general Gamma-Phi ($\gamma-\phi$) VLE relationship:

$$y_i \phi_i^V P = x_i \gamma_i P_i^{sat} \phi_i^{sat} \exp\left[ \frac{v_i^L (P - P_i^{sat})}{R T} \right]$$

+-----------------------------------------------------------------------------------+
|                             THE VLE REGIME SPECTRUM                               |
+-----------------------------------------------------------------------------------+
|  Ideal Gas + Ideal Liquid:       P * y_i = x_i * P_i^sat      (Raoult's Law)      |
|  Ideal Gas + Real Liquid:        P * y_i = x_i * gamma_i * P_i^sat (Modified RL)  |
|  Real Gas + Real Liquid:         y_i * phi_i * P = x_i * gamma_i * f_i^o (Full)   |
+-----------------------------------------------------------------------------------+

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2. Low-to-Moderate Pressure Simplifications

In standard chemical plant operations operating at low to moderate pressures ($P < 5\text{ bar}$ or $\sim 75\text{ psia}$):

1. The vapor phase obeys the ideal gas law: $\phi_i^V \approx 1.00$.
2. Pure saturated vapor is near-ideal: $\phi_i^{sat} \approx 1.00$.
3. Liquid volume is small, so the Poynting factor is negligible: $\mathcal{P}_i = \exp[v_i^L(P - P_i^{sat})/(R T)] \approx 1.000$.

Modified Raoult's Law

Under these standard conditions, the rigorous VLE expression reduces to Modified Raoult's Law:

$$P y_i = x_i \gamma_i P_i^{sat}(T)$$

Where the partial pressure of component $i$ in the vapor is $P_i = P y_i = x_i \gamma_i P_i^{sat}$.

Raoult's Law (Ideal Solution)

When the liquid mixture consists of chemically similar molecules with nearly identical sizes and intermolecular forces (e.g., benzene + toluene, n-hexane + n-heptane, o-xylene + p-xylene), the activity coefficients equal unity ($\gamma_i = 1.00$). The relationship collapses to Raoult's Law (François-Marie Raoult, 1887):

$$P y_i = x_i P_i^{sat}(T)$$

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3. Bubble Point, Dew Point, and Relative Volatility

Because the mole fractions in each phase must sum to unity ($\sum x_i = 1$ and $\sum y_i = 1$), Modified Raoult's Law establishes two foundational boundary calculations:

1. Bubble Point Pressure (Given $T$ and liquid composition $\{x_i\}$)

The bubble point pressure is the total pressure at which the first infinitesimal bubble of vapor forms from a subcooled liquid:

$$P = \sum_{i=1}^C P_i = \sum_{i=1}^C x_i \gamma_i P_i^{sat}(T)$$

The equilibrium vapor composition formed at this bubble pressure is:

$$y_i = \frac{x_i \gamma_i P_i^{sat}(T)}{P}$$

2. Dew Point Pressure (Given $T$ and vapor composition $\{y_i\}$)

The dew point pressure is the total pressure at which the first infinitesimal droplet of liquid condenses from a superheated vapor:

$$x_i = \frac{P y_i}{\gamma_i P_i^{sat}(T)}$$

Summing over all liquid mole fractions ($\sum x_i = 1$):

$$1 = \sum_{i=1}^C \frac{P y_i}{\gamma_i P_i^{sat}(T)} \implies P = \frac{1}{\sum_{i=1}^C \frac{y_i}{\gamma_i P_i^{sat}(T)}}$$

(Note: For non-ideal liquids, this calculation requires iteration because $\gamma_i$ depends on the unknown liquid composition $\{x_i\}$).

3. Relative Volatility ($\alpha_{ij}$)

The relative volatility ($\alpha_{12}$) measures the separability of component 1 relative to component 2 by distillation. It is defined as the ratio of their vapor-liquid distribution coefficients ($K$-values, where $K_i \equiv y_i / x_i$):

$$\alpha_{12} \equiv \frac{K_1}{K_2} = \frac{y_1 / x_1}{y_2 / x_2} = \frac{\gamma_1 P_1^{sat}}{\gamma_2 P_2^{sat}}$$

For an ideal binary system ($\gamma_1 = \gamma_2 = 1$), $\alpha_{12} = P_1^{sat} / P_2^{sat}$. For a binary system, rearranging the definition yields the equilibrium vapor mole fraction as a function of liquid composition:

$$y_1 = \frac{\alpha_{12} x_1}{1 + (\alpha_{12} - 1) x_1}$$

• If $\alpha_{12} > 1.0$: Component 1 is more volatile (light key) and concentrates in the vapor phase.
• If $\alpha_{12} = 1.0$: The vapor and liquid compositions are identical ($y_1 = x_1$). Separation by conventional distillation is physically impossible (an azeotrope exists).
• As $\alpha_{12} \to 1.0$: Separation requires an extraordinarily tall column with hundreds of stages and extreme reflux ratios.

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4. Excess Gibbs Free Energy ($G^E$) & Activity Coefficient Models

Non-ideality in real liquid solutions is formulated through the excess Gibbs free energy ($G^E$), defined as the difference between the actual molar Gibbs energy of the solution ($G$) and that of an ideal solution ($G^{id}$) at the same temperature, pressure, and composition:

$$G^E \equiv G - G^{id} = R T \sum_{i=1}^C x_i \ln \gamma_i$$

From Euler's theorem for homogeneous functions, the activity coefficient of any component $i$ is the partial molar excess Gibbs energy:

$$\ln \gamma_i = \left[ \frac{\partial (n G^E / R T)}{\partial n_i} \right]_{T, P, n_{j \ne i}}$$

The Gibbs-Duhem Equation

Thermodynamic consistency between activity coefficients in a mixture is governed by the Gibbs-Duhem equation. At constant temperature and pressure:

$$\sum_{i=1}^C x_i d\ln \gamma_i = 0$$

For a binary mixture: $x_1 \frac{d\ln \gamma_1}{dx_1} + x_2 \frac{d\ln \gamma_2}{dx_1} = 0$. Experimental VLE data must satisfy this area-integral constraint to be considered thermodynamically valid on the PE exam.

Overview of Major Activity Coefficient Models

1. One-Parameter Margules Equation

The simplest empirical excess Gibbs model, assuming symmetrical non-ideality:

$$\frac{G^E}{R T} = A x_1 x_2$$ $$\ln \gamma_1 = A x_2^2, \quad \ln \gamma_2 = A x_1^2$$

• At infinite dilution ($x_1 \to 0, x_2 = 1$): $\ln \gamma_1^\infty = A$.
• Symmetric: $\gamma_1^\infty = \gamma_2^\infty = \exp(A)$. Peak non-ideality occurs at equimolar composition ($x_1 = x_2 = 0.5$).

2. Two-Parameter Margules Equation

Accounts for molecular asymmetry between dissimilar components:

$$\frac{G^E}{R T} = x_1 x_2 \left( A_{21} x_1 + A_{12} x_2 \right)$$ $$\ln \gamma_1 = x_2^2 \left[ A_{12} + 2(A_{21} - A_{12}) x_1 \right]$$ $$\ln \gamma_2 = x_1^2 \left[ A_{21} + 2(A_{12} - A_{21}) x_2 \right]$$

• Infinite dilution activity coefficients: $\ln \gamma_1^\infty = A_{12}$ and $\ln \gamma_2^\infty = A_{21}$.

3. Van Laar Equation

Derived from van der Waals cohesive energy density concepts; relates non-ideality to molecular co-volumes:

$$\ln \gamma_1 = A_{12}' \left[ 1 + \frac{A_{12}' x_1}{A_{21}' x_2} \right]^{-2}, \quad \ln \gamma_2 = A_{21}' \left[ 1 + \frac{A_{21}' x_2}{A_{12}' x_1} \right]^{-2}$$

Where $A_{12}' = \ln \gamma_1^\infty$ and $A_{21}' = \ln \gamma_2^\infty$.

4. Wilson Equation (Local Composition Model)

Grant M. Wilson (1964) introduced the concept of local composition to account for non-random molecular orientation caused by differences in intermolecular interaction energies:

$$\frac{G^E}{R T} = -x_1 \ln\left( x_1 + x_2 \Lambda_{12} \right) - x_2 \ln\left( x_2 + x_1 \Lambda_{21} \right)$$ $$\ln \gamma_1 = -\ln(x_1 + x_2 \Lambda_{12}) + x_2 \left[ \frac{\Lambda_{12}}{x_1 + x_2 \Lambda_{12}} - \frac{\Lambda_{21}}{x_2 + x_1 \Lambda_{21}} \right]$$

• Key Advantage: Highly accurate for strongly non-ideal polar/non-polar mixtures (e.g., alcohols + hydrocarbons).
• CRITICAL PE LIMITATION: The Wilson equation is mathematically incapable of predicting liquid-liquid phase splitting (immiscibility) because its formulation always yields a positive second derivative $(\partial^2 G / \partial x_1^2) > 0$. It must never be used for decanters or liquid-liquid extraction (LLE).

5. NRTL (Non-Random Two-Liquid) Equation

Renon and Prausnitz (1968) extended local composition theory by adding an empirical non-randomness parameter ($\alpha_{12}$):

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

Where $G_{12} = \exp(-\alpha_{12} \tau_{12})$ and $\alpha_{12} = \alpha_{21}$ (typically $0.20$ to $0.47$).

• Key Advantage: Accurately models both VLE and LLE (partially miscible liquids). Excellent for aqueous-organic mixtures.

6. UNIQUAC & UNIFAC

• UNIQUAC (Universal Quasi-Chemical): Separates excess Gibbs energy into a combinatorial part (molecular size and shape based on van der Waals volume $r_i$ and surface area $q_i$) and a residual part (energetic interactions).
• UNIFAC (UNIQUAC Functional-group Activity Coefficients): A group contribution method that estimates activity coefficients by breaking molecules into structural functional groups (e.g., $-\text{CH}_3, -\text{CH}_2-, -\text{OH}, -\text{COOH}$). Enables prediction of VLE when no experimental mixture data exist.

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5. Comparison Table: Activity Coefficient Models

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6. Azeotropic Systems: Criteria and Classification

An azeotrope is a liquid mixture that boils at a constant temperature and produces a vapor phase with a composition identical to the liquid phase:

$$y_i = x_i \quad \text{for all } i = 1, 2, \dots, C$$

At the azeotropic point, relative volatility is identically unity ($\alpha_{12} = 1.00$). From Modified Raoult's Law:

$$P y_1 = x_1 \gamma_1 P_1^{sat} \implies P = \gamma_1 P_1^{sat}$$ $$P y_2 = x_2 \gamma_2 P_2^{sat} \implies P = \gamma_2 P_2^{sat}$$

Equating the two expressions for total pressure yields the Azeotropic Activity Ratio Criterion:

$$\frac{\gamma_1}{\gamma_2} = \frac{P_2^{sat}}{P_1^{sat}}$$

Azeotrope Classification

  P ^       Positive Deviation (gamma > 1)        P ^       Negative Deviation (gamma < 1)
    |               /---\                           |   P1_sat\                 /P2_sat
    |              /  |  \ Azeotrope                |          \               /
    |      P1_sat /   |   \                         |           \    /---\    /
    |            /    |    \ P2_sat                 |            \  /  |  \  / Azeotrope
    |           /     |     \                       |             \/   |   \/
    +------------------------------------> x1       +------------------------------------> x1
          Minimum-Boiling Azeotrope (P-max)               Maximum-Boiling Azeotrope (P-min)

1. Minimum-Boiling (Positive) Azeotrope:
   • Arises from positive deviations from Raoult's law ($\gamma_i > 1.0$). Unlike molecules experience repulsive or weaker intermolecular forces than like molecules ($A-B$ attraction is weaker than $A-A$ and $B-B$).
   • Produces a maximum in pressure on a $P-x-y$ diagram and a corresponding minimum in boiling temperature on a $T-x-y$ diagram.
   • Examples: Ethanol + Water ($89.4\text{ mol}\%$ ethanol, boils at $78.15^\circ\text{C}$ vs pure ethanol $78.3^\circ\text{C}$ and water $100^\circ\text{C}$), Isopropanol + Water, Acetone + Methanol.
2. Maximum-Boiling (Negative) Azeotrope:
   • Arises from negative deviations from Raoult's law ($\gamma_i < 1.0$). Unlike molecules attract each other more strongly than like molecules ($A-B$ hydrogen bonding or acid-base association).
   • Produces a minimum in pressure on a $P-x-y$ diagram and a maximum in boiling temperature on a $T-x-y$ diagram.
   • Examples: Acetone + Chloroform (strong hydrogen bonding between carbonyl oxygen and chloroform hydrogen; boils at $64.5^\circ\text{C}$ vs pure acetone $56.2^\circ\text{C}$ and chloroform $61.2^\circ\text{C}$), Nitric Acid + Water, Formic Acid + Water.

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7. Comprehensive Worked Numerical Example: Binary VLE, Bubble Pressure, and Azeotrope Screening

Problem Statement

A binary liquid mixture consists of Methyl Ethyl Ketone (MEK, Component 1) and Toluene (Component 2) at a constant temperature of $T = 60.0^\circ\text{C}$.

Thermodynamic Data at $60.0^\circ\text{C}$:

• Pure component vapor pressure of MEK: $P_1^{sat} = 51.00\text{ kPa}$
• Pure component vapor pressure of Toluene: $P_2^{sat} = 18.50\text{ kPa}$
• Liquid phase non-ideality is described by the One-Parameter Margules equation with parameter $A = 0.450$: $$\ln \gamma_1 = A x_2^2, \quad \ln \gamma_2 = A x_1^2$$

Calculate:

1. The activity coefficients $\gamma_1$ and $\gamma_2$ for a liquid mixture with $x_1 = 0.400$ ($40.0\text{ mol}\%$ MEK).
2. The total bubble point pressure ($P$) of this mixture in $\text{kPa}$.
3. The equilibrium vapor phase composition ($y_1$).
4. The relative volatility ($\alpha_{12}$) of MEK relative to Toluene at this composition.
5. Evaluate whether this binary system forms an azeotrope at $60.0^\circ\text{C}$.

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Step 1: Activity Coefficient Evaluation

Liquid mole fractions:

$$x_1 = 0.400 \implies x_2 = 1.000 - 0.400 = 0.600$$

Calculate $\gamma_1$:

$$\ln \gamma_1 = A x_2^2 = 0.450 \times (0.600)^2 = 0.450 \times 0.3600 = \mathbf{0.1620}$$ $$\gamma_1 = \exp(0.1620) = \mathbf{1.1759}$$

Calculate $\gamma_2$:

$$\ln \gamma_2 = A x_1^2 = 0.450 \times (0.400)^2 = 0.450 \times 0.1600 = \mathbf{0.0720}$$ $$\gamma_2 = \exp(0.0720) = \mathbf{1.0747}$$

Both $\gamma_1, \gamma_2 > 1.00$, confirming moderate positive deviation from Raoult's law.

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Step 2: Bubble Point Pressure Determination

Using Modified Raoult's Law:

$$P = x_1 \gamma_1 P_1^{sat} + x_2 \gamma_2 P_2^{sat}$$

Evaluate partial pressures:

$$P_1 = x_1 \gamma_1 P_1^{sat} = 0.400 \times 1.17586 \times 51.00\text{ kPa} = \mathbf{23.988\text{ kPa}}$$ $$P_2 = x_2 \gamma_2 P_2^{sat} = 0.600 \times 1.07466 \times 18.50\text{ kPa} = \mathbf{11.929\text{ kPa}}$$

Total bubble point pressure:

$$P = P_1 + P_2 = 23.988 + 11.929 = \mathbf{35.92\text{ kPa}}$$

(Comparison with ideal Raoult's law: $P_{ideal} = 0.40(51.0) + 0.60(18.5) = 20.40 + 11.10 = 31.50\text{ kPa}$. Non-ideality elevates the bubble pressure by $14.0\%$).

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Step 3: Equilibrium Vapor Composition

$$y_1 = \frac{P_1}{P} = \frac{23.988\text{ kPa}}{35.917\text{ kPa}} = \mathbf{0.6679} \approx \mathbf{0.668}$$ $$y_2 = 1.000 - 0.6679 = \mathbf{0.3321} \approx \mathbf{0.332}$$

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Step 4: Relative Volatility ($\alpha_{12}$)

$$\alpha_{12} = \frac{y_1 / x_1}{y_2 / x_2} = \frac{0.6679 / 0.400}{0.3321 / 0.600} = \frac{1.6698}{0.5535} = \mathbf{3.017} \approx \mathbf{3.02}$$

Alternatively, evaluate via modified Raoult's terms directly:

$$\alpha_{12} = \frac{\gamma_1 P_1^{sat}}{\gamma_2 P_2^{sat}} = \frac{1.17586 \times 51.00}{1.07466 \times 18.50} = \frac{59.969}{19.881} = \mathbf{3.016}$$

Because $\alpha_{12} = 3.02 \gg 1.0$, MEK is substantially more volatile than toluene, indicating an easy separation in a distillation column.

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Step 5: Azeotrope Screening Analysis

For an azeotrope to form at $60.0^\circ\text{C}$, there must exist a liquid mole fraction $x_1 \in (0, 1)$ where $\alpha_{12} = 1.00$:

$$\frac{\gamma_1}{\gamma_2} = \frac{P_2^{sat}}{P_1^{sat}} = \frac{18.50\text{ kPa}}{51.00\text{ kPa}} = \mathbf{0.3627}$$

For the One-Parameter Margules model:

$$\ln\left( \frac{\gamma_1}{\gamma_2} \right) = A \left( x_2^2 - x_1^2 \right) = A (x_2 - x_1)(x_2 + x_1) = A (1 - 2x_1)$$

Taking exponentials:

$$\frac{\gamma_1}{\gamma_2} = \exp\left[ A (1 - 2x_1) \right]$$

Determine the physical bounds of this ratio for $x_1 \in [0, 1]$:

• As $x_1 \to 0$ (pure Toluene): $(\gamma_1 / \gamma_2)_{max} = \exp(+A) = \exp(0.450) = \mathbf{1.568}$
• As $x_1 \to 1$ (pure MEK): $(\gamma_1 / \gamma_2)_{min} = \exp(-A) = \exp(-0.450) = \mathbf{0.6376}$

Therefore, for any mixture composition, the activity coefficient ratio is strictly bounded:

$$0.6376 \le \frac{\gamma_1}{\gamma_2} \le 1.568$$

However, the required azeotropic ratio is $\frac{P_2^{sat}}{P_1^{sat}} = 0.3627$. Because $0.3627 < 0.6376$, the required ratio lies completely outside the achievable physical range of the liquid mixture.

Conclusion: No azeotrope can form in the MEK-Toluene system at $60.0^\circ\text{C}$. The difference in pure component vapor pressures ($51.0$ vs $18.5\text{ kPa}$) is large enough to overwhelm the moderate liquid non-ideality ($A = 0.450$).

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8. Critical PE Exam Traps & Pitfalls

Trap 1: Blindly Applying Raoult's Law to Polar or Hydrogen-Bonding Mixtures
Raoult's law ($P y_i = x_i P_i^{sat}$) is valid only for chemically similar non-polar compounds. If an exam question involves an alcohol + water, alcohol + hydrocarbon, or acetone + water mixture, assuming $\gamma_i = 1.00$ introduces massive errors ($30-200\%$). Always check whether activity coefficients are given or must be evaluated.

Trap 2: The Wilson Equation Decanter Trap
A favorite theoretical question on the PE Chemical exam asks you to select an appropriate thermodynamic property package for a flowsheet containing an extraction column or decanter (two liquid phases). If the options include the Wilson Equation, remember that Wilson cannot predict liquid-liquid immiscibility! Selecting Wilson will cause process simulator decanters to crash or falsely predict a single liquid phase. The correct answer is always NRTL, UNIQUAC, or a two-parameter Margules model.

Trap 3: Inverting Compositions in Bubble vs. Dew Point Equations
In a bubble point calculation, the known composition is the liquid phase ($x_i$), and you multiply by activity coefficients: $P = \sum x_i \gamma_i P_i^{sat}$. In a dew point calculation, the known composition is the vapor phase ($y_i$), and you divide: $P = 1 / \sum [y_i / (\gamma_i P_i^{sat})]$. Confusing these leads to inverted bubble/dew pressures.