11.1 Vapor-Liquid Equilibrium and Relative Volatility in Distillation

11.1 Vapor-Liquid Equilibrium and Relative Volatility in Distillation

Distillation is the most prevalent thermal separation process in the chemical and petrochemical industries, accounting for roughly $40\%$ to $60\%$ of total industrial energy consumption in fluid separations. On the NCEES PE Chemical Exam, mastery of distillation begins with the thermodynamic foundations of Vapor-Liquid Equilibrium (VLE). Every distillation calculation—whether sizing a simple flash drum, stepping off trays using the McCabe-Thiele method, or executing multicomponent Fenske-Underwood-Gilliland shortcut designs—depends directly on accurate phase equilibrium relationships.

---

1. Thermodynamic Criteria for Vapor-Liquid Equilibrium

For a closed, heterogeneous multiphase system at constant temperature ($T$) and pressure ($P$), thermodynamic equilibrium is attained when the total Gibbs free energy is minimized. This requires three fundamental equalities across the liquid ($L$) and vapor ($V$) phases:

1. Thermal Equilibrium: $T^L = T^V = T$
2. Mechanical Equilibrium: $P^L = P^V = P$
3. Chemical Equilibrium: The chemical potential of each chemical species $i$ must be equal across all phases:

$$\mu_i^L(T, P, \mathbf{x}) = \mu_i^V(T, P, \mathbf{y}) \quad (i = 1, 2, \dots, C)$$

Because chemical potential approaches negative infinity as concentration approaches zero, chemical engineers work with the fugacity ($f_i$), which behaves as an effective thermodynamic pressure. The phase equilibrium criterion is therefore written:

$$f_i^L(T, P, \mathbf{x}) = f_i^V(T, P, \mathbf{y})$$

Where:

• $f_i^L = x_i \gamma_i f_i^{\circ, L}$ is the fugacity of component $i$ in the liquid mixture.
• $f_i^V = y_i \phi_i^V P$ is the fugacity of component $i$ in the vapor mixture.
• $x_i, y_i$ are the liquid and vapor mole fractions of species $i$.
• $\gamma_i$ is the liquid-phase activity coefficient, quantifying deviations from ideal solution behavior.
• $\phi_i^V$ is the vapor-phase fugacity coefficient, quantifying deviations from ideal gas behavior.
• $f_i^{\circ, L} = P_i^{sat} \phi_i^{sat} \exp\left[ \frac{V_i^L (P - P_i^{sat})}{R T} \right]$ is the standard-state pure liquid fugacity, incorporating the Poynting correction factor for high system pressures.

Low-to-Moderate Pressure Simplifications

At low to moderate operating pressures ($P < 5\text{ bar}$ or $\sim 75\text{ psia}$), vapor behaves essentially as an ideal gas mixture ($\phi_i^V \approx 1.0$, $\phi_i^{sat} \approx 1.0$), and the Poynting correction factor is practically unity ($\exp[V_i^L(P - P_i^{sat})/RT] \approx 1.0$). Under these conditions, the general equilibrium relation reduces to Modified Raoult's Law:

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

If the liquid phase also forms an ideal solution (chemically similar components with negligible excess volume and zero heat of mixing, such as benzene-toluene or n-hexane-n-heptane), the activity coefficient becomes unity ($\gamma_i = 1.0$), collapsing the relation to classical Raoult's Law:

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

Where $P_i^{sat}(T)$ is the pure component vapor pressure evaluated at the system temperature $T$, commonly computed using the three-parameter Antoine Equation:

$$\ln P_i^{sat} = A_i - \frac{B_i}{T + C_i} \quad \text{or} \quad \log_{10} P_i^{sat} = A_i - \frac{B_i}{T + C_i}$$

---

2. Vapor-Liquid Distribution Ratio ($K$-Value) & Relative Volatility ($\alpha$)

The Equilibrium Ratio ($K$-Value)

The equilibrium distribution of component $i$ between vapor and liquid is quantified by the $K$-value (equilibrium ratio):

$$K_i \equiv \frac{y_i}{x_i}$$

For systems obeying Modified Raoult's Law:

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

For ideal systems (Raoult's Law):

$$K_i = \frac{P_i^{sat}(T)}{P}$$

Components with $K_i > 1.0$ concentrate preferentially in the vapor phase (more volatile), whereas components with $K_i < 1.0$ concentrate in the liquid phase (less volatile).

Relative Volatility ($\alpha$)

Relative volatility ($\alpha_{ij}$) is the ratio of the $K$-values of two components, measuring their relative tendency to vaporize. By convention, species $1$ is chosen as the more volatile (light) component and species $2$ as the less volatile (heavy) component:

$$\alpha_{12} \equiv \frac{K_1}{K_2} = \frac{y_1 / x_1}{y_2 / x_2}$$

In a binary mixture, $x_2 = 1 - x_1$ and $y_2 = 1 - y_1$. Therefore:

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

For ideal systems obeying Raoult's law:

$$\alpha_{12} = \frac{P_1^{sat}(T) / P}{P_2^{sat}(T) / P} = \frac{P_1^{sat}(T)}{P_2^{sat}(T)}$$

For non-ideal systems obeying Modified Raoult's law:

$$\alpha_{12} = \frac{\gamma_1 P_1^{sat}(T)}{\gamma_2 P_2^{sat}(T)}$$

The Binary Equilibrium Curve Equation

Rearranging the definition of $\alpha$ yields the explicit relationship for the equilibrium vapor mole fraction ($y$) as a function of liquid mole fraction ($x$) and relative volatility ($\alpha$):

$$y = \frac{\alpha x}{1 + (\alpha - 1) x}$$

Inverting this expression allows calculation of the liquid composition in equilibrium with a known vapor composition:

$$x = \frac{y}{\alpha - (\alpha - 1) y}$$

   y (Vapor Mole Fraction)
 1.0 +----------------------------------+ /
     |                        ...--*''  |/ 
     |                 ...--''          /  
 0.8 |             .-''                /|
     |          .-'                   / |
 0.6 |        .'                     /  |
     |       /                      /   |
 0.4 |      /                      /    |  Equilibrium Curve
     |     /                      /     |  y = alpha*x / [1 + (alpha-1)x]
 0.2 |    /                      /      |
     |   /                      /       |  Diagonal Line (y = x)
 0.0 +--+----------------------+--------+/
    0.0        0.4            0.8      1.0
               x (Liquid Mole Fraction)

Practical Significance of Relative Volatility Thresholds

The numerical magnitude of $\alpha$ dictates the practical feasibility and capital cost of distillation:

• $\mathbf{\alpha > 2.0}$: Easy separation; requires few equilibrium stages ($N < 15$) and low reflux ratios ($R < 2$).
• $\mathbf{1.2 < \alpha \le 2.0}$: Moderate to standard separation; standard commercial distillation column sizing ($20\text{--}50$ stages).
• $\mathbf{1.05 \le \alpha \le 1.20}$: Close-boiling fractionation (e.g., propane/propylene, ethylbenzene/styrene); requires massive columns ($60\text{--}150+$ trays) and high reflux ratios ($R > 5\text{--}10$).
• $\mathbf{\alpha < 1.05}$: Conventional fractional distillation is economically prohibitive; requires extractive distillation, azeotropic distillation, or membrane pervaporation.
• $\mathbf{\alpha = 1.0}$: Azeotrope; separation by conventional distillation is thermodynamically impossible because vapor and liquid compositions are identical ($y = x$).

---

3. Bubble Point, Dew Point, and Flash Distillation Algorithms

On the PE Chemical exam, equilibrium temperature and pressure determinations fall into two standard categories: bubble point and dew point calculations.

Bubble Point Determination (Liquid at Boiling Threshold)

A liquid mixture at known composition $\mathbf{x} = [x_1, x_2, \dots, x_C]$ begins to vaporize at its bubble point. The first bubble of vapor formed is in equilibrium with the liquid. Because the mole fractions of the nascent vapor must sum to unity:

$$\sum_{i=1}^C y_i = \sum_{i=1}^C K_i x_i = 1.0$$

• Bubble Point Pressure ($P_{bubble}$) at specified temperature $T$: $$P_{bubble} = \sum_{i=1}^C x_i \gamma_i P_i^{sat}(T)$$ For an ideal solution, $P_{bubble} = \sum x_i P_i^{sat}(T)$.
• Bubble Point Temperature ($T_{bubble}$) at specified pressure $P$: Requires numerical or trial-and-error iteration on $T$ until: $$f(T) = \sum_{i=1}^C \frac{x_i \gamma_i P_i^{sat}(T)}{P} - 1.0 = 0$$

Dew Point Determination (Vapor at Condensation Threshold)

A vapor mixture at known composition $\mathbf{y} = [y_1, y_2, \dots, y_C]$ begins to condense at its dew point. The first drop of liquid formed is in equilibrium with the vapor. Because the liquid mole fractions must sum to unity:

$$\sum_{i=1}^C x_i = \sum_{i=1}^C \frac{y_i}{K_i} = 1.0$$

• Dew Point Pressure ($P_{dew}$) at specified temperature $T$: $$P_{dew} = \frac{1}{\sum_{i=1}^C \frac{y_i}{\gamma_i P_i^{sat}(T)}}$$ For an ideal solution, $P_{dew} = \left[ \sum_{i=1}^C \frac{y_i}{P_i^{sat}(T)} \right]^{-1}$.
• Dew Point Temperature ($T_{dew}$) at specified pressure $P$: Requires iteration on $T$ until: $$g(T) = \sum_{i=1}^C \frac{y_i P}{\gamma_i P_i^{sat}(T)} - 1.0 = 0$$

Equilibrium Flash Distillation (Rachford-Rice Equation)

When a feed stream with overall composition $\mathbf{z} = [z_1, z_2, \dots, z_C]$ is flashed at temperature $T$ and pressure $P$ into liquid ($L$) and vapor ($V$) streams, the molar flash fraction is $\psi = V / F$ (where $L/F = 1 - \psi$). Overall and component material balances yield:

$$F z_i = V y_i + L x_i = V (K_i x_i) + (F - V) x_i$$ $$x_i = \frac{z_i}{1 + \psi (K_i - 1)}, \quad y_i = \frac{K_i z_i}{1 + \psi (K_i - 1)}$$

Because $\sum y_i - \sum x_i = 0$, the governing Rachford-Rice Equation is formulated:

$$f(\psi) = \sum_{i=1}^C \frac{z_i (K_i - 1)}{1 + \psi (K_i - 1)} = 0$$

The root $\psi$ is bounded strictly between $0$ (bubble point) and $1$ (dew point) and is solved monotonically via Newton-Raphson iteration.

---

4. Non-Ideal Phase Behavior and Azeotropes

When molecules exhibit strong intermolecular interactions (e.g., hydrogen bonding, dipole-dipole forces, or steric hindrance), liquid-phase activity coefficients deviate substantially from unity:

Thermodynamic Origin of Azeotropes

At an azeotrope, the equilibrium vapor composition is identical to the liquid composition:

$$y_i = x_i \iff K_i = 1.0 \iff \alpha_{12} = 1.0$$

According to the Gibbs-Konovalov theorem, at constant pressure an azeotrope corresponds to a stationary point (minimum or maximum) in the boiling temperature-composition ($T\text{-}x\text{-}y$) diagram. Fractional distillation cannot cross an azeotropic composition because the driving force ($y - x$) collapses to zero.

Methods to Overcome Azeotropic Limitations

1. Pressure-Swing Distillation: Exploits the sensitivity of azeotropic composition to system pressure. By operating two connected columns at different pressures ($P_1$ and $P_2$), the shift in azeotrope allows high-purity recovery without chemical additives (e.g., THF/water or acetonitrile/water).
2. Extractive Distillation: Introduces a heavy, high-boiling polar solvent (entrainer) that selectively interacts with one component via hydrogen bonding, altering its activity coefficient and increasing relative volatility without forming a new azeotrope.
3. Heterogeneous Azeotropic Distillation: Adds an entrainer (e.g., cyclohexane or benzene added to ethanol-water) that forms a minimum-boiling ternary heteroazeotrope that separates into two liquid phases upon condensation in a decanter.

---

5. Summary Comparison: Ideal vs. Non-Ideal Distillation Systems

---

6. Comprehensive Worked Numerical Example: Benzene-Toluene VLE & Bubble Point Sizing

Problem Statement

A chemical process feed consists of an equimolar liquid mixture of benzene (1) and toluene (2) with $x_1 = 0.40$ and $x_2 = 0.60$. The mixture is fed to a continuous distillation column operating at a top pressure of $P = 101.325\text{ kPa}$ ($1.00\text{ atm} = 760.0\text{ mmHg}$).

Assume the liquid and vapor phases form an ideal system obeying Raoult's Law. The saturation vapor pressures (in $\text{kPa}$) are modeled by the Antoine equation:

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

Antoine Parameters:

• Benzene (1): $A_1 = 13.7819$, $B_1 = 2,726.81$, $C_1 = 217.57$
• Toluene (2): $A_2 = 13.9320$, $B_2 = 3,056.96$, $C_2 = 217.65$

Calculate:

1. The saturation vapor pressures of pure benzene and pure toluene at $T = 95.0^\circ\text{C}$.
2. The relative volatility $\alpha_{12}$ at $95.0^\circ\text{C}$.
3. The bubble point pressure of the liquid mixture at $95.0^\circ\text{C}$, and verify whether $95.0^\circ\text{C}$ is close to the true bubble point temperature at $101.325\text{ kPa}$.
4. The equilibrium vapor mole fraction of benzene ($y_1$) using Raoult's law and verify using the relative volatility formula.

---

Step 1: Saturation Vapor Pressures at $95.0^\circ\text{C}$

For Benzene (1) at $T = 95.0^\circ\text{C}$:

$$T + C_1 = 95.0 + 217.57 = 312.57$$ $$\frac{B_1}{T + C_1} = \frac{2,726.81}{312.57} = 8.72384$$ $$\ln P_1^{sat} = 13.7819 - 8.72384 = 5.05806$$ $$P_1^{sat} = \exp(5.05806) = \mathbf{157.28\text{ kPa}}$$

For Toluene (2) at $T = 95.0^\circ\text{C}$:

$$T + C_2 = 95.0 + 217.65 = 312.65$$ $$\frac{B_2}{T + C_2} = \frac{3,056.96}{312.65} = 9.77758$$ $$\ln P_2^{sat} = 13.9320 - 9.77758 = 4.15442$$ $$P_2^{sat} = \exp(4.15442) = \mathbf{63.72\text{ kPa}}$$

---

Step 2: Relative Volatility ($\alpha_{12}$)

Because the system is ideal:

$$\alpha_{12} = \frac{P_1^{sat}}{P_2^{sat}} = \frac{157.28\text{ kPa}}{63.72\text{ kPa}} = \mathbf{2.468}$$

---

Step 3: Bubble Point Pressure Verification

At $T = 95.0^\circ\text{C}$ and $x_1 = 0.40$, $x_2 = 0.60$:

$$P_{bubble} = x_1 P_1^{sat} + x_2 P_2^{sat}$$ $$P_{bubble} = (0.40 \times 157.28\text{ kPa}) + (0.60 \times 63.72\text{ kPa})$$ $$P_{bubble} = 62.91\text{ kPa} + 38.23\text{ kPa} = \mathbf{101.14\text{ kPa}}$$

Because $P_{bubble} = 101.14\text{ kPa} \approx 101.325\text{ kPa}$ ($0.18\%$ difference), the true bubble point temperature of this mixture at atmospheric pressure is $T_{bubble} = \mathbf{95.1^\circ\text{C}}$.

---

Step 4: Equilibrium Vapor Composition

Using Raoult's law:

$$y_1 = \frac{x_1 P_1^{sat}}{P} = \frac{0.40 \times 157.28\text{ kPa}}{101.14\text{ kPa}} = \frac{62.912}{101.14} = \mathbf{0.6220} \quad (62.20\text{ mol}\%)$$

Verifying with the relative volatility equation:

$$y_1 = \frac{\alpha_{12} x_1}{1 + (\alpha_{12} - 1) x_1} = \frac{2.468 \times 0.40}{1 + (2.468 - 1) \times 0.40} = \frac{0.9872}{1 + (1.468 \times 0.40)} = \frac{0.9872}{1 + 0.5872} = \frac{0.9872}{1.5872} = \mathbf{0.6220}$$

Both analytical routes produce identical results, confirming the thermodynamic consistency of the equilibrium curve formulation.

---

7. Critical PE Exam Traps & Pitfalls

Trap 1: Assuming Relative Volatility Is Constant Throughout the Column
In an actual column, temperature varies from the reboiler (hot, bottoms) to the condenser (cool, overhead). Because saturation pressures follow the non-linear Clausius-Clapeyron relation, relative volatility typically varies across trays. When using shortcut equations (Fenske), engineers must use the geometric mean relative volatility: $\alpha_{avg} = \sqrt{\alpha_{top} \cdot \alpha_{bottom}}$, rather than evaluating $\alpha$ only at feed conditions.

Trap 2: Inverting Bubble Point vs. Dew Point Objective Functions
A common exam error is writing the bubble point condition as $\sum x_i / K_i = 1$ instead of $\sum K_i x_i = 1$. Remember: Bubble point tests whether a liquid will boil; you know liquid mole fractions $x_i$, and multiplying by $K_i$ generates the prospective vapor mole fractions $y_i$. Dew point tests whether a vapor will condense; you know $y_i$, and dividing by $K_i$ produces prospective liquid mole fractions $x_i$.

Trap 3: Neglecting Activity Coefficients in Polar-Hydrocarbon Mixtures
If an exam question mentions ethanol-water, acetone-water, or alcohol-hydrocarbon systems, never use pure saturation pressures alone! Assuming Raoult's law for polar systems underestimates required stages by $200\text{--}400\%$ and misses azeotropic boundaries entirely.