Relative Volatility Calculation

Reference ID: MET-FC19 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Relative volatility, denoted as α, is a fundamental parameter in process engineering used to quantify the ease of separating a binary mixture via distillation. It represents the ratio of the distribution coefficients of two components between the vapor and liquid phases. In distillation column design, α dictates the minimum number of theoretical stages and the minimum reflux ratio required to achieve a target separation purity. A value of α close to 1.0 indicates a difficult separation, while higher values signify that the components have significantly different boiling points, making separation more energy-efficient.

Methodology & Formulas

The calculation of relative volatility relies on the vapor-liquid equilibrium (VLE) of the system. For a binary mixture consisting of a more volatile component (A) and a less volatile component (B), the relative volatility is defined by the ratio of their respective vapor-liquid distribution ratios.

The vapor pressure of each pure component at a given temperature T is determined using the Antoine equation:

\[ \log_{10}(p_i^\circ) = A_i - \frac{B_i}{T + C_i} \]

Where p_i^\circ is the saturation pressure of component i. Once the saturation pressures are obtained, the relative volatility α_AB is calculated by incorporating the liquid-phase activity coefficients (γ) to account for non-ideality:

\[ \alpha_{AB} = \frac{\gamma_A \cdot p_A^\circ}{\gamma_B \cdot p_B^\circ} \]

In systems where the liquid phase behaves ideally, the activity coefficients are assumed to be unity (γ_A = γ_B = 1), simplifying the expression to the ratio of pure-component vapor pressures.

Regime / Condition	Criteria	Engineering Implication
Ideal Mixture	γ_A ≈ γ_B ≈ 1	α is primarily a function of temperature; separation is governed by boiling point differences.
Non-Ideal Mixture	γ_i ≠ 1	Requires activity coefficient models (e.g., NRTL, UNIQUAC) to accurately predict α.
Azeotropic Point	α_AB = 1	Distillation is ineffective; y_i = x_i, preventing further enrichment of the distillate.
Operational Limit	T_min ≤ T ≤ T_max	Antoine constants are only valid within specific temperature ranges; extrapolation is prohibited.

Relative volatility, denoted as α, represents the ratio of the vapor-liquid distribution coefficients of two components. It serves as a measure of the ease of separation for a given mixture. The calculation follows these principles:

It is the ratio of the volatility of the more volatile component to the volatility of the less volatile component.
For an ideal system, it is calculated as the ratio of the vapor pressures of the two pure components at the system temperature.
A value of 1.0 indicates that the components cannot be separated by distillation.

Pressure significantly influences the vapor-liquid equilibrium of a system. As a process engineer, you should note the following:

Increasing system pressure generally decreases the relative volatility of most hydrocarbon mixtures.
Lowering the pressure typically enhances the separation factor, potentially reducing the required reflux ratio or the number of theoretical stages.
You must evaluate the trade-off between improved separation efficiency at low pressures and the increased equipment size required for vacuum operations.

The assumption of constant relative volatility is a common simplification used in the Fenske equation for preliminary design. You may apply this assumption under these conditions:

The mixture behaves as an ideal solution following Raoult's Law.
The operating temperature range across the column is relatively narrow.
The components are chemically similar, such as in close-boiling hydrocarbon isomers.
For non-ideal systems, you should use an average α value calculated from the top and bottom compositions of the column.

Worked Example: Relative Volatility of Ethanol–Water Mixture at 60°C

Scenario: A binary liquid mixture of ethanol (A) and water (B) is present in a distillation column at an equilibrium stage operating at 1.013 bar. At a liquid-phase mole fraction of ethanol of 0.5, the stage temperature is measured as 60.0°C. Activity coefficients at this composition are known from laboratory data: γ_A = 1.5 and γ_B = 1.05. The relative volatility is to be determined to assess the ease of separation.

Known Parameters

Temperature, T = 60.0 °C
Total pressure, P = 1.013 bar
Liquid mole fraction of ethanol, x_A = 0.5
Activity coefficient of ethanol, γ_A = 1.5
Activity coefficient of water, γ_B = 1.05
Antoine constants (ethanol, valid –4 to 62°C): A_A = 8.04494, B_A = 1554.3, C_A = 222.65 (units: °C, mmHg)
Antoine constants (water, valid 0 to 100°C): A_B = 8.07131, B_B = 1730.63, C_B = 233.426 (units: °C, mmHg)
Conversion factor: 1 mmHg = 1/750.06 bar so that 1 mmHg = 0.00133323 bar

Calculation Steps

Compute vapor pressures using Antoine equation. The Antoine equation is log₁₀ p_i^sat (mmHg) = A_i - B_i / (T + C_i). Substituting the values yields:
\[ \log_{10} p_{A}^{\text{sat}} = 8.04494 - \frac{1554.3}{60.0 + 222.65} = 2.54594 \]

\[ \log_{10} p_{B}^{\text{sat}} = 8.07131 - \frac{1730.63}{60.0 + 233.426} = 2.17362 \]
Convert to bar. Multiply the antilog by the conversion factor:
\[ p_{A}^{\text{sat}} = (10^{2.54594}) \times 0.00133323 = 0.4691 \text{ bar} \]

\[ p_{B}^{\text{sat}} = (10^{2.17362}) \times 0.00133323 = 0.1987 \text{ bar} \]
After rounding to three decimal places, p_A^sat = 0.469 bar and p_B^sat = 0.199 bar.
Apply the relative volatility formula for a non-ideal liquid phase. The governing equation is:
\[ \alpha_{AB} = \frac{\gamma_{A} \, p_{A}^{\text{sat}}}{\gamma_{B} \, p_{B}^{\text{sat}}} \]
Insert the known rounded values:
\[ \alpha_{AB} = \frac{1.5 \times 0.469}{1.05 \times 0.199} = \frac{0.7035}{0.20895} = 3.366 \]
Rounded to three significant figures, α_AB = 3.37.
Validity check. The temperature (60.0°C) lies within the recommended Antoine range for both components (ethanol: –4 to 62°C; water: 0 to 100°C). The mole fraction (0.5) is between 0 and 1. Both vapor pressures are positive. Since α_AB > 1, the mixture is not at an azeotrope, and the separation factor is sufficient for distillation.

Final Answer

The relative volatility of ethanol to water under these conditions is α_AB = 3.37. This value indicates that ethanol is approximately 3.37 times more volatile than water, implying a moderately easy separation with a modest number of theoretical stages.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle