Combined Raoult-Dalton VLE Calculation

Reference ID: MET-F3F5 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The Combined Raoult‑Dalton Vapor‑Liquid Equilibrium (VLE) calculation is a fundamental analytical tool in process engineering used to predict the distribution of chemical species between a liquid phase and its equilibrium vapor phase, and it is essential for the design and simulation of separation processes such as distillation columns, flash drums, and condensers, including the development of an effective batch distillation product collection strategy.

In practice, this model is applied to ideal or near‑ideal binary and multicomponent mixtures where the liquid phase behaves as an ideal solution (obeying Raoult's Law) and the vapor phase behaves as an ideal gas (obeying Dalton's Law); for systems that deviate from ideality, see our guide on activity coefficient calculation for non‑ideal solutions. It serves as the baseline for determining bubble points, dew points, and the composition of vapor streams exiting equilibrium stages.

Methodology & Formulas

The calculation relies on the determination of pure component saturation pressures followed by the application of partial pressure summation. The step-by-step physics are defined as follows:

1. Saturation Pressure Calculation: The saturation pressure \( p_{i}^{\circ} \) for each component at a given temperature \( T \) is determined using the Antoine Equation:

\[ \log_{10}(p_{i}^{\circ}) = A_{i} - \frac{B_{i}}{C_{i} + T} \]

Where \( A_{i} \), \( B_{i} \), and \( C_{i} \) are component-specific empirical constants. The temperature \( T \) must be within the valid range of the Antoine coefficients, typically in degrees Celsius.

2. Partial Pressure Determination: Based on Raoult's Law calculation for ideal solutions, the partial pressure \(P_{i}\) of each component in the vapor phase is the product of its liquid mole fraction \(x_{i}\) and its saturation pressure.

\[ P_{i} = x_{i} \cdot p_{i}^{\circ} \]

3. Vapor Phase Composition: According to Dalton's Law, the vapor mole fraction \( y_{i} \) is the ratio of the component partial pressure to the total system pressure \( P_{\text{total}} \):

\[ y_{i} = \frac{P_{i}}{P_{\text{total}}} \]

4. Equilibrium Constraint: The system is at equilibrium when the sum of the partial pressures equals the total system pressure:

\[ \sum P_{i} = P_{\text{total}} \]

Parameter	Constraint / Regime
Pressure Limit	\( P_{\text{total}} \leq 5 \text{ atm} \) (Above this, gas phase non-ideality requires fugacity coefficients)
Composition Limit	\( \sum x_{i} = 1.0 \) (Mole fractions must be normalized)
Temperature Range	Must be within the empirical validity range of the Antoine constants for the specific components
Equilibrium State	If \( \sum P_{i} \neq P_{\text{total}} \), the system is not at the bubble point for the specified temperature

To calculate the bubble point pressure for a multicomponent system, follow these steps:

Assume a system temperature and define the liquid phase mole fractions (\(x_i\)).
Retrieve the saturation pressure (\(p_i^{\circ}\)) for each component at the given temperature using the Antoine equation.
Calculate the total pressure (\(P\)) by summing the partial pressures: \(P = \sum (x_i \cdot p_i^{\circ})\).
Determine the vapor phase mole fractions from \(y_i = x_i \cdot p_i^{\circ} / P\). The sum of \(y_i\) will automatically equal unity, confirming consistency.

The Combined Raoult-Dalton law assumes that the vapor phase behaves as an ideal gas and the liquid phase forms an ideal solution. This approach is generally valid under the following conditions:

Low to moderate system pressures, typically below 5 atm.
Components that are chemically similar, such as homologous series of hydrocarbons.
Systems operating far from the critical point of the mixture.

While computationally efficient, the Combined Raoult-Dalton method has significant limitations for rigorous process engineering:

It fails to account for molecular interactions in non-ideal liquid mixtures, requiring activity coefficient models like NRTL or UNIQUAC for better accuracy.
It ignores vapor phase non-ideality, which becomes significant at high pressures where fugacity coefficients must be introduced.
It cannot accurately predict azeotropic behavior in polar or dissimilar chemical systems.

In the context of this model, the equilibrium ratio or K-value for a component is defined as the ratio of the vapor mole fraction to the liquid mole fraction (\(K_i = y_i / x_i\)). Under the Raoult-Dalton assumption, the K-value is simplified to:

\(K_i = p_i^{\circ} / P_{\text{total}}\)
This relationship demonstrates that the distribution of a component between phases is solely a function of the component saturation pressure and the total system pressure.

Worked Example: Combined Raoult-Dalton VLE for an Ideal Binary System (Bubble Point Calculation)

Consider a binary mixture of benzene (A) and toluene (B) at a temperature of \(T = 90.0 \,^\circ\text{C}\) and liquid-phase mole fractions \(x_A = 0.5\) and \(x_B = 0.5\). The objective is to determine the bubble point pressure and the equilibrium vapor-phase mole fractions \(y_A\) and \(y_B\) using the combined Raoult-Dalton relation.

Knowns:

Temperature, \(T = 90.0 \,^\circ\text{C}\)
Liquid mole fraction of benzene, \(x_A = 0.5\)
Liquid mole fraction of toluene, \(x_B = 0.5\)
Saturation pressure of benzene at \(90.0 \,^\circ\text{C}\), \(p_A^\circ = 1.343 \, \text{atm}\)
Saturation pressure of toluene at \(90.0 \,^\circ\text{C}\), \(p_B^\circ = 0.538 \, \text{atm}\)
Unknown: Bubble point pressure \(P_{\text{bubble}}\) and vapor mole fractions

Step-by-Step Calculation:

Compute pure-component saturation pressures.
The saturation pressures at \(T = 90.0 \,^\circ\text{C}\) are provided as:

\[ p_A^\circ = 1.343 \, \text{atm}, \quad p_B^\circ = 0.538 \, \text{atm} \]
Calculate partial pressures using Raoult's law.
The vapor-phase partial pressures of each component are:

\[ P_A = x_A \cdot p_A^\circ = 0.5 \times 1.343 = 0.672 \, \text{atm} \]

\[ P_B = x_B \cdot p_B^\circ = 0.5 \times 0.538 = 0.269 \, \text{atm} \]

(Values shown to three decimal places.)
Determine the bubble point pressure.
At the bubble point, the total pressure equals the sum of the partial pressures:

\[ P_{\text{bubble}} = P_A + P_B = 0.672 + 0.269 = 0.941 \, \text{atm} \]

This is the equilibrium total pressure for the given liquid composition and temperature.
Compute vapor mole fractions using Dalton's law.
Using the bubble point pressure, the vapor-phase mole fractions are:

\[ y_A = \frac{P_A}{P_{\text{bubble}}} = \frac{0.672}{0.941} = 0.714 \]

\[ y_B = \frac{P_B}{P_{\text{bubble}}} = \frac{0.269}{0.941} = 0.286 \]

(Values rounded to three decimal places.)
Verify consistency.
The sum of vapor mole fractions is:

\[ y_A + y_B = 0.714 + 0.286 = 1.000 \]

This confirms that the bubble point condition is satisfied and the Raoult-Dalton relation is self-consistent.

Final Answer:

The bubble point pressure is \(P_{\text{bubble}} = 0.941 \, \text{atm}\). The equilibrium vapor-phase mole fractions are \(y_A = 0.714\) and \(y_B = 0.286\).

"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

Combined Raoult-Dalton VLE Calculation

Introduction & Context

Methodology & Formulas

Worked Example: Combined Raoult-Dalton VLE for an Ideal Binary System (Bubble Point Calculation)

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