Introduction & Context

Raoult's Law is a fundamental principle in chemical and process engineering used to describe the phase equilibrium of ideal liquid mixtures. It establishes a linear relationship between the partial pressure of a component in the vapor phase and its mole fraction in the liquid phase. This calculation is essential for designing separation processes such as distillation columns, flash drums, and condensers. By predicting the bubble point and vapor composition of a mixture, engineers can determine the efficiency of mass transfer operations and ensure that process conditions remain within safe operating limits, while a complementary perspective on how partial pressures combine in the vapor phase can be found in Dalton's Law for vapor phase composition.

Methodology & Formulas

The calculation follows a systematic approach to determine the vapor‑liquid equilibrium (VLE) for a binary system; after obtaining pure‑component saturation pressures with the Antoine equation, you apply both Raoult's Law and Dalton's Law to compute the total system pressure and vapor‑phase composition, as detailed in the combined Raoult‑Dalton VLE calculation guide.

The saturation pressure for each component i is calculated using the Antoine equation:

\[ P_{\text{sat},i} = 10^{A_{i} - \frac{B_{i}}{T + C_{i}}} \cdot k_{\text{conv}} \]

Where k_conv represents the conversion factor from the pressure units of the Antoine constants (e.g., mmHg) to the desired engineering units (e.g., bar).

The partial pressure p_i of each component is determined by the liquid mole fraction x_i and the saturation pressure:

\[ p_{i} = x_{i} \cdot P_{\text{sat},i} \]

The total system pressure P_total is the sum of the partial pressures:

\[ P_{\text{total}} = \sum p_{i} = p_{1} + p_{2} \]

Finally, the vapor mole fraction y_i is calculated using Dalton's Law:

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

Condition/Parameter	Criteria/Regime
Solution Ideality	Valid only for ideal solutions where the activity coefficient γ_i = 1.
Mole Fraction Range	\(0 \leq x_{i} \leq 1\)
Pressure Regime	Valid for low to moderate pressures (typically \(P < 10\) bar) where vapor behaves as an ideal gas.
Temperature Range	Must be within the empirical range defined by the specific Antoine constants used.
Consistency Check	\(\sum y_{i} = 1.0\) must hold true for a valid equilibrium state.

To calculate the partial pressure of a component in an ideal liquid mixture, apply the following steps:

Identify the mole fraction of the component in the liquid phase, denoted as x_i.
Determine the vapor pressure of the pure component at the system temperature, denoted as P_sat,i.
Multiply the mole fraction by the pure component vapor pressure: \(p_{i} = x_{i} \cdot P_{\text{sat},i}\).

For Raoult's Law to provide an accurate representation of a system, the following conditions must be met:

The solution must behave as an ideal solution where intermolecular forces between unlike molecules are similar to those between like molecules.
The vapor phase must behave as an ideal gas.
The system must be at equilibrium, meaning the chemical potential of each component is equal in both the liquid and vapor phases.

The total pressure of the system is determined by summing the partial pressures of all individual components present in the mixture. The calculation follows these steps:

Calculate the partial pressure for each component i using \(p_{i} = x_{i} \cdot P_{\text{sat},i}\).
Sum all individual partial pressures to find the total pressure: \(P_{\text{total}} = \sum_{i} \left( x_{i} \cdot P_{\text{sat},i} \right)\).
Ensure that the sum of all mole fractions x_i equals 1.0 for the calculation to be valid.

Raoult's Law Worked Example: Ideal Ethanol-Water System

Consider a binary liquid mixture of ethanol (1) and water (2) at 78.0°C. The mixture is assumed to form an ideal solution, so liquid-phase activity coefficients are unity. The liquid contains 40.0 mol% ethanol. The pure-component saturation pressures at 78.0°C have been precomputed using the Antoine equation with constants supplied (in mmHg and then converted to bar). We apply Raoult's law to determine the bubble-point total pressure and the equilibrium vapor composition.

Knowns

Temperature: \(T = 78.0\,^{\circ}\text{C}\)
Liquid mole fraction of ethanol: \(x_{\text{ethanol}} = 0.4\)
Liquid mole fraction of water: \(x_{\text{water}} = 0.6\)
Saturation pressure of ethanol at 78.0°C: \(P_{\text{sat,ethanol}} = 1.0\,\text{bar}\)
Saturation pressure of water at 78.0°C: \(P_{\text{sat,water}} = 0.436\,\text{bar}\)

Step-by-Step Calculation

Compute partial pressures via Raoult's law:
\[ p_{\text{ethanol}} = x_{\text{ethanol}} \cdot P_{\text{sat,ethanol}} = 0.4 \cdot 1.0 = 0.4\,\text{bar} \]

\[ p_{\text{water}} = x_{\text{water}} \cdot P_{\text{sat,water}} = 0.6 \cdot 0.436 = 0.2616\,\text{bar} \]
Determine total pressure:
\[ P_{\text{total}} = p_{\text{ethanol}} + p_{\text{water}} = 0.4 + 0.2616 = 0.6616\,\text{bar} \]
Calculate vapor-phase mole fractions:
\[ y_{\text{ethanol}} = \frac{p_{\text{ethanol}}}{P_{\text{total}}} = \frac{0.4}{0.6616} \approx 0.6046 \quad (\text{rounding to } 0.605) \]

\[ y_{\text{water}} = \frac{p_{\text{water}}}{P_{\text{total}}} = \frac{0.2616}{0.6616} \approx 0.3954 \quad (\text{rounding to } 0.395) \]

Final Answer

For the given liquid composition and temperature under the ideal-solution assumption:

Bubble-point total pressure: \(P_{\text{total}} = 0.6616\,\text{bar} \; (\approx 0.662\,\text{bar})\)
Vapor mole fraction of ethanol: \(y_{\text{ethanol}} \approx 0.605\)
Vapor mole fraction of water: \(y_{\text{water}} \approx 0.395\)

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