Vapor-Liquid Equilibrium: Principles, Laws, and Calculations

Vapor-Liquid Equilibrium (VLE)

Introduction to Vapor-Liquid Equilibrium

Vapor-liquid equilibrium (VLE) describes the state in which the rate of vaporization of a liquid equals the rate of condensation of its vapor. This concept is fundamental in chemical engineering and physical chemistry, especially for the design and analysis of separation processes such as distillation.

• Equilibrium: Occurs when the rate of vaporization equals the rate of condensation.

• Vapor Pressure (Po): The pressure exerted by the vapor when the liquid and vapor phases of a pure component are in equilibrium.

• Applications: Used in the calculation of boiling points, distillation, and other phase equilibrium processes.

Evaluating Vapor Pressure (Po)

Vapor pressure can be determined using tabulated values or empirical equations. Two common methods are the Antoine equation and the DIPPR equation.

• Antoine Equation: Used to estimate vapor pressure over a limited temperature range.

• DIPPR Equation: Provides a more general approach for a wide range of substances.

Antoine Equation:

• $\log_{10} P^{\text{sat}} = A - \frac{B}{T + C}$

• Where $P^{\text{sat}}$ is the vapor pressure, $T$ is temperature (°C), and $A$, $B$, $C$ are substance-specific constants.

DIPPR Equation:

• $\ln P^{\text{sat}} = C_1 + \frac{C_2}{T} + C_3 \ln T + C_4 T^{C_5}$

• Where $P^{\text{sat}}$ is the vapor pressure, $T$ is temperature (K), and $C_1$ to $C_5$ are constants for each substance.

Boiling Point Calculations

The boiling point of a liquid is the temperature at which its vapor pressure equals the external pressure. DIPPR equation tables are commonly used to determine boiling points under various pressures.

• Boiling Point: The temperature at which $P^{\text{sat}} = P_{\text{external}}$.

• Calculation: Use vapor pressure equations to solve for temperature when $P^{\text{sat}}$ equals the given pressure.

Raoult's Law

Definition and Application

Raoult's Law relates the partial pressure of a component in a mixture to its mole fraction in the liquid phase and the vapor pressure of the pure component.

• Raoult's Law: $P_i = x_i P_i^o$

• Where $P_i$ is the partial pressure of component $i$, $x_i$ is its mole fraction in the liquid phase, and $P_i^o$ is the vapor pressure of pure $i$ at temperature $T$.

• Ideal Solutions: Raoult's Law applies best to ideal solutions where intermolecular forces are similar.

Vapor-Liquid Equilibrium in Multi-Component Systems

Dalton's Law of Partial Pressures

Dalton's Law states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual components.

• Dalton's Law: $P = P_A + P_B + \cdots + P_N$

• Where $P$ is the total pressure, and $P_A$, $P_B$, ..., $P_N$ are the partial pressures of each component.

Combining Raoult's and Dalton's Laws

• Combined Law: $P = x_A P_A^o + x_B P_B^o + \cdots + x_N P_N^o$

• Used to calculate the total pressure in a multi-component system at equilibrium.

Vapor Phase Composition

• Vapor Mole Fraction: $y_A = \frac{P_A}{P}$

• Where $y_A$ is the mole fraction of component $A$ in the vapor phase.

Sample Problems and Applications

Sample Problem 7

• Task: Calculate the vapor pressure of benzene at 42.5°C using the Antoine equation and tabulated constants.

Sample Problem 8

• Task: Calculate the vapor pressure of benzene at 42.5°C using the DIPPR equation and provided constants.

Sample Problem 9

• Task: Determine the temperature at which water will begin to boil if the total pressure is 125 kPa.

Sample Problem 10

• Task: Find the temperature at which water will begin to condense in a gas mixture at 755 mmHg, given 9.48 mol% water in the gas phase.

Sample Problem 11

• Task: For a mixture of 40% n-pentane and 60% n-hexane by mole, heated to 100°C and 3.5 atm, calculate the equilibrium composition of vapor and liquid phases and the percentage of liquid vaporized.

Sample Problem 12

• Task: For a solution with equimolar n-pentane, cyclopentane, and benzene at 60°C, calculate:

  • The total pressure of the gaseous phase in atm

  • The mass % composition of the vapor mixture in equilibrium

Representative Table: DIPPR Constants for Vapor Pressure Calculation

The following table provides DIPPR constants for selected substances, used in vapor pressure calculations:

Substance	MW	C1	C2	C3	C4	C5
CO2	44	140.54	-4736.0	-21.268	4.9096 × 10-2	1
Argon	40	42.157	-1093.4	-4.1425	4.7524 × 10-2	1
Nitrogen	28	19.241	-808.3	-1.8144	4.4175 × 10-2	1
n-pentane	72	14.741	-5.0203	-3.8443	9.6013 × 10-3	1
Cyclopentane	70	51.844	-4.5174	-3.1155	1.1125 × 10-2	1
Benzene	78	13.918	-4.5174	-3.1155	7.1125 × 10-3	1
Helium	4	8.031	-2.8972	-0.6474	2.7430 × 10-2	1
Ethyl acetate	88	16.661	-7.021	-4.1943	1.1663 × 10-2	1
Water	18	22.589	-49.309	-4.5758	4.5758 × 10-2	1
Air	29	21.629	-49.309	-4.5758	4.5758 × 10-2	1

Additional info:

• These notes are based on chemical engineering lecture slides and are relevant for students studying phase equilibria, especially in the context of organic and physical chemistry.

• Sample problems illustrate practical applications of VLE concepts in real-world chemical engineering scenarios.