Problem 8 - Chemistry in 30 days

Question

The total vapour pressure of a solution of components \(A\) and \(B\) is 600 torr. The mole fraction of component \(A\) in liquid and vapour phase are \(0.70\) and \(0.35\), respectively. The vapour pressures of pure \(A\) and \(B\) are (a) 300 torr, 130 torr (b) 1300 torr, 300 torr (d) 300 torr, 300 torr (c) 300 torr, 1300 torr

Step-by-Step Solution

Answer

The vapor pressures of pure components are 300 torr for A and 1300 torr for B.

Step 1: Understanding Given Information

We are given the total vapor pressure of the solution as 600 torr. The mole fraction of component \(A\) in the liquid phase \(X_A = 0.70\) and in the vapor phase \(Y_A = 0.35\). We need to determine the vapor pressures of the pure components \(A\) and \(B\).

Step 2: Raoult’s Law for Total Vapor Pressure

According to Raoult's Law, the total vapor pressure, \(P_{total}\), of a solution containing two components can be calculated using the formula: \[ P_{total} = P_A^0 \cdot X_A + P_B^0 \cdot X_B \]where \(P_A^0\) and \(P_B^0\) are the vapor pressures of the pure components \(A\) and \(B\), respectively, and \(X_A\) and \(X_B\) are their mole fractions in the liquid phase. Given \(P_{total} = 600\) torr, we'll use this relationship.

Step 3: Applying Dalton's Law for Vapor Composition

Dalton's Law of Partial Pressures relates the mole fraction in the vapor to the vapor pressure of the pure components:\[ Y_A = \frac{P_A}{P_{total}} = \frac{P_A^0 \cdot X_A}{P_{total}} \]Given \(Y_A = 0.35\), we substitute to find the relationship between \(P_A^0\), \(X_A\), and \(P_{total}\):\[ 0.35 = \frac{P_A^0 \cdot 0.70}{600} \]

Step 4: Calculating Pure Component Vapor Pressure \(P_A^0\)

Solve for the vapor pressure of component \(A\):\[ P_A^0 = \frac{0.35 \times 600}{0.70} \]\[ P_A^0 = 300 \text{ torr} \]

Step 5: Determine Vapor Pressure of Component \(B\) \(P_B^0\)

Substitute \(P_A^0\) into the Raoult's Law equation to find \(P_B^0\):\[ 600 = 300 \cdot 0.70 + P_B^0 \cdot (1 - 0.70) \]\[ 600 = 210 + P_B^0 \cdot 0.30 \]\[ P_B^0 \cdot 0.30 = 390 \]\[ P_B^0 = \frac{390}{0.30} \]\[ P_B^0 = 1300 \text{ torr} \]

Step 6: Conclusion

The vapor pressures of pure components are \(P_A^0 = 300\) torr and \(P_B^0 = 1300\) torr. Hence, the correct option is (c) 300 torr, 1300 torr.

Key Concepts

Dalton's Law of Partial Pressures

Dalton’s Law of Partial Pressures is a fundamental principle in the study of gases. It states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. Each gas in the mixture exerts a pressure as if it alone occupied the entire volume. This individual pressure is called the partial pressure of the gas.

For a solution in a vapor phase, Dalton’s Law can be applied to determine the composition of the vapor. In a mixture that includes components like in the exercise, this law is essential for relating the mole fractions and the vapor pressures of each component.

In practice, Dalton’s Law is expressed mathematically as:

• \[ P_{total} = P_A + P_B + \ldots + P_n \]

Here, \(P_{A}, P_{B}\), etc., denote the partial pressures of the individual gases present in the vapor phase.

Applying Dalton's Law to a vapor-liquid equilibrium allows us to derive important relationships that make it possible to calculate the distribution of components between phases using their known properties.

vapor pressure

Vapor pressure is a critical concept in understanding the behavior of liquids and gases in equilibrium. It refers to the pressure exerted by the vapor in equilibrium with its liquid at a given temperature. Every liquid has a unique vapor pressure based on its specific properties and temperature.

In the case of pure components, the vapor pressure is constant at a given temperature. When you mix two liquids, like components \(A\) and \(B\) from the exercise, each component contributes to the overall vapor pressure of the solution based on its proportion and its own pure form vapor pressure.

One way to calculate vapor pressure is by using Raoult's Law, which ties the mole fraction of each component in a solution to its vapor pressure:

• \[ P_{total} = P_A^0 \times X_A + P_B^0 \times X_B \]

Here, \(P_{A}^0\) and \(P_{B}^0\) are the vapor pressures of the pure components; \(X_A\) and \(X_B\) are their mole fractions in the liquid phase. Understanding vapor pressure is vital as it determines how components in a solution will behave under various conditions.

mole fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a specific component to the total number of moles of all components in the mixture.

Mathematically, the mole fraction \(X\) of a component \(A\) can be represented as:

• \[ X_A = \frac{n_A}{n_A + n_B} \]

Where \(n_A\) is the number of moles of component \(A\) and \(n_B\) is the number of moles of component \(B\). Mole fraction is a unitless quantity and is especially useful when dealing with solutions and mixtures, as it provides a straightforward way to express the component concentrations.

In the context of vapor-liquid equilibrium, the mole fraction serves to describe both the composition of the liquid and vapor phases, allowing for deeper insights into how the mixture behaves under different conditions.

vapor-liquid equilibrium

Vapor-liquid equilibrium (VLE) is a condition where the liquid and vapor phases of a substance, or combination of substances, are in balance. This means the rate at which liquid molecules evaporate equals the rate at which vapor molecules condense. When VLE is achieved, the system is stable and no further net evaporation or condensation occurs.

The study of VLE is critical in chemical engineering and thermodynamics as it determines how components distribute themselves between the liquid and vapor phases. This equilibrium is described by the diagram of pressure versus composition, often used to optimize separation processes such as distillation.

Key parameters in VLE include:

• Consistency of temperature
• Compositions in both phases that do not change over time
• The equality of chemical potentials in vapor and liquid phases for each component

Achieving VLE involves satisfying certain conditions where the chemical potential, which is a measure of chemical energy, is the same in both phases for a particular component. Understanding VLE is crucial for manipulating and predicting the behavior of chemical substances in various industrial processes.