Problem 162 - A Complete Resource Book in Chemistry for JEE Main

Question

A mixture of ethyl alcohol and propyl alcohol has a vapour pressure of \(290 \mathrm{~mm}\) at \(300 \mathrm{~K}\). The vapour pressure of propyl alcohol is \(200 \mathrm{~mm}\). If the mole fraction of ethyl alcohol is \(0.6\), its vapour pressure (in \(\mathrm{mm}\) ) at the same temperature will be (a) 300 (b) 700 (c) 360 (c) 350

Step-by-Step Solution

Answer

The vapor pressure of ethyl alcohol is 350 mm.

Step 1: Understand the Problem

We're given a mixture of ethyl alcohol and propyl alcohol with a total vapor pressure of \(290\, \text{mm}\) at \(300 \, \text{K}\). The vapor pressure of propyl alcohol alone is \(200\, \text{mm}\). The mole fraction of ethyl alcohol is given as \(0.6\). We need to determine the vapor pressure of ethyl alcohol at the same temperature.

Step 2: Apply Raoult's Law

According to Raoult’s Law, the total vapor pressure of the mixture is the sum of the partial pressures of each component in the solution. The equation can be expressed as: \[ P = P_Ax_A + P_Bx_B \]Where \(P\) is the total vapor pressure, \(P_A\) and \(P_B\) are the vapor pressures of pure ethyl alcohol and propyl alcohol respectively, and \(x_A\) and \(x_B\) are their mole fractions.

Step 3: Calculate the Mole Fraction of Propyl Alcohol

The mole fraction of ethyl alcohol is \(0.6\), hence, the mole fraction of propyl alcohol \(x_B\) is:\[ x_B = 1 - x_A = 1 - 0.6 = 0.4 \]

Step 4: Set Up the Equation

Substitute the known values into Raoult’s Law:\[ 290 = P_A \cdot 0.6 + 200 \cdot 0.4 \]

Step 5: Solve for the Vapor Pressure of Ethyl Alcohol \(P_A\)

First, calculate the contribution from propyl alcohol:\[ 200 \times 0.4 = 80 \]Next, substitute and rearrange to find \(P_A\):\[ 290 = P_A \times 0.6 + 80 \]\[ P_A \times 0.6 = 290 - 80 = 210 \]\[ P_A = \frac{210}{0.6} \]\[ P_A = 350 \text{ mm} \]

Step 6: Verify the Solution

Check the calculations to ensure no arithmetic errors were made. The result of \(P_A = 350 \text{ mm}\) matches the expected choices.

Key Concepts

Vapor Pressure

Vapor pressure is an important concept when it comes to understanding the behavior of liquids in mixtures. It refers to the pressure exerted by the vapor present above a liquid in a closed system. When the molecules of a liquid have enough kinetic energy, they escape from the liquid's surface and become vapor. This balance between the liquid and vapor phases constitutes the vapor pressure.
Vapor pressure is temperature-dependent; as temperature increases, so does the energy of the liquid's molecules, leading to higher vapor pressure. Different liquids have different intrinsic vapor pressures due to variations in molecular structure and intermolecular forces.

• Combining liquids, as seen in mixtures, leads to a total vapor pressure that reflects the individual pressures each component can exert—adjusted for their mole fractions in the mixture.
• This combination behavior is best explained using Raoult's Law, which states that the partial vapor pressure of each component in a solution is equal to the vapor pressure of the pure component multiplied by its mole fraction.

Mole Fraction

Mole fraction is a way to express the concentration of a particular component in a mixture. It is calculated by dividing the number of moles of a component by the total number of moles of all components in the mixture.
The mole fraction, denoted as \(x\), has no units and ranges from 0 to 1. A component's mole fraction enables us to use Raoult's Law to determine its contribution to the total vapor pressure of the mixture.

• In a binary mixture, if the mole fraction of one component is known, the other can be easily deduced since the sum of the mole fractions equals 1.
• This principle is particularly useful when calculating the vapor pressures of components in a mixture, such as ethyl and propyl alcohol.

By understanding the mole fractions, we can properly use Raoult's Law to find each component's influence on the vapor pressure.

Ethyl Alcohol

Ethyl alcohol, also known as ethanol, is a common alcohol widely used in beverages, as a solvent, and in numerous industrial applications. It has significant properties that affect its behavior in mixtures, such as its moderate vapor pressure at various temperatures.
Being a volatile compound, understanding its vapor pressure is crucial for various chemical and industrial processes where evaporation, boiling, and mixing are involved.

• In mixtures, ethanol will contribute to the total vapor pressure based on its mole fraction and its own pure vapor pressure as per Raoult's Law.
• For example, in a mixture with a given mole fraction, calculating its vapor pressure contribution requires knowing these two variables.

The role of ethyl alcohol in mixed solutions, like the one with propyl alcohol, can determine the mixture's evaporation rate, boiling point, and overall stability.

Propyl Alcohol

Propyl alcohol, less commonly used than ethyl alcohol, is another type of alcohol with its unique properties and applications, especially in the chemical industry. Although similar to ethyl alcohol, propyl alcohol generally has a lower vapor pressure, indicating a lesser tendency to evaporate at the same temperature.
When present in a mixture, such as with ethyl alcohol, propyl alcohol contributes to the total vapor pressure based on its mole fraction and inherent vapor pressure. This interaction is fluently explained through Raoult’s Law.

• Knowing its pure vapor pressure helps determine how much it contributes to a mixture's vapor pressure, influencing the mixture's evaporation behavior.
• Propyl alcohol's lower intrinsic vapor pressure typically means it will have a smaller effect on increasing the total vapor pressure of the solution compared to a more volatile substance.

This distinction in vapor pressure behavior is essential for understanding and predicting the properties of mixtures in both theoretical calculations and practical applications.