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 Raoult's Law

Raoult's Law states that the partial vapor pressure of each component in a liquid mixture is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture. Mathematically, for a binary mixture: \( P_{total} = P_{A}^{0} imes X_{A} + P_{B}^{0} imes X_{B} \), where \( P_{total} \) is the total vapor pressure, \( P_{A}^{0} \) and \( P_{B}^{0} \) are the vapor pressures of pure components A and B, and \( X_{A} \) and \( X_{B} \) are their mole fractions.

Step 2: Define Variables

In this problem, ethyl alcohol is component A with a vapor pressure we are trying to find, denoted as \( P_{ethyl}^{0} \), and propyl alcohol is component B with a known vapor pressure of \( P_{propyl}^{0} = 200 \) mm. The total vapor pressure \( P_{total} \) is given as 290 mm, and the mole fraction of ethyl alcohol, \( X_{ethyl} \), is 0.6.

Step 3: Calculate Mole Fraction of Propyl Alcohol

Since the sum of mole fractions in a mixture is 1, we have: \( X_{ethyl} + X_{propyl} = 1 \). Thus, the mole fraction of propyl alcohol, \( X_{propyl} \), is \( 1 - 0.6 = 0.4 \).

Step 4: Substitute Values into Raoult's Law Equation

Substitute the known values into the Raoult's Law equation: \[ 290 = P_{ethyl}^{0} imes 0.6 + 200 imes 0.4 \].

Step 5: Solve for Vapor Pressure of Ethyl Alcohol

Rearrange and solve the equation for \( P_{ethyl}^{0} \): \[ 290 = P_{ethyl}^{0} imes 0.6 + 80 \] \[ 290 - 80 = P_{ethyl}^{0} imes 0.6 \] \[ 210 = P_{ethyl}^{0} imes 0.6 \] \[ P_{ethyl}^{0} = \frac{210}{0.6} \] \[ P_{ethyl}^{0} = 350 \text{ mm} \].

Key Concepts

Partial Vapor Pressure

Partial vapor pressure refers to the pressure that a single component of a mixture would exert if it occupied the entire volume on its own, at the same temperature. In a liquid mixture, each component contributes to the total vapor pressure of the system, depending on its concentration. According to Raoult's Law, the partial vapor pressure of a component can be calculated using the formula:

• \( P_A = P_A^0 \times X_A \), where \( P_A^0 \) is the vapor pressure of the pure component, and \( X_A \) is its mole fraction in the mixture.

This allows us to predict how each component behaves in the mixture and contributes to its overall vaporization ability. When we know the partial vapor pressures, we can find the total vapor pressure by simply summing them. This relationship is useful in designing processes where separation of components is needed or predicting boiling points of mixtures.

Binary Mixture

A binary mixture consists of two different substances combined. In cases like our example with ethyl alcohol and propyl alcohol, we deal with a simple two-component system.

• Here, each component behaves in line with its concentration or 'fraction' in the mix, influencing the mixture's physical properties.

In binary mixtures, like other mixtures, Raoult's Law is typically applied to predict the behavior of the components such as vapor pressure. By understanding how each component contributes to the properties of the mixture, one is able to better anticipate behavior under different conditions. This predictability is a cornerstone for chemical processes and various industrial applications that involve separating, distilling, or chemically reacting components.

Mole Fraction

The mole fraction of a component in a mixture is its proportion relative to the total number of moles of all components combined. It is an essential concept in understanding the magnitude of a substance's presence in a mixture, represented as a dimensionless number.

• For instance, in our problem, the mole fraction of ethyl alcohol is \(0.6\), indicating it covers 60% of the entire mixture in terms of moles.

The sum of all mole fractions in any mixture equals to 1. This means that if you have the mole fraction of one component, you can always find the other by subtracting it from one.Mole fractions provide an easier understanding of component distribution in a mixture, and are vital for applying Raoult's Law to predict the resulting properties, such as vapor pressure in the given setup.