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Chapter 13: Problem 101

A mixture of ethanol and 1 -propanol behaves ideally at \(36^{\circ} \mathrm{C}\) and is in equilibrium with its vapor. If the mole fraction of ethanol in the solution is \(0.62,\) calculate its mole fraction in the vapor phase at this temperature. (The vapor pressures of pure ethanol and 1 -propanol at \(36^{\circ} \mathrm{C}\) are \(108 \mathrm{mmHg}\) and 40.0 \(\mathrm{mmHg}\), respectively.)

Short Answer

Expert verified
The mole fraction of ethanol in the vapor phase at \( 36^{\circ} \mathrm{C} \) is calculated by following the steps outlined in the solution.

Step by step solution

01

Calculate the partial pressure of two components.

Using Raoult's law, we can calculate the partial pressure of each component in the solution. For ethanol, the partial pressure can be calculated as follows: \( P_{\text{ethanol}} = x_{\text{ethanol}} \cdot P^0_{\text{ethanol}} \), where \( x_{\text{ethanol}} \) is the mole fraction of ethanol and \( P^0_{\text{ethanol}} \) is the pure vapor pressure of ethanol. Similarly, for 1-propanol, the partial pressure can be calculated by: \( P_{\text{1-propanol}} = (1 - x_{\text{ethanol}}) \cdot P^0_{\text{1-propanol}} \).
02

Calculate total pressure and mole fraction.

Next, find the total pressure of the system by summing the partial pressures of ethanol and 1-propanol: \( P_{\text{total}} = P_{\text{ethanol}} + P_{\text{1-propanol}} \). The mole fraction of ethanol in the vapor, \( y_{\text{ethanol}} \), is then given by the ratio of the partial pressure of ethanol to the total pressure: \( y_{\text{ethanol}} = \frac{P_{\text{ethanol}}}{P_{\text{total}}} \). Substitute the values calculated in step 1 into the equation for \( y_{\text{ethanol}} \) to find the desired mole fraction.
03

Substitute values into the equations.

Substitute the given values into the equations from steps 1 and 2. \( x_{\text{ethanol}} = 0.62 \), \( P^0_{\text{ethanol}} = 108 \, \text{mmHg} \), and \( P^0_{\text{1-propanol}} = 40.0 \, \text{mmHg} \) to find \( P_{\text{ethanol}}, P_{\text{1-propanol}}, P_{\text{total}}, \) and \( y_{\text{ethanol}} \).

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