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Chapter 12: Problem 86

Two liquids A and B have vapor pressures of \(76 \mathrm{mmHg}\) and \(132 \mathrm{mmHg},\) respectively, at \(25^{\circ} \mathrm{C}\) What is the total vapor pressure of the ideal solution made up of (a) 1.00 mole of \(\mathrm{A}\) and 1.00 mole of \(\mathrm{B}\), and (b) 2.00 moles of \(\mathrm{A}\) and 5.00 moles of \(\mathrm{B}\) ?

Short Answer

Expert verified
The total vapor pressure of the solution is (a) \(104 \mathrm{mmHg}\) and (b) \(116.0 \mathrm{mmHg}\).

Step by step solution

01

Calculation of Mole Fractions: Scenario (a)

First calculate the mole fraction of each component. Since we have 1.00 mole of A and 1.00 mole of B, the mole fraction \(\chi_{A}\) of A is \(1.00 / (1.00 + 1.00) = 0.5\), and the mole fraction \(\chi_{B}\) of B is also \(0.5\).
02

Calculation of Total Vapor Pressure: Scenario (a)

Next, use Raoult's law to calculate the partial pressure of each component in the solution and add them up to get the total vapor pressure of the solution. The partial vapor pressure \(P_{A}\) of A is \(\chi_{A} \times P_{A}^{0} = 0.5 \times 76 \mathrm{mmHg} = 38 \mathrm{mmHg}\), and the partial vapor pressure \(P_{B}\) of B is \(\chi_{B} \times P_{B}^{0} = 0.5 \times 132 \mathrm{mmHg} = 66 \mathrm{mmHg}\). The total vapor pressure \(P_{total}\) is \(P_{A} + P_{B} = 38 \mathrm{mmHg} + 66 \mathrm{mmHg} = 104 \mathrm{mmHg}\)
03

Calculation of Mole Fractions: Scenario (b)

Again, calculate the mole fractions first. Here we have 2.00 moles of A and 5.00 moles of B, so \(\chi_{A} = 2.00 / (2.00 + 5.00) \approx 0.286\), and \(\chi_{B} = 5.00 / (2.00 + 5.00) \approx 0.714\).
04

Calculation of Total Vapor Pressure: Scenario (b)

We can now calculate the total vapor pressure in the same way as before. Here \(P_{A} = \chi_{A} \times P_{A}^{0} \approx 0.286 \times 76 \mathrm{mmHg} \approx 21.7 \mathrm{mmHg}\), and \(P_{B} = \chi_{B} \times P_{B}^{0} \approx 0.714 \times 132 \mathrm{mmHg} \approx 94.3 \mathrm{mmHg}\). So \(P_{total} = P_{A} + P_{B} \approx 21.7 \mathrm{mmHg} + 94.3 \mathrm{mmHg} \approx 116.0 \mathrm{mmHg}\).

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