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Chapter 10: Problem 62

Consider a mixture of two gases, \(\mathrm{A}\) and \(\mathrm{B}\), confined in a closed vessel. A quantity of a third gas, \(\mathrm{C}\), is added to the same vessel at the same temperature. How does the addition of gas \(\mathrm{C}\) affect the following: (a) the partial pressure of gas \(A,(\mathbf{b})\) the total pressure in the vessel, (c) the mole fraction of gas B?

Short Answer

Expert verified
(a) The partial pressure of gas A remains unchanged. (b) The total pressure in the vessel increases. (c) The mole fraction of gas B decreases.

Step by step solution

01

Understand the concepts of partial pressure and mole fraction

Partial pressure is the pressure exerted by each individual gas in a mixture as if it were occupying the container by itself. Mole fraction is the ratio of the moles of a particular gas to the total moles of the gaseous mixture.
02

Use Ideal Gas Law for the initial system of gases A and B

We can write the Ideal Gas Law for the initial system containing gases A and B as follows: \[P_{initial}V = n_{A}RT + n_{B}RT\] Where: - \(P_{initial}\) is the initial total pressure of the system - \(V\) is the volume of the vessel - \(n_A\) and \(n_B\) are the moles of gases A and B, respectively - \(R\) is the gas constant - \(T\) is the temperature
03

Add gas C and write the Ideal Gas Law for the final system

Now, we add gas C to the system. Let's write the Ideal Gas Law for the final system containing gases A, B, and C: \[P_{final}V = n_{A}RT + n_{B}RT + n_{C}RT\] Where: - \(P_{final}\) is the final total pressure of the system after adding gas C - \(n_C\) is the moles of gas C added to the system
04

Analyze the effect of adding gas C on the partial pressure of gas A

As the addition of gas C does not change the moles of gas A, its partial pressure, which is given by: \[P_{A} = \dfrac{n_{A}RT}{V}\] will not change because the number of moles, volume, gas constant, and temperature remain the same after adding gas C. (a) Effect on partial pressure of gas A: No change.
05

Analyze the effect of adding gas C on the total pressure in the vessel

Based on the comparison of the equation from step 2 and step 3, we can see that the final pressure increases due to the additional moles of gas C. This means the total pressure in the vessel after adding gas C will be greater than the initial total pressure. (b) Effect on the total pressure in the vessel: Increases.
06

Analyze the effect of adding gas C on the mole fraction of gas B

Mole fraction of gas B is given by the formula: \[\chi_{B} = \dfrac{n_{B}}{n_{A} + n_{B} + n_{C}}\] Comparing the mole fraction of gas B before and after the addition of gas C, initially: \[\chi_{B,initial} = \dfrac{n_{B}}{n_{A} + n_{B}}\] After adding gas C: \[\chi_{B,final} = \dfrac{n_{B}}{n_{A} + n_{B} + n_{C}}\] Since the addition of gas C increases the denominator in the mole fraction expression without affecting the numerator, the mole fraction of gas B will decrease after adding gas C. (c) Effect on the mole fraction of gas B: Decreases.

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Most popular questions from this chapter

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