Problem 62 - Chemistry The Central Science

Question

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

Step-by-Step Solution

Answer

The addition of gas C (a) does not affect the partial pressure of gas A, (b) increases the total pressure in the vessel, and (c) decreases the mole fraction of gas B.

Step 1: Understand the given information

In a closed vessel, there are two gases A and B with a certain pressure, volume, and temperature. A third gas C is added, and we have to determine the effect of its addition on the partial pressure of gas A, total pressure, and mole fraction of gas B.

Step 2: Consider the ideal gas law

The ideal gas law is defined as \(PV=nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature. The pressure and volume of the container remain unchanged with the addition of gas C because the container is closed.

Step 3: Calculate the partial pressure of gas A

The partial pressure of gas A (P_A) is defined as the portion of the total pressure exerted by gas A. According to Dalton's law of partial pressures, the addition of gas C will not affect the partial pressure of gas A (P_A) if the volume and temperature remain constant. Therefore, the partial pressure of gas A remains the same. Answer (a): The addition of gas C does not affect the partial pressure of gas A.

Step 4: Calculate the total pressure

To find the total pressure, we can use Dalton's law of partial pressures: \(P_{total}=P_A+P_B+P_C\). When gas C is added to the vessel, the pressure (P_C) exerted by gas C contributes to the total pressure. Therefore, the total pressure in the vessel will increase by the pressure of gas C. Answer (b): The addition of gas C increases the total pressure in the vessel.

Step 5: Calculate the mole fraction of gas B

The mole fraction of gas B (χ_B) is defined as the ratio of the number of moles of gas B (n_B) to the total number of moles (n_A + n_B + n_C) in the vessel: χ_B = \(\frac{n_B}{n_A + n_B + n_C}\) Since gas C is added to the vessel and the moles of gas A and B remain constant, the denominator (n_A + n_B + n_C) will increase as the moles of gas C are added. As the denominator increases, the mole fraction of gas B (χ_B) will decrease. Answer (c): The addition of gas C decreases the mole fraction of gas B.

Key Concepts

Understanding Partial Pressure

Partial pressure is a crucial concept when studying gas mixtures. It represents the pressure that a single gas component would exert if it were alone in the entire volume occupied by the gas mixture, at the same temperature. In other words, it's the individual contribution of a gas to the total pressure of the mixture.

Considering a scenario where a mixture contains two gases, A and B, each gas has its own partial pressure, denoted as \(P_A\) and \(P_B\) respectively. If a third gas, C, is introduced, the total pressure increases, but the partial pressure of gas A, \(P_A\), remains the same, provided the volume and temperature of the system do not change. This is an essential aspect of gas behavior in closed systems and is directly tied to Dalton's law of partial pressures, which we'll explore next.

Dalton's Law of Partial Pressures

Named after John Dalton, Dalton's law of partial pressures is grounded in the idea that in a mixture of non-reacting gases, each gas acts independently of the others. Dalton's law states that the total pressure exerted by a gas mixture is equal to the sum of the partial pressures of each individual gas in the mixture.

Mathematically, for a mixture of gases A, B, and C, it is expressed as \(P_{total} = P_A + P_B + P_C\). The law highlights that the addition of gas C will not change the partial pressures of gases A or B, but it will contribute to the total pressure. This concept is pivotal in understanding gas mixtures and provides insight into how different gases contribute to pressure in a system. For students, visualizing each gas expanding to fill the entire volume of the container separately can help in comprehending how partial pressures combine to form the total pressure.

Practical Applications of Dalton's Law

From calculating the concentration of gases in the atmosphere to designing chemical reactors, Dalton's law aids in predicting the behavior of gas mixtures under various conditions.

Mole Fraction and Its Implications

The mole fraction is defined as the ratio of the number of moles of one component in a mixture to the total number of moles of all components. Represented by \(\chi\), the mole fraction is dimensionless and provides a way to express the composition of a mixture without concern for the amount of the mixture present.

For a given component B in a mixture with components A and C, the mole fraction of B is calculated using the formula \(\chi_B = \frac{n_B}{n_A + n_B + n_C}\). With the addition of gas C, while the number of moles of B remains constant, the total number of moles increases, leading to a decrease in \(\chi_B\). Understanding mole fraction is important for predicting the properties of gas mixtures and for calculating concentrations in reactions.

Significance in Chemical Reactions

Mole fractions are used in stoichiometry and thermodynamics to assess reaction yields and the behavior of gas mixtures in various conditions, from industrial processes to biological systems.