Problem 62 - Chemistry The Central Science In Si Units

Question

Consider a mixture of two gases, \(A\) and \(B\), 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 \(C\) affect the following: (a) the partial pressure of gas \(\mathrm{A},(\mathbf{b})\) the total pressure in the vessel, (c) the mole fraction of gas B?

Step-by-Step Solution

Answer

(a) Partial pressure of gas A remains unchanged. (b) Total pressure increases. (c) Mole fraction of gas B decreases.

Step 1: Understand Dalton's Law of Partial Pressures

Dalton's Law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. The partial pressure of each gas is the pressure it would exert if it were alone in the container. Thus, initially, the partial pressure of gas A is determined only by its amount and the container's volume.

Step 2: Consider Adding Gas C

When gas C is added to the vessel, it increases the total number of moles of gas in the vessel but does not change the individual moles of gases A and B. Therefore, according to Dalton's Law, the addition of gas C doesn't alter the partial pressures of gases A or B, since these depend solely on their respective amounts and the volume of the vessel.

Step 3: Analyze Effect on Total Pressure

The addition of gas C increases the total number of moles of gas, thus increasing the total pressure in the vessel. Total pressure is the sum of all partial pressures, so with the added moles from gas C, the total pressure will increase even though the partial pressures of A and B remain unchanged.

Step 4: Determine Effect on Mole Fraction of Gas B

The mole fraction of a gas is the ratio of the moles of that gas to the total moles of all gases in the mixture. Adding gas C increases the total number of moles in the vessel, thus decreasing the mole fraction of gas B since the denominator (total moles) increases while the numerator (moles of gas B) stays the same.

Key Concepts

Partial Pressure

The concept of partial pressure is central to understanding gas mixtures. Each gas in a mixture exerts a pressure independently of the others. This "partial pressure" is defined as the pressure a gas would exert if it occupied the container alone. For example, in our scenario with gases A and B, each has its own partial pressure based on its particular amount within the vessel's volume. These pressures are additive according to Dalton's Law. Even with the introduction of a third gas, C, into the vessel, the partial pressures of A and B remain unchanged. Each gas's partial pressure depends solely on its own quantity and the container's volume, not on the presence of other gases.

Total Pressure

Total pressure in a gas mixture is the sum of the partial pressures of all the individual gases present. In our case, when gas C is added, the total pressure is affected because more gas molecules are present in the same volume. According to Dalton's Law of Partial Pressures, the total pressure is calculated by adding up the partial pressures of all gases. So, if C is added to gases A and B, the total pressure increases due to the additional molecules. The partial pressures of A and B remain constant because they depend on the number of moles of each specific gas, not on the contribution of other gases.

Mole Fraction

The mole fraction is a way to express the concentration of a gas in a mixture relative to the total amount of gas. It is calculated by dividing the number of moles of a particular gas by the total number of moles of all gases in the mixture. For example, if gas B has 2 moles and the total moles of all gases is 10, the mole fraction of B is 0.2. When gas C is added, while the moles of B remain the same, the total moles increase. Hence, the mole fraction of B decreases because its ratio to the total moles has changed. Understanding how the mole fraction is influenced by the presence of other gases can help in various practical applications involving gas mixtures.

Gas Mixtures

Gas mixtures are common in both natural environments and industrial processes. Each gas in such a mixture contributes differently to the overall properties. Unlike a simple compound, where substances combine chemically, gases in a mixture retain their individual properties. The behavior of these gas mixtures is often described using Dalton’s Law, which helps us understand how partial and total pressures work together. In our exercise, adding gas C changes the overall properties of the gas mixture without affecting some individual components. Handling and manipulating gas mixtures effectively requires a clear understanding of these fundamental concepts. They are crucial for fields such as chemistry, environmental science, and chemical engineering.