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

Short answer

In summary, the addition of gas C to the closed vessel has the following effects: (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

Dalton's Law of Partial Pressures

Recall Dalton's Law of Partial Pressures, which states that the total pressure of a mixture of non-reacting gases is equal to the sum of their individual partial pressures. Mathematically, this can be represented as: \(P_{total} = P_A + P_B + P_C\), where \(P_A, P_B\) and \(P_C\) are the partial pressures of gases A, B, and C, respectively, and \(P_{total}\) is the total pressure inside the vessel.

Definition of Mole Fraction

Recall the concept of mole fraction, which is a measure of the amount of a substance compared to the total amount of substance in a mixture. Mathematically, the mole fraction of gas i is denoted by \(X_i\) and can be computed as: \(X_i = \frac{n_i}{n_{total}}\), where \(n_i\) is the amount (in moles) of gas i, and \(n_{total}\) is the total amount of gas in the mixture.

(a) Partial Pressure of Gas A

Since the temperature and volume of the vessel remain constant, the addition of gas C does not affect the moles of gas A in the mixture. Therefore, the partial pressure of gas A remains unchanged.

(b) Total Pressure in the Vessel

Since we have added a third gas C to the mixture, there will now be an additional partial pressure associated with it. Thus, the total pressure in the vessel will increase due to the presence of the third gas. Mathematically, the total pressure will be: \(P_{total(new)} = P_A + P_B + P_C\), where \(P_C\) is greater than zero, hence increasing the total pressure.

(c) Mole Fraction of Gas B

With the addition of gas C, the total number of moles in the mixture has increased. This will affect the mole fractions of gases A, B, and C in the mixture. Let's consider the mole fraction of gas B. Initially, \(X_B = \frac{n_B}{n_A + n_B}\), After adding gas C to the mixture, we now have: \(X_{B(new)} = \frac{n_B}{n_A + n_B + n_C}\), Since \(n_C > 0\), the denominator is now larger than before. This means that the mole fraction of gas B, \(X_{B(new)}\), will be smaller than its original value, indicating a decrease in the mole fraction of gas B.

Key concepts

Partial Pressure

Partial pressure refers to the pressure contributed by a single gas in a mixture of gases. It reflects how each gas, if it alone occupied the volume of the container at the same temperature, would exert pressure on the walls of the container.

In a mixture, each gas behaves independently and contributes to the total pressure as if no other gases were present. This fundamental principle is encapsulated in Dalton’s Law of Partial Pressures: For a mixture of non-reacting gases, the total pressure is simply the sum of their individual partial pressures.

For example, if a container holds three gases, the total pressure \(P_{total}\) can be calculated as \(P_{total} = P_A + P_B + P_C\), where \(P_A\), \(P_B\), and \(P_C\) are the partial pressures of gases A, B, and C, respectively. In a real-world scenario, this means when you blow up a balloon with a mixture of helium and oxygen, the pressure pushing against the balloon's inner surface is the sum of the pressures that each gas would exert on its own.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is a dimensionless number that represents the proportion of moles of one substance relative to the total number of moles of all substances present.

The mole fraction \(X_i\) of any component \(i\) is given by the formula \(X_i = \frac{n_i}{n_{total}}\), where \(n_i\) is the number of moles of the substance and \(n_{total}\) is the total number of moles in the mixture. For example, if a container has 2 moles of nitrogen and 1 mole of oxygen, the mole fraction of nitrogen would be \(\frac{2}{3}\) and that of oxygen would be \(\frac{1}{3}\).

Mole fraction is important because it is used in calculating partial pressures, determining relative quantities in chemical reactions, and describing the composition of solutions and mixtures in terms of their molecular constituents. It’s crucial for understanding how the behavior of individual gases in a mixture compares to that of pure gases.

Total Pressure in a Gas Mixture

The total pressure in a gas mixture is the sum of all the partial pressures of the individual gases present. Each gas contributes to the total pressure proportionally to its fraction of the total number of moles, according to Dalton's Law.

When calculating the total pressure, if the volume and temperature of the container are held constant, any addition of gas will lead to an increase in the total pressure of the mixture. As we add more gas molecules into the space, more collisions occur between the molecules and the walls of the container, thereby increasing the pressure. Mathematically, the new total pressure \(P_{total(new)}\) is the sum of the individual pressures of all present gases.

For instance, if a new gas is introduced into a vessel already containing a mixture of two other gases, the total pressure rises as \(P_{total(new)} = P_A + P_B + P_C\). This is because we have added the partial pressure of gas C, \(P_C\), to the initial combined pressures of gases A and B.