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

Short answer

(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

Recall Dalton's Law of Partial Pressures

According to Dalton's Law of Partial Pressures, the total pressure in a mixture of non-reacting ideal gases is equal to the sum of the partial pressures of the individual gases. The partial pressure of a gas is the pressure it would exert if it occupied the entire volume by itself, and it's equal to the product of its mole fraction and the total pressure.

Analyze the scenario before adding gas C

Initially, we have a closed vessel containing two gases, A and B, with a total pressure \(P_{total}\) and partial pressures \(P_A\) and \(P_B\), respectively. We can represent these pressures as: \(P_{total} = P_A + P_B\) \(P_A = x_A P_{total}\) \(P_B = x_B P_{total}\) Where \(x_A\) and \(x_B\) are the mole fractions of gas A and gas B, respectively.

Analyze the scenario after adding gas C

When we add gas C to the vessel, the total pressure (\(P_{total}'\)) and partial pressures (\(P_A'\), \(P_B'\), \(P_C\)) will be affected. However, since the molar amounts of A and B and the volume of the closed vessel remain constant, their mole fractions and partial pressures should not change. Let's find the new total pressure and mole fractions.

Determine the effect on total pressure

The total pressure in the vessel after adding gas C (\(P_{total}'\)) can be written as: \(P_{total}' = P_A' + P_B' + P_C\) Since we already established that the mole fractions and partial pressures of A and B remain constant, and the pressure of gas C contributes to the total pressure, it must be the case that \(P_{total}' > P_{total}\).

Determine the effect on partial pressure of gas A

As we discussed earlier, the molar amount and partial pressure of gas A remain constant because neither the gas nor the volume of the vessel is changing. Therefore, \(P_A = P_A'\).

Determine the effect on mole fraction of gas B

Now we'll find the new mole fraction for gas B (\(x_B'\)). From Step 3, we know that the partial pressure of gas B remains constant. We can write the expression for the mole fraction of gas B after adding gas C as: \(x_B' = \frac{P_B'}{P_{total}'}\) Since \(P_B' = P_B\) and \(P_{total}' > P_{total}\), it follows that \(x_B' = \frac{P_B}{P_{total}'} < \frac{P_B}{P_{total}} = x_B\). This means that the mole fraction of gas B decreases after adding gas C. In summary: (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.

Key concepts

Partial Pressure

Partial pressure is a fundamental concept in understanding gaseous mixtures and how individual gases interact within a confined space. In a mixture, each gas exerts a pressure as if it were alone in the entire volume of the container; this is known as its partial pressure. When considering real-life applications such as scuba diving or the administration of medical gases, understanding partial pressure is critical in ensuring safety and efficacy.

Let's illustrate this with an example similar to our exercise: imagine a balloon filled with a mix of oxygen and helium. The pressure exerted by the oxygen alone is its partial pressure, irrespective of the helium present. Hence, if the balloon's total pressure is measured, it represents the sum of the partial pressures of both gases according to Dalton's Law of Partial Pressures. In mathematical terms, if oxygen has a partial pressure of 300 mmHg and helium 200 mmHg, ignoring any other factors, the balloon's total pressure would be 500 mmHg.

Moreover, partial pressures are directly linked to mole fractions, another crucial concept. Due to their interdependence, any change in the mole fraction directly affects the partial pressure and vice versa, given a constant volume and temperature. Importantly, when a new gas is introduced into the mixture, like gas C in our exercise, it adds to the total pressure but does not affect the existing gases' partial pressures if the amount and volume of those gases stay constant.

Mole Fraction

Mole fraction is another central concept when analyzing gas mixtures, providing insight into the proportion of each component. It is defined as the ratio of the number of moles of a particular gas to the total number of moles of all gases present. The mole fraction is a unitless number, which allows chemists and physicists to discuss the makeup of a mixture without concern for the overall size of the sample.

For instance, in a container with 2 moles of nitrogen (N₂) and 3 moles of oxygen (O₂), the mole fraction of nitrogen is 2/5 or 0.4, and oxygen is 3/5 or 0.6. This directly informs us about the presence of each gas in the mixture. Moreover, mole fractions are used to calculate partial pressures. If the total pressure of the container is known, the partial pressure of each gas can be determined by multiplying its mole fraction with the total pressure.

When examining the effects of adding a third gas C to the mixture, as in our exercise scenario, while the actual amount of gases A and B remains unchanged, the introduction of gas C increases the total number of moles, thereby altering the mole fractions. The mole fraction of gas B decreases because it now constitutes a smaller proportion of the total mix. This concept is vital for various applications, ranging from chemical reactions to industrial processes involving gas mixtures.

Ideal Gas Laws

The ideal gas laws are a set of relationships that describe the behavior of ideal gases. An ideal gas is a theoretical gas composed of many randomly moving point particles that do not interact except when they collide elastically. The laws are often encapsulated in the equation PV=nRT, where P denotes pressure, V is volume, n represents the number of moles, R is the ideal gas constant, and T stands for temperature.

Applying these laws, we can predict how a gas will behave under different conditions. For example, if we increase the temperature while keeping the volume constant, the pressure will increase. Likewise, if we increase the number of moles of gas in a container (at constant temperature and volume), the pressure will also increase because each additional mole of gas particles contributes to the pressure exerted on the container’s walls.

In the case of the exercise, the addition of gas C increases the total number of moles in a fixed volume at the same temperature, which according to the ideal gas laws, must result in an increase in pressure. It is important to note that while real gases do not always follow the ideal gas laws perfectly, they are often close enough that the laws serve as an excellent approximation for many practical calculations and understandings in chemistry and physics.