Problem 61 - Chemistry: The Central Science

Question

Consider the apparatus shown in the following drawing. (a) When the valve between the two containers is opened and the gases allowed to mix, how does the volume occupied by the \(\mathrm{N}_{2}\) gas change? What is the partial pressure of \(\mathrm{N}_{2}\) after mixing? (b) How does the volume of the \(\mathrm{O}_{2}\) gas change when the gases mix? What is the partial pressure of \(\mathrm{O}_{2}\) in the mixture? (c) What is the total pressure in the container after the gases mix?

Step-by-Step Solution

Answer

When the valve between the two containers is opened and the gases mix, the volume occupied by the N2 gas remains the same, as the total volume of the container is unchanged. The partial pressure of N2 after mixing is \(P_{N_2}' = x_{N_2} P\), where \(x_{N_2}\) is the mole fraction of N2 and P is the total pressure of the mixture. Similarly, the volume occupied by the O2 gas remains the same when the gases mix. The partial pressure of O2 after mixing is \(P_{O_2}' = x_{O_2}P\), where \(x_{O_2}\) is the mole fraction of O2. The total pressure in the container after the gases mix is given by the sum of the partial pressures of N2 and O2: \(P_{total} = P_{N_2}' + P_{O_2}'\).

Step 1: Analyze the initial conditions for both gases

Initially, both the gases are in separate containers. We'll need to find the volume occupied by each gas before mixing. Let's denote the initial volume of N2 gas as \(V_{N_2}\) and the initial volume of O2 gas as \(V_{O_2}\). The initial volume can be found using the ideal gas law: \[ PV = nRT \] where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.

Step 2: Calculate the initial volumes of N2 and O2

Using the ideal gas law, we can find the initial volumes occupied by each gas: \(V_{N_2} = \frac{n_{N_2}RT}{P_{N_2}}\) \(V_{O_2} = \frac{n_{O_2}RT}{P_{O_2}}\)

Step 3: Analyze the gas mixture

When the two gases are mixed, the total volume V of the container remains the same. So, the volume occupied by each gas remains the same. The partial pressures of the gases will change, based on the mole fractions of the gases in the mixture. We can use the following formula for partial pressures: \(P_i = x_iP\) where \(P_i\) is the partial pressure of the gas i, \(x_i\) is the mole fraction of the gas i, and P is the total pressure.

Step 4: Calculate the partial pressure of N2 after mixing

The mole fraction of N2 gas, \(x_{N_2}\), is given by: \(x_{N_2} = \frac{n_{N_2}}{n_{N_2} + n_{O_2}}\) The total pressure of the mixture P can be found using the ideal gas law for the overall mixture: \(P = \frac{(n_{N_2} + n_{O_2})RT}{V}\) The partial pressure of N2 gas, \(P_{N_2}'\), is given by: \(P_{N_2}' = x_{N_2} P\)

Step 5: Calculate the partial pressure of O2 after mixing

The mole fraction of O2 gas, \(x_{O_2}\), is given by: \(x_{O_2} = \frac{n_{O_2}}{n_{N_2} + n_{O_2}}\) The partial pressure of O2 gas, \(P_{O_2}'\), is given by: \(P_{O_2}' = x_{O_2}P\)

Step 6: Calculate the total pressure in the container after the gases mix

The total pressure of the mixture after mixing can be found by summing the partial pressures of N2 and O2: \(P_{total} = P_{N_2}' + P_{O_2}'\) This is the final answer for the total pressure in the container after the gases mix.

Key Concepts

Partial Pressure

When multiple gases are mixed in a container, each gas exerts its own pressure as if it were alone. This is what we call partial pressure.

For example, if you mix nitrogen (\(N_2\)) and oxygen (\(O_2\)) in a container, the gases don't change their volume, but each exerts individual pressure. We calculate this using the formula:

\[P_i = x_iP\]
where \\(P_i\)\ is the partial pressure of a specific gas, \\(x_i\)\ is the mole fraction, and \(P\)\ is the total pressure of the mixture.

This means each gas's pressure depends on its proportion in the mixture. Partial pressure is crucial for understanding how different gases contribute to the total pressure in a system.

Mole Fraction

The mole fraction is a way to express the concentration of a specific gas in a mixture. It's calculated by comparing the moles of one gas to the total moles in the mixture.
It's expressed as \[x_i = \frac{n_i}{n_{total}}\]
where \\(n_i\)\ is the moles of a particular gas, and \\(n_{total}\)\ is the total moles of all gases combined in the mixture.

The mole fraction has no units, making it a simple ratio.
Using mole fraction makes it easy to determine the partial pressure when combining gases, since it represents each gas's share of the total number of particles in the mixture.

• Mole fraction allows us to focus on the relative quantities of gases.
• This concept helps in further calculations, like partial pressures and reactions involving gases.

Gas Mixture

A gas mixture is created when different gases are combined in the same container, like our nitrogen and oxygen example. When mixed, each gas occupies the total container volume, but they do not influence each other’s behavior. This happens because gas particles are far apart, allowing free movement.

The gas mixture's behavior is predicted by combining the properties of the individual gases. When mixing gases:

• The temperature remains constant.
• Each gas acts as if it were alone.
• The volume occupied by each gas is the volume of the container.

Mixing gases doesn't change their individual characteristics, such as volume or pressure exerted. Instead, it allows us to study their collective properties and impacts.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures helps explain how gas mixtures behave. It states that the total pressure of a gas mixture equals the sum of the partial pressures of each individual gas.
Expressed as \[P_{total} = P_1 + P_2 + P_3 + ... + P_n\]
where each \(P\)\ is the partial pressure of a gas.

Dalton’s Law is immensely helpful when calculating how different gases contribute to a mixture’s total pressure in a container. Each gas's pressure remains unaffected by the presence of other gases. This principle allows chemists to predict and balance how gases will interact in mixtures.

By understanding Dalton's Law, one can efficiently calculate the pressures in any gas system, which is crucial in both experimental and theoretical chemistry contexts.