Question

A mixture of helium and argon contains 3 moles of He for every two moles of Ar. The partial pressure of argon is (1) \(2 / 3\) of the total pressure (2) \(1 / 3\) of the total pressure (3) \(2 / 5\) of the total pressure (4) \(1 / 5\) of the total pressure

Short answer

Option (3) \( \frac{2}{5} \) of the total pressure.

Step-by-step solution

Determine the total number of moles

We are given that there are 3 moles of He for every 2 moles of Ar. Let's denote the moles of He as 3 and the moles of Ar as 2. Therefore, the total number of moles is the sum of moles of He and Ar: Total moles = 3 (He) + 2 (Ar) = 5 moles.

Define the partial pressure formula

The partial pressure of a gas in a mixture is given by the fraction of the number of moles of that gas over the total number of moles, multiplied by the total pressure. Formula: \[ P_{Ar} = \frac{n_{Ar}}{n_{total}} \times P_{total} \]

Calculate the partial pressure of Ar

Given: \( n_{Ar} = 2 \) \( n_{total} = 5 \) Using the formula from step 2: \[ P_{Ar} = \frac{2}{5} \times P_{total} \]Hence, the partial pressure of argon is \( \frac{2}{5} \) of the total pressure.

Identify the correct option

From the calculations in step 3, we can see that the partial pressure of argon is \( \frac{2}{5} \) of the total pressure. Therefore, the correct option is (3) \( \frac{2}{5} \) of the total pressure.

Key concepts

helium and argon mixture

A mixture of helium (He) and argon (Ar) involves combining the gases in specific proportions. In this example, we have 3 moles of helium for every 2 moles of argon. Understanding the behavior of a gas mixture involves considering the individual gases that make up the mixture and their amounts.

Key points about gas mixtures:
- Each type of gas in a mixture behaves independently.
- The total pressure of the mixture is determined by the sum of the partial pressures of each gas.
- The proportion of each gas in the mixture often impacts factors such as density and volume.

In this problem, with 3 moles of helium and 2 moles of argon, we first calculate the total number of moles present. Summing these gives us a total of 5 moles. This total will help us determine how pressure is contributed by each gas.

mole fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a particular component to the total number of moles of all components.

Formula for mole fraction: \( X_i = \frac{n_i}{n_{total}} \)
where:
- \( X_i \) is the mole fraction of component i.
- \( n_i \) is the number of moles of component i.
- \( n_{total} \) is the total number of moles in the mixture.

For argon in our helium and argon mixture:
- \( n_{Ar} = 2 \)
- \( n_{total} = 5 \)
\( X_{Ar} = \frac{2}{5} = 0.4 \)

Similarly, for helium:
- \( n_{He} = 3 \)
- \( n_{total} = 5 \)
\( X_{He} = \frac{3}{5} = 0.6 \)

Mole fractions are useful because they allow us to calculate the partial pressures of gases in the mixture.

partial pressure formula

Partial pressure refers to the contribution of each gas to the total pressure in a gas mixture. It can be found using the mole fraction and the total pressure of the mixture.

The general formula for partial pressure is:
\( P_i = X_i \times P_{total} \)
where:
- \( P_i \) is the partial pressure of component i.
- \( X_i \) is the mole fraction of component i.
- \( P_{total} \) is the total pressure of the mixture.

Applying this to our problem, to find the partial pressure of argon (\( P_{Ar} \)):
- We already know \( X_{Ar} = 0.4 \)
- We don't have the exact total pressure, but we express the partial pressure in terms of \( P_{total} \)
\( P_{Ar} = 0.4 \times P_{total} \)

This simplifies to:
\( P_{Ar} = \frac{2}{5} \times P_{total} \)

Therefore, the partial pressure of argon is \( \frac{2}{5} \) or 40% of the total pressure. This matches the provided solution and confirms that option (3) is correct.