Question

A mixture of gases contains \(0.31 \mathrm{~mol} \mathrm{CH}_{4}, 0.25 \mathrm{~mol}\) \(\mathrm{C}_{2} \mathrm{H}_{6}\), and \(0.29 \mathrm{~mol} \mathrm{C}_{3} \mathrm{H}_{8}\). The total pressure is \(1.50 \mathrm{~atm} .\) Calculate the partial pressures of the gases.

Short answer

The partial pressures are: \(\mathrm{CH}_4\) is 0.547 atm, \(\mathrm{C}_2\mathrm{H}_6\) is 0.441 atm, and \(\mathrm{C}_3\mathrm{H}_8\) is 0.511 atm.

Step-by-step solution

Calculate Total Moles of Gas

First, we need to determine the total number of moles in the gas mixture. Add the moles of each gas together to get the total n (in moles):\[n = 0.31 + 0.25 + 0.29 = 0.85 \text{ mol}\]

Determine Mole Fraction

Next, we calculate each gas's mole fraction \(X_i\). For each gas \(i\), use the formula:\[X_i = \frac{\text{moles of } i}{n}\]E.g., for \(\mathrm{CH}_4\):\[X_{\mathrm{CH}_4} = \frac{0.31}{0.85} \approx 0.3647\]Repeat for \(\mathrm{C}_2\mathrm{H}_6\) and \(\mathrm{C}_3\mathrm{H}_8\).

Calculate Partial Pressures

Now use the mole fraction and total pressure to find each gas's partial pressure \(P_i\) using:\[P_i = X_i \cdot P_{\text{total}}\]For \(\mathrm{CH}_4\):\[P_{\mathrm{CH}_4} = 0.3647 \times 1.50 = 0.547 \text{ atm}\]Repeat this for \(\mathrm{C}_2\mathrm{H}_6\) and \(\mathrm{C}_3\mathrm{H}_8\).

Key concepts

Gas Mixture

A gas mixture consists of two or more gases blended together without any specific chemical reactions occurring between them. In this concept, understanding how different gases combine and share a container is essential for calculating properties like pressure and volume.

In our example, the mixture is composed of methane (\(\mathrm{CH}_4\)), ethane (\(\mathrm{C}_2\mathrm{H}_6\)), and propane (\(\mathrm{C}_3\mathrm{H}_8\)). Each gas contributes to the overall properties of the mixture. The total behavior of the mixture is affected by:

• The number of moles of each gas present
• The temperature and pressure of the mixture
• The interactions between the different gas molecules

A crucial aspect is calculating the contribution of each gas to the overall mixture, which is directly related to its mole fraction in the gas mixture.

Mole Fraction

Mole fraction is a way to express the concentration of a component in a mixture. It is defined as the ratio of the moles of a particular component to the total moles in the mixture. This method of expression is particularly useful when dealing with gas mixtures because it allows us to easily calculate each gas's influence on the total pressure.

The formula used to calculate the mole fraction \(X_i\) of a particular gas \(i\) in a mixture is:

• \(X_i = \frac{\text{moles of } i}{\text{total moles}}\)

For instance, in our problem, methane has a mole fraction calculated as \(X_{\mathrm{CH}_4} = \frac{0.31}{0.85}\). This tells us how much of the total mixture is comprised of methane, and similarly, this can be done for ethane and propane. Knowing each gas's mole fraction, we can determine each gas's contribution to pressure, utilizing Dalton's Law.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a critical principle in gas mixture calculations. It states that the total pressure exerted by a gas mixture is the sum of the partial pressures of each individual gas within the mix. Each gas's partial pressure is its mole fraction times the total pressure of the gas mixture.

This can be mathematically represented as:

• \(P_i = X_i \cdot P_{\text{total}}\)

where \(P_i\) is the partial pressure of gas \(i\), \(X_i\) is its mole fraction, and \(P_{\text{total}}\) is the overall pressure of the gas mixture. For example, the partial pressure of methane in our problem is \(0.547 \text{ atm}\). Knowing how each gas's partial pressure contributes to the total gives us insights into the behavior of the whole mixture and is crucial when analyzing or designing systems where gas mixtures are involved, like chemical reactions or industrial applications.