Question

In a mixture of \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2}\), the \(\mathrm{P}_{\mathrm{CO}_{2}}\) is \(0.4 \mathrm{~atm}\) and \(\mathrm{P}_{\text {Total }}\) is 2 atm. The percentage composition of the mixture by volume can be given as: (a) \(\mathrm{CO}_{2}=20 \%, \mathrm{H}_{2}=80 \%\) (b) \(\mathrm{CO}_{2}=40 \%, \mathrm{H}_{2}=60 \%\) (c) \(\mathrm{CO}_{2}=80 \%, \mathrm{H}_{2}=20 \%\) (d) \(\mathrm{CO}_{2}=60 \%, \mathrm{H}_{2}=40 \%\)

Short answer

The mixture is 20% CO2 and 80% H2, so the answer is (a).

Step-by-step solution

Understanding Total Pressure

In this mixture, the total pressure \( P_{\text{Total}} \) is given as \( 2 \text{ atm} \). This pressure represents the sum of the partial pressures of each component in the gas mixture. Thus, \( P_{\text{Total}} = P_{\text{CO}_2} + P_{\text{H}_2} \).

Identifying Partial Pressure of CO2

We know that the partial pressure of \( \text{CO}_2 \) is \( 0.4 \text{ atm} \). Thus, \( P_{\text{CO}_2} = 0.4 \text{ atm} \).

Calculating Partial Pressure of H2

Using the relation for total pressure, \( P_{\text{H}_2} = P_{\text{Total}} - P_{\text{CO}_2} = 2 \text{ atm} - 0.4 \text{ atm} = 1.6 \text{ atm} \).

Finding Volume Ratio

Volume percentages are directly proportional to the partial pressures in a gas mixture if measured under the same conditions. Therefore, the percentage of each gas by volume can be calculated using \( \frac{P_x}{P_{\text{Total}}} \times 100 \% \).

Calculate CO2 Percentage

For \( \text{CO}_2 \), the volume percentage is \( \frac{0.4}{2} \times 100\% = 20\% \).

Calculate H2 Percentage

For \( \text{H}_2 \), the volume percentage is \( \frac{1.6}{2} \times 100\% = 80\% \).

Conclusion

Therefore, the composition of the mixture by volume is \( 20\% \text{CO}_2 \) and \( 80\% \text{H}_2 \). Thus the correct answer is (a) \( \text{CO}_2 = 20\% , \text{H}_2 = 80\% \).

Key concepts

Partial Pressure

In any gas mixture, the pressure exerted by an individual gas is known as its partial pressure. This concept is crucial for understanding how gases in a mixture contribute to the total pressure. For instance, consider a mixture of carbon dioxide (CO\(_2\)) and hydrogen (H\(_2\)). Each gas exerts its own partial pressure. The total pressure of the gas mixture is simply the sum of these individual partial pressures. This is expressed by the equation:\[ P_{\text{Total}} = P_{\text{CO}_2} + P_{\text{H}_2} \]Given that the partial pressure of CO\(_2\) is known to be 0.4 atm, we can determine the partial pressure of H\(_2\) by subtracting the CO\(_2\) partial pressure from the total pressure. For this exercise, the total pressure is provided as 2 atm:

• \( P_{\text{H}_2} = P_{\text{Total}} - P_{\text{CO}_2} \)
• \( P_{\text{H}_2} = 2\, \text{atm} - 0.4\, \text{atm} = 1.6\, \text{atm} \)

This calculation shows how knowing one partial pressure and the total pressure allows us to find the other partial pressures in the mixture.

Volume Percentage

To find the composition of gases in a mixture, we often use volume percentage. This percentage gives us an understanding of how much space each gas occupies in comparison to the entire mix. In a gas mixture under the same conditions, volume percentages are directly related to the partial pressures. This is given by the expression:\[\text{Volume Percentage} = \frac{P_x}{P_{\text{Total}}} \times 100\%\]Using this formula, we calculate the volume percentages for CO\(_2\) and H\(_2\):

• For CO\(_2\), with a partial pressure of 0.4 atm:\[ \text{Volume Percentage of CO}_2 = \frac{0.4}{2} \times 100\% = 20\% \]
• For H\(_2\), with a partial pressure of 1.6 atm:\[ \text{Volume Percentage of H}_2 = \frac{1.6}{2} \times 100\% = 80\% \]

This tells us that 20% of the mixture by volume is CO\(_2\) and 80% is H\(_2\). Volume percentages are particularly useful in fields such as chemistry and environmental sciences where gas composition needs precise analysis.

Ideal Gas Law

The Ideal Gas Law is an essential equation in chemistry that connects pressure, volume, temperature, and the number of moles of a gas. Though this particular problem does not require using the Ideal Gas Law directly, understanding it helps grasp gas behaviors. The Ideal Gas Law is expressed as:\[ PV = nRT \]Where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles.
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.

While solving gas mixture problems, realizing that the law assumes that gases behave ideally, it can apply when considering the behavior of gases at a range of conditions. The connections it explains display how changes in one variable affect the others, which is foundational to conducting any gas-related calculations.

Chemical Composition Analysis

Chemical Composition Analysis is the practice of quantitatively evaluating the makeup of a chemical sample. It is integral in determining what substances are present and in what proportions. For gas mixtures, this analysis often involves finding volume or mass percentages, as shown in this exercise.
The ability to calculate volume percentages from partial pressures is a fundamental part of this analysis. Techniques such as chromatography or spectrometry can provide actual mole or mass fractions. However, the simplicity of the relation between partial pressure and volume percentage helps in rapid assessment of gaseous samples.
This skill is pivotal in various scientific fields, from atmospheric studies, where knowing the exact gas composition can predict chemical reactions, to engineering, where it assists in process control and compliance with safety and environmental standards. Understanding chemical composition paves the way for theoretical and practical developments in chemistry and beyond.