Question

Equal masses of methane and hydrogen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). The fraction of the total pressure exerted by hydrogen is: (a) \(1 / 2\) (b) \(8 / 9\) (c) \(1 / 9\) (d) \(16 / 17\)

Short answer

The fraction of the total pressure exerted by hydrogen is \(\frac{8}{9}\).

Step-by-step solution

Identify the Known Values

We are given that equal masses of methane \(\text{(CH}_4\text{)}\) and hydrogen \(\text{(H}_2\text{)}\) are mixed at \(25^{\circ} \text{C}\). We need to calculate the fraction of the total pressure exerted by hydrogen.

Determine the Molar Masses

Calculate the molar masses of methane and hydrogen. The molar mass of \(\text{CH}_4\) is \(12 + 4 = 16\, \text{g/mol}\), and the molar mass of \(\text{H}_2\) is \(2\, \text{g/mol}\).

Calculate Moles for Each Gas Using Equal Masses

Since the masses are equal, let's assume each has a mass of \(m\) grams. The number of moles of methane \(n_{\text{CH}_4}\) is given by \(n_{\text{CH}_4} = \frac{m}{16}\).The number of moles of hydrogen \(n_{\text{H}_2}\) is given by \(n_{\text{H}_2} = \frac{m}{2}\).

Determine Total Moles in the Mixture

Total moles \(n_{\text{total}}\) is the sum of moles of methane and hydrogen:\[n_{\text{total}} = n_{\text{CH}_4} + n_{\text{H}_2} = \frac{m}{16} + \frac{m}{2}\]

Simplify the Expression for Total Moles

Combine the terms:\[n_{\text{total}} = \frac{m}{16} + \frac{8m}{16} = \frac{9m}{16}\]

Calculate the Mole Fraction of Hydrogen

The mole fraction of hydrogen \(X_{\text{H}_2}\) is given by the ratio:\[X_{\text{H}_2} = \frac{n_{\text{H}_2}}{n_{\text{total}}} = \frac{\frac{m}{2}}{\frac{9m}{16}} = \frac{16}{18} = \frac{8}{9}\]

Relate Mole Fraction to Pressure Fraction

In a gas mixture, the partial pressure of a component is directly proportional to its mole fraction. Thus, the fraction of the total pressure exerted by hydrogen is \(\frac{8}{9}\).

Key concepts

Mole Fraction

The mole fraction is a way to express the concentration of a single component in a mixture. It tells us how many parts of a particular gas are present in the total mixture. To calculate the mole fraction, you divide the number of moles of one component by the total number of moles in the mixture.

For instance, if you have a mixture containing hydrogen (H_2) and methane (CH_4), and you want to find the mole fraction of hydrogen (X_{ ext{H}_2}), you need to measure how many moles of hydrogen there are compared to the entire mixture. The formula is:\[X_{ ext{H}_2} = \frac{n_{ ext{H}_2}}{n_{ ext{total}}}\]

Where:

• \(n_{ ext{H}_2}\) is the number of moles of hydrogen.
• \(n_{ ext{total}}\) is the total number of moles in the mixture.

In our problem, the mole fraction helps determine how much each gas contributes to the mixture's behavior.

Partial Pressure

Partial pressure is the pressure that a single gas in a mixture of gases would exert if it occupied the entire volume alone, at the same temperature. It's an essential concept for understanding gas behavior within mixtures.

Each gas in a mixture contributes to the total pressure. This contribution is based on its mole fraction. The partial pressure of any gas is calculated using:\[ P_{ ext{gas}} = X_{ ext{gas}} \times P_{ ext{total}}\]Where:

• \(P_{ ext{gas}}\) is the partial pressure of the gas.
• \(X_{ ext{gas}}\) is the mole fraction of the gas.
• \(P_{ ext{total}}\) is the total pressure of the gas mixture.

By understanding partial pressure, you can accurately assess how each gas in a mixture affects total pressure. In this context, knowing the mole fraction of hydrogen helped compute its partial pressure, showcasing its significant contribution to the total pressure.

Molar Mass Calculation

Molar mass is a critical factor when dealing with gases because it allows us to convert between mass and moles, which is fundamental for many gas calculations.

To find the molar mass, sum the atomic masses of all atoms in a molecule. For example:

• Methane (CH_4) has a molar mass obtained by adding the atomic mass of carbon (12 g/mol) and hydrogen (4 atoms x 1 g/mol), resulting in 16 g/mol.
• Hydrogen (H_2), just two hydrogen atoms, has a molar mass of 2 g/mol.

Knowing the molar mass lets you convert a known mass to moles using:\[n = \frac{m}{M}\]Where:

• \(n\) is the number of moles.
• \(m\) is the mass.
• \(M\) is the molar mass.

Understanding molar mass and this conversion is pivotal because it ties directly into calculating mole fractions and, ultimately, partial pressures.

Ideal Gas Law

The ideal gas law is a powerful tool in understanding gas behavior. It describes how gases behave under different conditions of pressure, volume, and temperature, following the equation:\[PV = nRT\]Where:

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

In our context, we aren't solving directly with the ideal gas law, but understanding it is key because it underlines why and how the mole fraction translates to pressure contributions. Every gas within the container follows the same principles, and by knowing moles and partial pressures, you ensure the gas system behaves as expected. This allows for accurate predictions and comprehensive analysis of gas behavior within mixtures.