Question

Equal masses of methane and oxygen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). The fraction of the total pressure exerted by oxygen is: (a) \(1 / 3 \times 273 / 298\) (b) \(1 / 3\) (c) \(1 / 2\) (d) \(2 / 3\)

Short answer

The fraction of total pressure exerted by oxygen is \(\frac{1}{3}\).

Step-by-step solution

Understanding the Problem

We need to find the fraction of the total pressure exerted by oxygen in a mixture of equal masses of methane (\(CH_4\)) and oxygen (\(O_2\)). First, calculate the moles of each gas to determine their partial pressures.

Determine Molar Masses

The molar mass of methane (\(CH_4\)) is approximately 16 g/mol (12 for carbon and 4 for hydrogen), and that of oxygen (\(O_2\)) is approximately 32 g/mol.

Calculate Moles of Each Gas

Let the mass of each gas be \(m\) grams. The number of moles of methane is \(\frac{m}{16}\) and the number of moles of oxygen is \(\frac{m}{32}\).

Calculate Mole Ratio

The mole ratio of methane to oxygen is \(\frac{m}{16} : \frac{m}{32}\), which simplifies to \(2:1\) when you divide both terms by the number of moles of oxygen, \(\frac{m}{32}\).

Calculate Pressure Ratio

According to Dalton's Law of Partial Pressures, the partial pressure of each gas in a mixture is proportional to its mole fraction. The total mole fraction is \(2 + 1 = 3\). The fraction of total pressure exerted by oxygen is therefore \(\frac{1}{3}\).

Key concepts

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It represents the portion of the total moles that a particular gas constitutes in the mixture. To calculate the mole fraction, you divide the number of moles of the gas of interest by the total number of moles in the mixture.
For example, in the mixture of methane (CH_4) and oxygen (O_2), the mole fraction of oxygen would be calculated as:

• Calculate moles of methane: \( \frac{m}{16} \)
• Calculate moles of oxygen: \( \frac{m}{32} \).
• Total moles: \( \frac{m}{16} + \frac{m}{32} \)
• Mole fraction of oxygen: \( \frac{\frac{m}{32}}{\frac{m}{16} + \frac{m}{32}} \).

These calculations help us understand how much 'space' each component occupies in the mixture when considering the entire collection of gases. Remember, the sum of mole fractions for all components in a mixture should equal 1.

Molar Mass Calculation

Calculating molar mass is a fundamental step in converting mass to moles, which is crucial in determining concentrations and mole fractions. The molar mass of a compound is the sum of the atomic masses of its constituent elements.
Let's break it down using our substances:

• Methane (CH_4):
  The molar mass is calculated by adding the mass of one carbon atom (12 \, \text{g/mol}) and four hydrogen atoms (1 \, \text{g/mol} \times 4 = 4 \, \text{g/mol}), giving us \(16 \, \text{g/mol} \).
• Oxygen (O_2):
  The molar mass is derived from two oxygen atoms, each weighing 16 \, \text{g/mol}, resulting in \(32 \, \text{g/mol} \).

Accurate molar mass calculations help ensure that we can determine the number of moles present when equal masses are mixed, a crucial step in evaluating the mole fraction and subsequent partial pressures.

Partial Pressure

Dalton’s Law of Partial Pressures states that in a gas mixture, each gas exerts pressure independently of the others. This pressure is proportional to the mole fraction of the gas in the mixture. The sum of all partial pressures is the total pressure of the gas mixture.
When solving our exercise, the partial pressure of oxygen in the mixture is computed based on its mole fraction. Here's the process:

• After finding the mole ratio and mole fraction, we note that the total number of moles is understood as \(2+1=3\).
• Oxygen, contributing to one part of this, thus makes up \(\frac{1}{3}\) of the total moles.
• Therefore, according to Dalton's Law, oxygen's partial pressure is also \(\frac{1}{3}\) of the total pressure.

By understanding partial pressure, we can predict how much pressure each gas component contributes, which is especially useful in various applications from chemistry to environmental science.