Question

Equal weights 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) \(1 / 2\) (b) \(2 / 3\) (c) \(1 / 3 \times 273 / 298\) (d) \(1 / 3\)

Short answer

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

Step-by-step solution

Determine Molar Masses

First, we need to determine the molar masses of methane (CH extsubscript{4}) and oxygen (O extsubscript{2}). The molar mass of methane is 16 g/mol, while the molar mass of oxygen is 32 g/mol.

Calculate Moles from Equal Weight

Since equal weights of methane and oxygen are mixed and the molar mass of methane is 16 g/mol and oxygen is 32 g/mol, there are more moles of methane than oxygen for the same mass. For 16 grams, we have 1 mole of CH extsubscript{4} and 0.5 moles of O extsubscript{2} (because 32 g/mol for O extsubscript{2}).

Calculate Mole Fraction of Oxygen

The total moles of the gas mixture is the sum of the moles of CH extsubscript{4} and O extsubscript{2}. With 1 mole of CH extsubscript{4} and 0.5 moles of O extsubscript{2}, the total is 1.5 moles. The mole fraction of oxygen is the moles of O extsubscript{2} divided by the total moles: \(\text{Mole fraction of O}_2 = \frac{0.5}{1.5} = \frac{1}{3}\).

Determine Fraction of Pressure

Since the fraction of the total pressure exerted by a gas in a mixture is equal to its mole fraction, the fraction of the total pressure exerted by oxygen is \(\frac{1}{3}\).

Key concepts

Understanding Mole Fraction

When dealing with gas mixtures, it is essential to understand the concept of mole fraction. The mole fraction of a component in a mixture is the number of moles of that component divided by the total number of moles of all components in the mixture. This gives us a fraction that represents the proportion of the total mixture that one specific component occupies.

This is a dimensionless quantity, making it very straightforward to use in calculations. For instance, in our methane-oxygen exercise, the mole fraction of oxygen is calculated by dividing the moles of oxygen by the total moles in the mixture. By finding the mole fraction, you know what fraction of the mixture a particular gas represents in terms of its amount, making it a fundamental concept in gas laws and many chemical calculations.

Understanding Partial Pressure

In a gaseous mixture, each gas exerts pressure like it would if it occupied the space alone. This pressure is called the partial pressure of the gas. The total pressure of a gas mixture is the sum of the partial pressures of all the gases present. This concept is derived from Dalton's Law of Partial Pressures.

Partial pressure is directly proportional to the mole fraction of the gas in the mixture. Therefore, in calculations, if you know the mole fraction, you immediately know the fraction of the total pressure each gas contributes. For example, in the exercise, the partial pressure exerted by oxygen is directly equal to its mole fraction, which is one-third of the total pressure.

Exploring Molar Mass

Molar mass is a crucial metric in chemistry, providing the mass of one mole of a given substance. It plays a pivotal role when dealing with stoichiometry and converting mass into moles, especially in chemical reactions and calculations.

The molar mass is typically expressed in grams per mole (g/mol), and it is calculated by summing the atomic masses of all atoms in a molecule. For methanol (CH extsubscript{4}), the molar mass is 16 g/mol, while for oxygen (O extsubscript{2}), it is 32 g/mol. Understanding molar mass allows efficient conversion between mass of a substance and the number of moles, which is essential for further calculations in chemistry, such as determining mole fractions.

Chemical Calculations: From Mass to Pressure

Chemical calculations often involve converting between different properties such as mass, moles, and pressure to solve problems related to gas mixtures. In such calculations, understanding how to calculate moles from a given mass using the molar mass is critical.

For example, equal masses of methane and oxygen are considered. By using their molar masses, one can convert these masses to moles. From here, mole fractions can be determined, leading us to the partial pressures. Essentially, by understanding the molar relationships, the behavior of gases in the mixture (such as the fraction of pressure each gas exerts) can be precisely calculated. These calculations highlight the interconnectedness of mass, moles, and pressure in understanding and predicting the physical properties of gas mixtures.