Problem 72 - Chemistry The Central Science In Si Units

Question

(a) What are the mole fractions of each component in a mixture of \(15.08 \mathrm{~g}\) of \(\mathrm{O}_{2}, 8.17 \mathrm{~g}\) of \(\mathrm{N}_{2},\) and \(2.64 \mathrm{~g}\) of \(\mathrm{H}_{2} ?\) (b) What is the partial pressure in atm of each component of this mixture if it is held in a 15.50-L vessel at \(15^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

Mole fractions: \(x_{O_2} = 0.2277\), \(x_{N_2} = 0.1409\), \(x_{H_2} = 0.6314\). Partial pressures: \(P_{O_2} = 0.718 \text{ atm}\), \(P_{N_2} = 0.444 \text{ atm}\), \(P_{H_2} = 1.992 \text{ atm}\).

Step 1: Calculate Moles of Each Gas

To find the moles, use the formula: \( n = \frac{m}{M} \), where \( m \) is the mass and \( M \) is the molar mass of the gas. The molar masses are: \( O_2 = 32.00 \text{ g/mol} \), \( N_2 = 28.02 \text{ g/mol} \), \( H_2 = 2.02 \text{ g/mol} \). \( n_{O_2} = \frac{15.08 \text{ g}}{32.00 \text{ g/mol}} = 0.47125 \text{ mol} \), \( n_{N_2} = \frac{8.17 \text{ g}}{28.02 \text{ g/mol}} = 0.2916 \text{ mol} \), \( n_{H_2} = \frac{2.64 \text{ g}}{2.02 \text{ g/mol}} = 1.30693 \text{ mol} \).

Step 2: Total Moles in the Mixture

Add the moles of each gas: \( n_{\text{total}} = n_{O_2} + n_{N_2} + n_{H_2} = 0.47125 + 0.2916 + 1.30693 = 2.06978 \text{ mol} \).

Step 3: Calculate Mole Fractions

Mole fraction \( x_i \) is calculated as \( x_i = \frac{n_i}{n_{\text{total}}} \). For \( O_2: x_{O_2} = \frac{0.47125}{2.06978} = 0.2277 \). For \( N_2: x_{N_2} = \frac{0.2916}{2.06978} = 0.1409 \). For \( H_2: x_{H_2} = \frac{1.30693}{2.06978} = 0.6314 \).

Step 4: Calculate Total Pressure using Ideal Gas Law

First, convert temperature to Kelvin: \( 15^\circ C = 273.15 + 15 = 288.15 K \). Then use the Ideal Gas Law: \( PV = nRT \). Rearrange to find \( P: P = \frac{nRT}{V} \), using \( R = 0.0821 \text{ L atm K}^{-1} \text{mol}^{-1} \). Thus, \( P = \frac{2.06978 \times 0.0821 \times 288.15}{15.50} = 3.154 \text{ atm} \).

Step 5: Calculate Partial Pressures

Multiply the mole fraction of each gas by the total pressure to find partial pressure. For \( O_2: P_{O_2} = x_{O_2} \times P_{\text{total}} = 0.2277 \times 3.154 = 0.718 \text{ atm} \). For \( N_2: P_{N_2} = 0.1409 \times 3.154 = 0.444 \text{ atm} \). For \( H_2: P_{H_2} = 0.6314 \times 3.154 = 1.992 \text{ atm} \).

Key Concepts

Partial Pressure

Partial pressure is a critical concept in understanding mixtures of gases. It describes the contribution each gas exerts on the total pressure of a mixture. When you have several gases sharing a container, each gas behaves as if it occupies the space alone, even though they are actually mixed. This pressure exerted by an individual gas in a mixture is known as partial pressure.
To calculate the partial pressure of a specific gas, you can use the formula:

• \( P_i = x_i \times P_{\text{total}} \)

where \( P_i \) is the partial pressure of gas \( i \), \( x_i \) is its mole fraction, and \( P_{\text{total}} \) is the total pressure of the gas mixture.
In this exercise, you multiplied each gas's mole fraction by the calculated total pressure, demonstrating how each component contributes to the mixture's overall pressure. This helps to illustrate the behavior of gases according to Dalton's Law of Partial Pressures.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as:

• \( PV = nRT \)

In this equation, \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) is the ideal gas constant \((0.0821 \ \text{L atm K}^{-1} \text{mol}^{-1})\), and \( T \) is the temperature in Kelvin.
When analyzing gas mixtures, the Ideal Gas Law comes in handy to determine the total pressure of the gases within a set volume. We used this equation to find the total pressure in the vessel by plugging in the total moles of gases, temperature in Kelvin, and the volume of the container. This application helps us understand how gases behave under different conditions of temperature and volume.

Molar Mass

Molar mass is the mass of one mole of a substance, typically measured in grams per mole (g/mol). It is a crucial factor in converting between the mass of a substance and the amount of substance present in moles. This conversion is essential because gases are often measured in moles rather than grams when using equations like the Ideal Gas Law.
For example, in this exercise, we began by calculating the number of moles for each gas using their respective masses and molar masses. The formula for this conversion is:

• \( n = \frac{m}{M} \)

where \( n \) represents the number of moles, \( m \) is the mass in grams, and \( M \) is the molar mass. This step is vital as it allows us to work with amounts of gases quantitatively in subsequent calculations, like determining mole fractions or total pressure.

Mixture of Gases

A mixture of gases refers to a collection of several gases sharing the same physical space but not chemically reacting with each other. Each gas retains its own properties and influences the overall behavior of the mixture.When you deal with gas mixtures, it's crucial to calculate specific properties like mole fractions and partial pressures. The mole fraction \((x_i)\) of each gas is calculated as the ratio of the number of moles of that gas to the total moles of all gases present:

• \( x_i = \frac{n_i}{n_{\text{total}}} \)

This fractional representation helps understand the composition of gases in the mixture.
In the exercise, after calculating the moles of \( O_2, N_2, \) and \( H_2 \), their individual influences on the total mixture were understood through their mole fractions. These fractions are crucial for calculating individual partial pressures, illustrating how each gas's presence impacts total pressure within a shared volume.