Problem 68 - Chemistry Central Science

Question

(a) What are the mole fractions of each component in a mixture of \(5.08 \mathrm{~g}\) of \(\mathrm{O}_{2}, 7.17 \mathrm{~g}\) of \(\mathrm{N}_{2}\), and \(1.32 \mathrm{~g}\) of \(\mathrm{H}_{2}\) ? (b) What is the partial pressure in \(a \operatorname{tm}\) of each component of this mixture if it is held in a 12.40-L vessel at \(15^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The mole fractions of O2, N2, and H2 in the mixture are approximately 0.254, 0.716, and 0.030, respectively. The partial pressures of O2, N2, and H2 in the mixture at 15°C and 12.40 L are approximately 0.689 atm, 1.943 atm, and 0.081 atm, respectively.

Step 1: Convert masses to moles

To find the moles of each gas, we need to use the molar mass of each gas. The molar masses are as follows: O2: 32 g/mol N2: 28 g/mol H2: 2 g/mol Now we can calculate the moles of each component: moles of O2 = mass of O2 / molar mass of O2 moles of N2 = mass of N2 / molar mass of N2 moles of H2 = mass of H2 / molar mass of H2

Step 2: Calculate the mole fractions

To find the mole fraction, we need to add the moles of all components and then divide the moles of each component by the total moles: Mole fraction of O2 = moles of O2 / (moles of O2 + moles of N2 + moles of H2) Mole fraction of N2 = moles of N2 / (moles of O2 + moles of N2 + moles of H2) Mole fraction of H2 = moles of H2 / (moles of O2 + moles of N2 + moles of H2)

Step 3: Calculate the total pressure using the Ideal Gas Law

First, we need to convert the temperature from Celsius to Kelvin, then we will use the Ideal Gas Law: T(K) = 15 °C + 273.15 = 288.15 K PV = nRT Where: P = total pressure in atm V = volume in liters (12.40 L) n = total moles of gas (we found it in step 2) R = gas constant (0.0821 atm.L/mol.K) T = temperature in Kelvin (288.15 K) Now we can solve for P (total pressure): P = nRT / V

Step 4: Calculate the partial pressures of each component

To find the partial pressure of each component, we will use the mole fraction we calculated in Step 2 and the total pressure we found in Step 3: Partial pressure of O2 = Mole fraction of O2 * Total pressure Partial pressure of N2 = Mole fraction of N2 * Total pressure Partial pressure of H2 = Mole fraction of H2 * Total pressure

Key Concepts

Partial Pressure

In a gas mixture, each gas contributes to the total pressure in proportion to its amount or mole fraction. The pressure contribution from a single type of gas is known as its partial pressure. Understanding this concept is vital for dealing with gas mixtures. The partial pressure can be calculated using the formula:

• Partial pressure of a gas = Mole fraction of the gas × Total pressure of the mixture

This formula means that each gas's partial pressure is derived from what fraction of the total moles it represents, multiplied by the total pressure of the entire mixture. This relationship underscores the idea that in an ideal gas mixture, each component behaves as if it alone occupies the entire volume. When solving problems, make sure to find the mole fraction first and then apply it to determine the respective partial pressures.

Ideal Gas Law

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

• \[PV = nRT\]

where:

• \(P\) is the pressure in atmospheres (atm)
• \(V\) is the volume in liters (L)
• \(n\) is the amount of substance in moles (mol)
• \(R\) is the gas constant, 0.0821 atm·L/mol·K
• \(T\) is the temperature in Kelvin (K)

To use the Ideal Gas Law effectively, always convert all temperatures to Kelvin and make sure volume and pressure are in consistent units. This equation is fundamental for calculating any missing quantity when the other three are known, such as determining the total pressure when volume, temperature, and moles of gas are given.

Molar Mass Calculation

Calculating molar mass is crucial when converting between grams and moles. The molar mass is the total mass in grams of one mole of molecules or atoms of a substance. For any chemical element, it is equivalent to the atomic mass expressed in units of grams per mole (g/mol). For example:

• For \(\text{O}_2\), molar mass = 32 g/mol
• For \(\text{N}_2\), molar mass = 28 g/mol
• For \(\text{H}_2\), molar mass = 2 g/mol

To convert from mass to moles, the formula is:

• \(\text{Moles} = \frac{\text{Mass (g)}}{\text{Molar mass (g/mol)}}\)

By understanding how to calculate molar masses and apply conversions, you can handle various chemical calculations effectively, whether determining reaction quantities or identifying unknown substances.