Problem 72 - Chemistry: The Central Science in SI Units, Global Edition

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 \(15.50-\mathrm{L}\) vessel at \(15^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The mole fractions of O2, N2, and H2 in the mixture are 0.22624, 0.14008, and 0.63368, respectively. The partial pressures of O2, N2, and H2 are 0.55001 atm, 0.34075 atm, and 1.54161 atm, respectively.

Step 1: Find moles of each component

In order to find the moles of each gas, we will use their molar masses: Molar Mass of O2: 32 g/mol Molar Mass of N2: 28 g/mol Molar Mass of H2: 2 g/mol Now, we will find the moles of each gas: moles of O2 = \( \frac{15.08 g}{32 g/mol} = 0.47125 mol\) moles of N2 = \( \frac{8.17 g}{28 g/mol} = 0.29196 mol\) moles of H2 = \( \frac{2.64 g}{2 g/mol} = 1.32 mol\)

Step 2: Calculate mole fractions of each component

Mole fractions of each component in the mixture can be found using the following formula: \(Mole\,Fraction\,(X_i) = \frac{moles\,of\,component\,i}{total\,moles\,of\,all\,components}\) Total moles of all components = moles of O2 + moles of N2 + moles of H2 = 0.47125 + 0.29196 + 1.32 = 2.08321 Mole Fraction of O2 (\(X_{O_2}\)) = \( \frac{0.47125}{2.08321} = 0.22624\) Mole Fraction of N2 (\(X_{N_2}\)) = \( \frac{0.29196}{2.08321} = 0.14008 \) Mole Fraction of H2 (\(X_{H_2}\)) = \( \frac{1.32}{2.08321} = 0.63368 \)

Step 3: Find the total pressure of the mixture

We can find the total pressure of the mixture using the ideal gas law: \(PV = nRT\) where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant (0.0821 L⋅atm/mol⋅K), and T is the temperature in Kelvin. First, we need to convert the temperature from degrees Celsius to Kelvin: Temperature in Kelvin = 15 + 273.15 = 288.15 K Now we can find the total pressure of the mixture. We will use the total moles from Step 2 and the given volume of the vessel (15.50 L): Total Pressure (P) = \( \frac{(2.08321\,mol)(0.0821\,\frac{L\cdot atm}{mol\cdot K})(288.15\,K)}{15.5\,L} = 2.43237\, atm\)

Step 4: Calculate partial pressures of each component

To find the partial pressure of each component, we will use the mole fraction and the total pressure: Partial Pressure (\(P_i\)) = Mole Fraction (\(X_i\)) × Total Pressure (P) Partial Pressure of O2 (\(P_{O_2}\)) = 0.22624 × 2.43237 atm = 0.55001 atm Partial Pressure of N2 (\(P_{N_2}\)) = 0.14008 × 2.43237 atm = 0.34075 atm Partial Pressure of H2 (\(P_{H_2}\)) = 0.63368 × 2.43237 atm = 1.54161 atm The partial pressures of O2, N2, and H2 are 0.55001 atm, 0.34075 atm, and 1.54161 atm, respectively.

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry, allowing us to convert between the mass of a substance and the amount of substance (in moles). Molar mass is expressed in units of grams per mole (g/mol). Each element or compound has a unique molar mass, which is crucial for calculations involving chemical reactions or gas mixtures.

To calculate the molar mass of a compound, sum the molar masses of all the atoms present in it. For instance, the molar mass of oxygen gas ( \(O_2\)) is 32 g/mol because an individual oxygen atom has a molar mass of 16 g/mol. Hence, a molecule of oxygen gas comprises two oxygen atoms: 16 g/mol + 16 g/mol = 32 g/mol.

Knowing the molar mass allows us to determine the amount of substance given by its mass, using the formula:

• Moles of substance = \( \frac{Mass}{Molar\ Mass} \)

For example, to find the number of moles of oxygen in 15.08 grams, divide the mass by the molar mass of oxygen: \( \frac{15.08}{32} \approx 0.47125 \ mol \). Similar calculations can be done for nitrogen and hydrogen using their respective molar masses.

Partial Pressure

Partial pressure is the pressure exerted by a single component of a gas mixture, considering it occupies the entire volume alone at a given temperature. Understanding partial pressures is crucial because it helps in determining how different gases contribute to the overall pressure of a mixture.

According to Dalton's Law of Partial Pressures, in a gas mixture, the total pressure is the sum of the partial pressures of all components. The partial pressure of a gas can be calculated using its mole fraction and the total pressure of the mixture:

• Partial Pressure (\(P_i\)) = Mole Fraction (\(X_i\)) × Total Pressure (\(P\))

In practice, if the total pressure of a gas mixture is known, and the mole fraction of a gas component is determined, calculating its partial pressure is straightforward. For instance, in the provided exercise, the partial pressure of oxygen can be found using its mole fraction (\(0.22624\)) and the total pressure (\(2.43237\ atm\)), yielding \(0.55001\ atm\). This approach applies to other components like nitrogen and hydrogen as well.

Ideal Gas Law

The Ideal Gas Law is a mathematical relationship between the pressure, volume, and temperature of a gas, connecting them to the number of moles of gas present. It's succinctly represented by the equation:

• \(PV = nRT\)

In this equation, \(P\) stands for pressure, \(V\) for volume, \(n\) for moles, \(R\) for the ideal gas constant (approximately 0.0821 L·atm/mol·K), and \(T\) for temperature in Kelvin. This law is useful in determining unknown gas properties when the other properties are known.

To use the Ideal Gas Law, remember to convert all units appropriately, such as converting Celsius to Kelvin by adding 273.15. This conversion is essential because temperature must always be in Kelvin for gas law calculations.

For instance, if a gas mixture is held at 15°C in a 15.50 L container, convert that temperature to Kelvin (288.15 K) to use in calculations. In the provided exercise, the Ideal Gas Law was applied to find the total pressure of the mixture using the total moles and the given temperature and volume.