Problem 66 - Chemistry The Central Science

Question

A deep-sea diver uses a gas cylinder with a volume of \(10.0 \mathrm{~L}\) and a content of \(51.2 \mathrm{~g}\) of \(\mathrm{O}_{2}\) and \(32.6 \mathrm{~g}\) of He. Calculate the partial pressure of each gas and the total pressure if the temperature of the gas is \(19^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The number of moles of \(\mathrm{O}_{2}\) and He are \(n_{O_2} = \frac{51.2 \mathrm{~g}}{32.00 \frac{\mathrm{g}}{\mathrm{mol}}}\) and \(n_{He} = \frac{32.6 \mathrm{~g}}{4.00 \frac{\mathrm{g}}{\mathrm{mol}}}\) respectively. With the temperature in Kelvin, \(T = 292.15 \mathrm{K}\), we can calculate the partial pressure of each gas using the ideal gas law: \(P_{O_2} = \frac{n_{O_2} \times R \times T}{V}\) and \(P_{He} = \frac{n_{He} \times R \times T}{V}\). Finally, the total pressure is found by summing the partial pressures: \(P_{total} = P_{O_2} + P_{He}\).

Step 1: Find the number of moles of \(\mathrm{O}_{2}\) and He.

To find the number of moles of each gas, we need to divide the mass of the gas by its molar mass: Number of moles of \(\mathrm{O}_{2}\): \(n_{O_2} = \frac{51.2 \mathrm{~g}}{32.00 \frac{\mathrm{g}}{\mathrm{mol}}} \) Number of moles of He: \(n_{He} = \frac{32.6 \mathrm{~g}}{4.00 \frac{\mathrm{g}}{\mathrm{mol}}}\) Calculate the values of \(n_{O_2}\) and \(n_{He}\).

Step 2: Convert the temperature to Kelvin.

The ideal gas law requires the temperature to be in Kelvin. To convert the temperature from Celsius to Kelvin, add 273.15: \(T(K) = 19^{\circ} \mathrm{C} + 273.15 = 292.15 \mathrm{K}\)

Step 3: Calculate the partial pressure of each gas.

To find the partial pressure of each gas, we'll use the ideal gas law \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin. Rearranging the equation to find \(P\), we get: \(P = \frac{nRT}{V}\) Calculate the partial pressure of each gas, using the number of moles found in Step 1, the temperature in Kelvin from Step 2, the gas constant \(R = 0.0821 \frac{\mathrm{L} \times \mathrm{atm}}{\mathrm{mol} \times \mathrm{K}}\), and the volume of the gas cylinder \(V = 10.0 \mathrm{L}\): Partial pressure of \(\mathrm{O}_{2}\): \(P_{O_2} = \frac{n_{O_2} \times R \times T}{V}\) Partial pressure of He: \(P_{He} = \frac{n_{He} \times R \times T}{V}\)

Step 4: Calculate the total pressure.

To find the total pressure, add the partial pressures of each gas: Total pressure: \(P_{total} = P_{O_2} + P_{He}\) Now, the partial pressure of each gas, as well as the total pressure in the gas cylinder, are found.