Problem 64 - 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 partial pressure of oxygen (O2) is \(P_{O2} = 66.62 \mathrm{~kPa}\), the partial pressure of helium (He) is \(P_{He} = 169.14 \mathrm{~kPa}\), and the total pressure is \(P_{total} = 235.76 \mathrm{~kPa}\).

Step 1: Convert mass to moles

First, we need to convert the mass of each of the gases to moles. We can do this by dividing the mass of each gas by its respective molar mass. For oxygen (O2): Molar mass of O2: \(\mathrm{M_{O2}} = 32~g/mol\) Number of moles of O2: \(n_{O2} =\frac{51.2 \mathrm{~g}}{32 \mathrm{~g/mol}} \) For helium (He): Molar mass of He: \(\mathrm{M_{He}} = 4~g/mol\) Number of moles of He: \(n_{He} =\frac{32.6 \mathrm{~g}}{4 \mathrm{~g/mol}} \)

Step 2: Apply Ideal Gas Law to find partial pressure of each gas

Now, we can use the ideal gas law to find the partial pressure of each gas inside the cylinder. Ideal Gas Law: \(PV = nRT\) Where: P = pressure V = volume n = number of moles R = Ideal Gas Constant, \(8.314 \mathrm{J/(mol \cdot K)}\) T = temperature in Kelvin We need to convert the temperature given in Celsius to Kelvin: \(T_K = T_C + 273.15\) For oxygen (O2): \(P_{O2}V = n_{O2}RT\) \(P_{O2} = \frac{n_{O2}RT}{V}\) For helium (He): \(P_{He}V = n_{He}RT\) \(P_{He} = \frac{n_{He}RT}{V}\)

Step 3: Calculate the total pressure

According to Dalton's Law of Partial Pressures, the total pressure in the cylinder is equal to the sum of the partial pressures of each gas. Total Pressure (P_total) = \(P_{O2} + P_{He}\)

Step 4: Substitute values and calculate

Now, we can substitute the known values into the equations and calculate the partial pressures of each gas and the total pressure. For oxygen (O2): \(P_{O2} = \frac{(\frac{51.2 \mathrm{~g}}{32 \mathrm{~g/mol}})(8.314 \mathrm{J/(mol \cdot K)})(19 + 273.15 \mathrm{~K})}{10.0 \mathrm{~L}}\) Calculate \(P_{O2}\). For helium (He): \(P_{He} = \frac{(\frac{32.6 \mathrm{~g}}{4 \mathrm{~g/mol}})(8.314 \mathrm{J/(mol \cdot K)})(19 + 273.15 \mathrm{~K})}{10.0 \mathrm{~L}}\) Calculate \(P_{He}\). Total Pressure (P_total) = \(P_{O2} + P_{He}\) Calculate the total pressure, \(P_{total}\).