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 \(19^{\circ} \mathrm{C}\).

Short answer

The partial pressure of \(\mathrm{O}_{2}\) in the gas cylinder is 3.87 atm, the partial pressure of \(\mathrm{He}\) is 19.58 atm, and the total pressure is 23.45 atm.

Step-by-step solution

Convert the given information into usable units

First, we need to convert the given information into usable units. The volume of the gas cylinder is already given in liters, and we remember that 1 L equals 1,000 mL. The masses of \(\mathrm{O}_{2}\) and \(\mathrm{He}\) are given in grams. The temperature is given as a Celsius, so it must be converted to Kelvin by adding 273 to it: T(K) = 19 + 273 = 292 K.

Find the number of moles of O2 and He

Now, we need to find the number of moles (\(n\)) of each gas separately. To do this, we use the molar mass (MM) of each element in grams per mole. For \(\mathrm{O}_{2}\), MM = 32 g/mol, and for \(\mathrm{He}\), MM = 4 g/mol. We have: n_O2 = (51.2 g) / (32 g/mol) = 1.6 mol n_He = (32.6 g) / (4 g/mol) = 8.15 mol

Calculate the partial pressure of each gas

Now that we have the number of moles of each gas, we can apply the Ideal Gas Law to find the partial pressures of each gas separately. Ideal Gas Law states, PV = nRT We are given the volume (V = 10 L) and temperature (T = 292 K), and the ideal gas constant (R = 0.0821 L·atm/mol·K). Thus, the Ideal Gas Law for each gas becomes: For \(\mathrm{O}_{2}\): P_O2 = (n_O2 × R × T) / V P_O2 = (1.6 mol × 0.0821 L·atm/mol·K × 292 K) / (10 L) = 3.87 atm For \(\mathrm{He}\): P_He = (n_He × R × T) / V P_He = (8.15 mol × 0.0821 L·atm/mol·K × 292 K) / (10 L) = 19.58 atm

Calculate the total pressure

The total pressure can be found by adding the partial pressures of both gases: Total Pressure = P_O2 + P_He = 3.87 atm + 19.58 atm = 23.45 atm So, the partial pressure of \(\mathrm{O}_{2}\) in the gas cylinder is 3.87 atm, the partial pressure of \(\mathrm{He}\) is 19.58 atm, and the total pressure is 23.45 atm.