Question

A gas mixture in a tank at \(550 \mathrm{R}\) and 25 psia consists of $1 \mathrm{lbm}\( of \)\mathrm{CO}_{2}\( and \)3 \mathrm{lbm}\( of \)\mathrm{CH}_{4}$. Determine the volume of the tank and the partial pressure of each gas.

Short answer

Answer: The volume of the tank is approximately 22,846 ft³. The partial pressure of CO2 is approximately 6.25 psia, and the partial pressure of CH4 is approximately 18.75 psia.

Step-by-step solution

Calculate the number of moles for each gas

Since we're given the masses of both gases, we can easily find the number of moles of each gas using their respective molar masses. For CO2, the molar mass is 44.01 g/mol and for CH4, it is 16.04 g/mol. Moles of CO2 = (1 lbm * 453.59 g/lbm) / 44.01 g/mol = 10.3077 mol Moles of CH4 = (3 lbm * 453.59 g/lbm) / 16.04 g/mol = 84.9772 mol

Use the Ideal Gas Law for each gas separately

Now, we'll apply the Ideal Gas Law (PV=nRT) for both gases separately. For CO2: P(V_CO2) = n_CO2 * R * T For CH4: P(V_CH4) = n_CH4 * R * T Here, R is the ideal gas constant with a value of 10.73 [(psia*ft^3)/(lbmol*R)] and T is the temperature in Rankine, which is 550 R.

Find the partial pressures of each gas

We'll use the property of partial pressures and express the total pressure as the sum of the partial pressures of both gases: P_total = P_CO2 + P_CH4. Also, the volumes of each gas are equal to the volume of the tank, since they are both contained within the tank: V_CO2 = V_CH4 = V_tank. We can now rewrite the Ideal Gas Law equations and sum them up: P_total * V_tank = (n_CO2 + n_CH4) * R * T Substitute the values we obtained earlier: 25 psia * V_tank = (10.3077 mol + 84.9772 mol) * 10.73 [(psia*ft^3)/(lbmol*R)] * 550 R

Calculate the volume of the tank

Solve the resulting equation for the volume of the tank: V_tank = (95.2849 mol * 10.73 [(psia*ft^3)/(lbmol*R)] * 550 R) / 25 psia ≈ 22,846 ft^3

Calculate the partial pressures of CO2 and CH4

To find the partial pressures, we use the Ideal Gas Law and the volume of the tank we just found. For CO2: P_CO2 = n_CO2 * R * T / V_tank For CH4: P_CH4 = n_CH4 * R * T / V_tank P_CO2 ≈ (10.3077 mol * 10.73 [(psia*ft^3)/(lbmol*R)] * 550 R) / 22,846 ft^3 ≈ 6.25 psia P_CH4 ≈ (84.9772 mol * 10.73 [(psia*ft^3)/(lbmol*R)] * 550 R) / 22,846 ft^3 ≈ 18.75 psia #-} The volume of the tank is approximately 22,846 ft^3. The partial pressure of CO2 is approximately 6.25 psia, and the partial pressure of CH4 is approximately 18.75 psia.