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Chapter 14: Problem 158

A gas mixture in a tank at \(600 \mathrm{R}\) and 20 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

Expert verified
Answer: The volume of the tank containing the gas mixture is 60.13 ft^3. The partial pressure of CO2 is 312.73 lbf/ft^2, and the partial pressure of CH4 is 2567.27 lbf/ft^2.

Step by step solution

01

Determine the total mass of the gas mixture

First, we need to calculate the total mass of the gas mixture by adding the mass of each individual gas. Total mass = Mass of CO2 + Mass of CH4 Total mass = 1 lbm + 3 lbm = 4 lbm
02

Calculate the moles of each gas

Next, we need to find the moles of each gas, using their respective molar masses. For this, we need to know the molar mass of CO2 and CH4. Molar mass of CO2: 44.01 lbm/lbmol Molar mass of CH4: 16.04 lbm/lbmol Now calculate moles of each gas: Moles of CO2 = (Mass of CO2) / (Molar mass of CO2) = 1 lbm / 44.01 lbm/lbmol = 0.02272 lbmol Moles of CH4 = (Mass of CH4) / (Molar mass of CH4) = 3 lbm / 16.04 lbm/lbmol = 0.18728 lbmol
03

Calculate the total moles and find the gas constant R

Now add the moles of each gas to find the total moles in the mixture: Total moles = Moles of CO2 + Moles of CH4 = 0.02272 lbmol + 0.18728 lbmol = 0.21 lbmol From the given pressure and temperature, we need to find the specific gas constant for the mixture (R). Since the mixture is composed of CO2 and CH4, we can express R as a weighted average of the individual gas constants: R = (Molar mass of CO2 * R_CO2 + Molar mass of CH4 * R_CH4) / (Molar mass of CO2 + Molar mass of CH4) We know R_CO2 = 55.2 ft*lbf/lbmol*R, and R_CH4 = 124.8 ft*lbf/lbmol*R. Therefore, R = (44.01 * 55.2 + 16.04 * 124.8) / (44.01 + 16.04) = 32330.56 / 60.05 = 538.45 ft*lbf/lbmol*R
04

Apply the Ideal Gas Law to find the volume of the tank

Now that we have the total moles and the gas constant for the mixture, we can use the Ideal Gas Law to find the volume of the tank. The Ideal Gas Law is given by: PV = nRT Volume of the tank: V = (nRT) / P We are given the pressure (P) as 20 psia, which we need to convert to lbf/ft^2: P = 20 psia * 144 lbf/ft^2/psi = 2880 lbf/ft^2 Using the values of P, n, R, and T, we can calculate the volume: V = (0.21 lbmol * 538.45 ft*lbf/lbmol*R * 600 R) / 2880 lbf/ft^2 = 173183.4 ft*lbf / 2880 lbf/ft^2 = 60.13 ft^3
05

Apply Dalton's Law of partial pressures to find the partial pressure of each gas

To find the partial pressure of each gas, we use Dalton's Law of partial pressures, which states that the total pressure is the sum of the individual partial pressures. Partial pressure of CO2: P_CO2 = (Moles of CO2 / Total moles) * Total pressure Partial pressure of CH4: P_CH4 = (Moles of CH4 / Total moles) * Total pressure Now, we can find the partial pressure of each gas: P_CO2 = (0.02272 lbmol / 0.21 lbmol) * 2880 lbf/ft^2 = 312.727 lbf/ft^2 P_CH4 = (0.18728 lbmol / 0.21 lbmol) * 2880 lbf/ft^2 = 2567.273 lbf/ft^2 The volume of the tank containing the gas mixture is 60.13 ft^3. The partial pressure of CO2 is 312.73 lbf/ft^2, and the partial pressure of CH4 is 2567.27 lbf/ft^2.

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