Question

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

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

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

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

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

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

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.

Key concepts

Dalton's Law of Partial Pressures

When dealing with gas mixtures, understanding Dalton's Law of Partial Pressures is essential. This law states that in a mixture of non-reacting gases, the total pressure exerted by the mixture is equal to the sum of the partial pressures of each individual gas. A partial pressure is the pressure that a single gas in a mixture would exert if it occupied the entire volume alone. This concept helps us understand how each component contributes to the overall pressure.

To apply Dalton's Law, you simply calculate each gas's partial pressure by multiplying its mole fraction by the total pressure. For example:

• Partial pressure of gas A: \( P_A = \left( \frac{\text{moles of gas A}}{\text{total moles}} \right) \times \text{total pressure} \)
• Partial pressure of gas B: \( P_B = \left( \frac{\text{moles of gas B}}{\text{total moles}} \right) \times \text{total pressure} \)

Understanding Dalton's Law allows you to break down complex gas mixtures into simpler, more manageable pressures of individual gases.

Gas Mixtures

Gas mixtures consist of two or more different gases present in the same volume without reacting chemically. Each gas in the mixture behaves as if it were alone in the container, a fundamental principle from the ideal gas laws. This assumption is key in both Dalton's and the Ideal Gas Law.

A gas mixture can be any combination of gases, such as air, which primarily consists of nitrogen, oxygen, and smaller amounts of other gases. The properties of a gas mixture, like pressure and volume, depend on the combined effect of all gases present.

When discussing gas mixtures, it is critical to remember:

• The total number of moles is the sum of moles of all gases.
• The total pressure is the sum of each gas's partial pressures.

Gas mixtures are prevalent in both natural and industrial processes, making the understanding of their behavior crucial in various scientific and engineering applications.

Molar Mass

Molar mass is the mass of one mole of a substance, usually measured in grams per mole (g/mol) or pound-moles (lbm/lbmol). It is a fundamental property in chemistry and physics that links mass to the amount of substance, allowing conversion between mass and moles.

To calculate the moles from a given mass, you use the formula:

• \( ext{Moles} = \frac{ ext{Mass}}{ ext{Molar Mass}} \)

For instance, in the exercise provided, the molar mass of \( \text{CO}_2 \) is 44.01 lbm/lbmol, and that of \( \text{CH}_4 \) is 16.04 lbm/lbmol. These values help determine the quantity of substance in moles from their respective masses given in pounds.

Understanding molar mass is crucial for working with chemical reactions and calculations, especially when dealing with gas laws where moles determine pressure and volume.

Partial Pressure Calculation

Calculating the partial pressure of each gas in a mixture is a straightforward application of Dalton's Law. This involves determining how much pressure each gas component contributes to the total pressure of the mixture. The formula to calculate the partial pressure of a gas is:

• \( P = \frac{ ext{Moles of the gas}}{ ext{Total moles}} \times ext{Total pressure} \)

In the exercise, to find the partial pressure of \( \text{CO}_2 \) and \( \text{CH}_4 \) within the gas mixture, the moles of each gas were divided by the total moles, then multiplied by the total pressure. This gives the share of pressure from each gas.

Partial pressure calculations are crucial in fields like chemistry and meteorology, where understanding individual gas behavior in a mixture can help predict both natural and industrial processes more accurately.