Question

The contents of a tank of natural gas at 1.20 atm is analyzed. The analysis showed the following mole percents: 88.6% CH4, 8.9% C2H6, and 2.5% C3H8. What is the partial pressure of each gas in the tank?

Short answer

Answer: The partial pressures are as follows: - Partial Pressure of CH4 (P_CH4): 1.0632 atm - Partial Pressure of C2H6 (P_C2H6): 0.1068 atm - Partial Pressure of C3H8 (P_C3H8): 0.03 atm

Step-by-step solution

Calculate the mole fraction of each gas

The mole fraction of a gas is the ratio of the moles of a particular gas to the total moles of gas in the mixture. In this case, we have the mole percent of each gas, which we can use to find the mole fraction of each gas. Let's calculate the mole fraction for each gas: Mole fraction of CH4 (X_CH4) = (88.6/100) = 0.886 Mole fraction of C2H6 (X_C2H6) = (8.9/100) = 0.089 Mole fraction of C3H8 (X_C3H8) = (2.5/100) = 0.025

Calculate the partial pressure of each gas using the mole fraction and total pressure

Now that we have the mole fraction of each gas, we can use the formula for calculating the partial pressure of each gas using the relation: Partial Pressure of a gas = Mole fraction of the gas × Total Pressure Total Pressure (P_total) = 1.20 atm Let's calculate the partial pressure of each gas: Partial Pressure of CH4 (P_CH4) = X_CH4 × P_total = 0.886 × 1.20 = 1.0632 atm Partial Pressure of C2H6 (P_C2H6) = X_C2H6 × P_total = 0.089 × 1.20 = 0.1068 atm Partial Pressure of C3H8 (P_C3H8) = X_C3H8 × P_total = 0.025 × 1.20 = 0.03 atm

Report the partial pressures of each gas

Now we have calculated the partial pressures of each gas in the tank: Partial Pressure of CH4 (P_CH4): 1.0632 atm Partial Pressure of C2H6 (P_C2H6): 0.1068 atm Partial Pressure of C3H8 (P_C3H8): 0.03 atm

Key concepts

Mole Fraction

Understanding the concept of mole fraction is critical when working with mixtures, especially gases in this context. The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a particular substance to the total number of moles of all substances present.

For example, if we have a mixture of gases containing 2 moles of nitrogen and 3 moles of oxygen, the mole fraction of nitrogen would be calculated as: \[ X_{\text{N}_2} = \frac{2}{2 + 3} = \frac{2}{5} \]and the mole fraction of oxygen as: \[ X_{\text{O}_2} = \frac{3}{2 + 3} = \frac{3}{5} \].

When mole fractions are given in percentage form, like in the exercise provided, they simply represent the percentage of the total moles that a particular gas comprises. The importance of the mole fraction comes into play when calculating other properties of the mixture, such as partial pressures.

Gas Mixture

A gas mixture consists of two or more different gases occupying the same volume. In a mixture, each gas behaves independently and contributes to the overall pressure of the mixture. Natural gas, for example, is a mixture primarily of methane (CH4), with smaller amounts of other hydrocarbons like ethane (C2H6) and propane (C3H8).

The behavior of each gas in a mixture can be predicted by understanding its individual properties and how it interacts with other gases. The conditions of the mixture, such as temperature and pressure, also play a significant role in determining the behavior of the gases. By assuming ideal behavior (under the conditions set by the Ideal Gas Law), the individual properties of each gas can help us calculate their individual effects on the overall properties of the mixture, such as total pressure.

Dalton's Law of Partial Pressures

When learning about gas mixtures, Dalton's Law of Partial Pressures is a fundamental principle. This law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of each individual gas. A gas's partial pressure is the pressure that gas would exert if it were alone in the container.

The law can be mathematically expressed as: \[ P_{\text{total}} = P_1 + P_2 + P_3 + \ldots \] where \( P_{\text{total}} \) is the total pressure of the mixture, and \( P_1, P_2, P_3, \ldots \) are the partial pressures of the individual gases.

For the given exercise, we used the formula \[ P_{\text{gas}} = X_{\text{gas}} \times P_{\text{total}} \] to calculate the partial pressure of each gas in the mixture. This formula is a direct application of Dalton's Law, as the mole fraction of gas represents the ratio of that gas's contribution to the total pressure. By multiplying the mole fraction by the total pressure, we obtain the partial pressure for that specific gas.