Question

A mixture of gases contains \(\mathrm{CH}_{4}, \mathrm{C}_{2} \mathrm{H}_{6},\) and \(\mathrm{C}_{3} \mathrm{H}_{8}\). If the total pressure is 1.50 atm and the numbers of moles of the gases present are 0.31 mole for \(\mathrm{CH}_{4}\) 0.25 mole for \(\mathrm{C}_{2} \mathrm{H}_{6},\) and 0.29 mole for \(\mathrm{C}_{3} \mathrm{H}_{8},\) calculate the partial pressures of the gases.

Short answer

The partial pressures of the gases are approximately \(0.5470 \, atm\) for \(CH_{4}\), \(0.4411 \, atm\) for \(C_{2}H_{6}\), and \(0.5118 \, atm\) for \(C_{3}H_{8}\)

Step-by-step solution

Calculate the total number of moles

First, it is important to calculate the total number of moles present in the mixture. This is done by adding the moles of each individual gas. So, it will be \(0.31 + 0.25 + 0.29 = 0.85 \, moles\)

Calculate the proportion of each gas

Next, we need to calculate the proportion of each gas. This is done by dividing the moles of each individual gas by the total moles calculated in Step 1. So for \(CH_{4}\) it is \(0.31/0.85 = 0.3647\), for \(C_{2}H_{6}\) it is \(0.25/0.85 = 0.2941\) and for \(C_{3}H_{8}\) it's \(0.29/0.85 = 0.3412\)

Calculate the partial pressures of each gas

Here we use these calculated proportions (from step 2) to work out the actual pressures. This is achieved by multiplying the mole ratio of each gas by the total pressure (1.50 atm). Therefore, the partial pressure of \(CH_{4}\) is \(0.3647 * 1.50 = 0.5470 \, atm\), the partial pressure of \(C_{2}H_{6}\) is \(0.2941 * 1.50 = 0.4411 \, atm\), and the partial pressure of \(C_{3}H_{8}\) is \(0.3412 * 1.50 = 0.5118 \, atm\)

Key concepts

Understanding Partial Pressure

In a gas mixture, partial pressure stands for the contribution each gas makes to the total pressure. It's like each type of gas has its own mini-pressure within the total pressure of the mix. Imagine a room filled with several types of balloons. Each balloon type adds pressure to the walls of the room. Similarly, gases in a container each exert their own partial pressure.

According to Dalton’s Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each component gas. We determine the partial pressure by knowing two things: the proportion of the gas in the mixture and the overall pressure. By multiplying the partial pressure ratio (or mole fraction, which we'll discuss further) of each gas by the total pressure, we can find out that gas's partial pressure.

This principle is often key in real-world applications such as calculating how much of each gas is in a mixture, which could be important when considering pressure settings for chemical reactions or even filling up a scuba tank.

Insight into the Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry used to understand the behavior of gases. This law combines three previously established laws: Boyle's Law, Charles's Law, and Avogadro's Law. It is expressed in the formula: \[ PV = nRT \] Where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

This equation provides a way to relate the physical properties of an ideal gas. While real gases deviate slightly from ideal behavior, especially under high pressure and low temperature, the Ideal Gas Law still describes their behavior quite accurately overall.

The Ideal Gas Law can help calculate unknown properties when you have some information available. For example, if you know the pressure, volume, and temperature, you can find out how many moles of a gas there are, which directly ties into finding the total moles in a gas mixture.

What is Mole Fraction?

The mole fraction is a way of expressing the concentration of a component in a mixture in terms of the number of moles. It's a dimensionless number that gives insight into how much of each substance is present in a mixture without being affected by pressure or temperature. The total of all mole fractions in a mixture always adds up to 1.

To find the mole fraction of a gas in a mixture, divide the number of moles of that specific gas by the total number of moles of all gases present. It is a ratio that represents the abundance of one component compared to the entire mixture.

For example, in our gas mixture, the mole fraction helps us calculate how much each gas is contributing to the total pressure. Knowing the mole fraction is crucial when calculating partial pressures, as it acts as the scaling factor between the total pressure of the gas mixture and the partial pressures of individual gases.