Question

Find the partial pressure of carbon dioxide in a gas mixture with a total pressure of 30.4 kPa if the partial pressures of the other two gases in the mixture are 16.5 kPa and 3.7 kPa

Short answer

The partial pressure of carbon dioxide in the gas mixture can be found using the formula \(P_3 = P_{total} - P_1 - P_2\). Plugging in the given values, we get \(P_3 = 30.4 - 16.5 - 3.7\), which gives us a partial pressure of carbon dioxide of \(P_3 = 10.2\) kPa.

Step-by-step solution

Write the formula for the total pressure in a gas mixture

We can use the formula for the total pressure in a gas mixture, which states that the total pressure (P_total) is the sum of the partial pressures of all component gases. Mathematically, the formula is given by: \(P_{total} = P_1 + P_2 + P_3\) Where P_total is the total pressure of the mixture, and P_1, P_2, and P_3 are the partial pressures of the component gases.

Assign the known values to the variables

In this exercise, we are given the following values: Total pressure (P_total) = 30.4 kPa Partial pressure of the first gas (P_1) = 16.5 kPa Partial pressure of the second gas (P_2) = 3.7 kPa We need to find the partial pressure of carbon dioxide (P_3).

Rearrange the formula to solve for the unknown variable

We need to find the partial pressure of carbon dioxide (P_3), so we rearrange the formula to solve for P_3: \(P_3 = P_{total} - P_1 - P_2\)

Plug in the known values into the formula and solve for P_3

Now, we plug in the known values of the total pressure and the partial pressures of the first and second gases into the formula to find the partial pressure of carbon dioxide: \(P_3 = 30.4 - 16.5 - 3.7\) After performing the calculations, we get: \(P_3 = 10.2\)

Write the final answer

The partial pressure of carbon dioxide in the gas mixture is 10.2 kPa.

Key concepts

Understanding Gas Mixtures

When we talk about gas mixtures, we refer to a combination of different gases occupying the same volume and essentially mixed thoroughly. Each component gas in a mixture exerts a pressure, which would be the pressure the gas would exert if it were alone in the container. This is known as its partial pressure. In any given mixture, gases behave independently of each other, and their individual pressures depend on the amount of gas and the temperature of the mixture.

For instance, the air we breathe is a gas mixture mostly consisting of nitrogen and oxygen, along with other gases like argon and carbon dioxide in much smaller amounts. When evaluating gas mixtures, it's essential to consider the interactions (or lack thereof) between the different gases, as these interactions can be minimal, allowing us to apply the ideal gas law individually to each component gas.

Total Pressure in Gas Mixtures

The total pressure of a gas mixture is defined as the sum of the partial pressures of all the component gases within the mixture. An easy way to visualize this is by picturing a container with various gases; each gas particle contributes to the total force exerted on the walls of the container. The more gas particles present, the higher the total pressure.

Now, consider a balloon - the pressure inside the balloon (total pressure) is due to each individual gas contributing its share (partial pressures). The concept of total pressure is important because it allows us to understand and predict how gas mixtures will behave under different conditions, such as changes in volume or temperature. Calculating the total pressure is vital in many scientific and industrial applications, such as scuba diving, where the pressure of individual gases must be managed to prevent health hazards like decompression sickness.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a fundamental principle in chemistry that states that the total pressure of a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases. The law was formulated by John Dalton, an English chemist, in the 1800s.

Dalton's Law can be mathematically represented as \(P_{total} = P_1 + P_2 + P_3 + ... + P_n\), where \(P_{total}\) is the total pressure, and \(P_1, P_2, P_3\), etc., are the individual partial pressures of each gas. This law is particularly useful when dealing with gas mixtures, as it provides a straightforward way to determine the behavior and pressure of each component in the mixture.

For students solving problems related to gas mixtures, it's important to be comfortable using Dalton's Law. It's essential not only for calculating unknown partial pressures but also for understanding the behavior of gases when they are mixed. Additionally, comprehension of Dalton's Law is crucial in fields like meteorology, respiratory physiology, and anesthesiology, where gas mixtures play a significant role.