Question

Gas Laws and Kinetic Theory of Gases I Shown here are two identical containers labeled \(\mathrm{A}\) and \(\mathrm{B}\). Container A contains a molecule of an ideal gas, and container B contains two molecules of an ideal gas. Both containers are at the same temperature. (Note that small numbers of molecules and atoms are being represented in these examples in order that you can easily compare the amounts. Real containers with so few molecules and atoms would be unlikely.) How do the pressures in the two containers compare? Be sure to explain your answer. Shown below are four different containers \((\mathrm{C}, \mathrm{D}, \mathrm{E}\) and \(\mathrm{F}\) ), each with the same volume and at the same temperature. How do the pressures of the gases in the containers compare? Container \(\mathrm{H}\) below has twice the volume of container G. How will the pressure in the containers compare? Explain your reasoning. How will the pressure of containers \(\mathrm{G}\) and \(\mathrm{H}\) compare if you add two more gas molecules to container \(\mathrm{H}\) ? Consider containers I and J below. Container J has twice the volume of container \(\mathrm{I}\). Container \(\mathrm{I}\) is at a temperature of \(100 \mathrm{~K},\) and container \(\mathrm{J}\) is at \(200 \mathrm{~K}\). How does the pressure in container I compare with that in container \(\mathrm{J} ?\) Include an explanation as part of your answer.

Short answer

Container B has double the pressure of A. Container pressures depend on molecule count and provided temperatures.

Step-by-step solution

Understanding Gas Laws

Pressure in a gas can be determined using the Ideal Gas Law given by the formula \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. For two gases in identical containers at the same temperature, the pressure is directly related to the number of molecules, as \( V \), \( R \), and \( T \) are constant.

Comparing Pressures in Containers A and B

Container A contains 1 molecule, while Container B contains 2 molecules. Given the ideal gas law relationship \( P \propto n \) with equal \( V \) and \( T \), the pressure in container B will be higher because it has more molecules. Therefore, the pressure in container B is twice that of container A.

Comparing Pressures in Containers C, D, E, and F

Since containers C, D, E, and F all have the same volume and temperature, the pressure in each container would depend solely on the number of molecules present. If the number of molecules in each container is not given or assumed equal, we cannot further specify their pressures without such details.

Comparing Pressures in Containers G and H

Container H has twice the volume of container G. Assuming the same number of molecules and temperature, and applying the ideal gas law \( P \propto \frac{1}{V} \), the pressure in container G would be twice that in container H since \( V_{H} = 2V_{G} \).

Impact of Additional Molecules in Container H

Adding two more molecules to container H increases the number of molecules, \( n \), thus increasing the pressure. For the same temperature and twice the volume of G, the additional molecules in H could balance or exceed G's pressure, contingent on the exact count of molecules and their distribution.

Comparing Pressures Considering Temperature Differences

For containers I and J, container J has twice the volume of container I. Container I is at \( 100 \mathrm{~K} \) and container J at \( 200 \mathrm{~K} \). According to the ideal gas law rearranged as \( P \propto \frac{T}{V} \), container J, having both a higher temperature and volume, will have a pressure comparable to that of container I, possibly higher, depending on molecular count.

Key concepts

Gas Pressure

Gas pressure is a fundamental concept in understanding the behavior of gases. It is defined as the force exerted by gas particles when they collide with the walls of their container. These collisions are a result of the kinetic energy possessed by the gas molecules.
The pressure (\(P\)) of a gas is directly proportional to the number of molecules (\(n\)). This means that as the number of molecules increases, the pressure increases, assuming the volume and temperature remain constant. Conversely, if you decrease the number of gas molecules, the gas pressure decreases. This relationship is described by the ideal gas law, which is expressed as:\[ PV = nRT \]where:

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

Thus, to compare pressures in different containers, like containers A and B from the exercise, we look at the number of gas molecules each one has, provided the volume and temperature are constant.

Kinetic Theory of Gases

The kinetic theory of gases is a crucial framework for understanding the behavior of gas particles. It explains gas properties in terms of the motion of molecules and how these motions result in observable characteristics such as pressure and temperature.
According to this theory:

• Gas particles are in constant, random motion.
• They frequently collide with each other and the walls of their container.
• These collisions are perfectly elastic, meaning no kinetic energy is lost.
• The average kinetic energy of gas molecules is proportional to the temperature of the gas.

These concepts help explain why gas pressure increases with an increase in the number of molecules, as more particles mean more collisions. The kinetic theory also illustrates why gas pressure decreases if the volume of the container increases (assuming constant temperature and number of molecules), because the particles have more space to move, resulting in fewer collisions with the walls per unit time.
Therefore, when comparing pressures in containers G and H, we can predict that a larger volume (H) would lead to a lower pressure than a smaller volume (G) if the number of molecules and temperature remains constant.

Temperature Effects on Gases

Temperature has a profound impact on the behavior of gases. When the temperature of a gas increases, the kinetic energy of its molecules increases. This means the molecules move faster and collide with the walls of the container more frequently and with greater force.
According to the ideal gas law:\[ P \propto \frac{T}{V} \]With a constant volume, an increase in temperature will result in an increase in pressure. Conversely, if the temperature decreases, gas pressure will also decrease, provided the volume and the number of gas molecules are unchanged.
This principle was illustrated in the comparison between containers I and J, where container J was at a higher temperature, 200 K, compared to container I at 100 K. Given that container J also has a larger volume, the combined effects of increased temperature and volume will influence the pressure. Yet, because pressure is proportional to temperature, container J might still have a pressure higher than or comparable to that of container I, depending on the molecular count and distribution in each container.