Question

Consider a $1.0$ -$\mathrm{L}$ container of neon gas at STP. Will the average kinetic energy, average velocity, and frequency of collisions of gas molecules with the walls of the container increase, decrease, or remain the same under each of the following conditions? a. The temperature is increased to ${100}^{\circ }\mathrm{C}$ b. The temperature is decreased to $-{50}^{\circ }\mathrm{C}$ c. The volume is decreased to $0.5\mathrm{L}$ d. The number of moles of neon is doubled.

Short answer

In summary: a. Temperature increased to ${100}^{\circ }\mathrm{C}$: - Average Kinetic Energy: Increase - Average Velocity: Increase - Frequency of Collisions: Increase b. Temperature decreased to $-{50}^{\circ }\mathrm{C}$: - Average Kinetic Energy: Decrease - Average Velocity: Decrease - Frequency of Collisions: Decrease c. Volume decreased to $0.5\mathrm{L}$: - Average Kinetic Energy: No change - Average Velocity: No change - Frequency of Collisions: Increase d. Number of moles of neon doubled: - Average Kinetic Energy: No change - Average Velocity: No change - Frequency of Collisions: Increase

Step-by-step solution

a. The temperature is increased to ${100}^{\circ }\mathrm{C}$

We start by calculating the new temperature in Kelvin, which is $100+273.15=373.15K$. When temperature increases, we expect the following changes in the gas properties: 1. Average Kinetic Energy: Since the average kinetic energy of gas molecules is directly proportional to the temperature (in Kelvin), it will increase. 2. Average Velocity: The average velocity of the gas molecules will also increase, as the square root of the temperature increases. 3. Frequency of Collisions: As the temperature increases, the gas molecules move faster. Consequently, the frequency of collisions with the walls of the container will increase.

b. The temperature is decreased to $-{50}^{\circ }\mathrm{C}$

We calculate the new temperature in Kelvin: $-50+273.15=223.15K$. When temperature decreases, we expect the following changes in the gas properties: 1. Average Kinetic Energy: Since average kinetic energy of gas molecules is directly proportional to the temperature (in Kelvin), it will decrease. 2. Average Velocity: The average velocity of gas molecules will also decrease, as the square root of the temperature decreases. 3. Frequency of Collisions: As the temperature decreases, the gas molecules move slower. Consequently, the frequency of collisions with the walls of the container will decrease.

c. The volume is decreased to $0.5\mathrm{L}$

When the volume of the container is decreased to $0.5\mathrm{L}$, we expect the following changes in the gas properties without changing the temperature: 1. Average Kinetic Energy: No change, as average kinetic energy depends only on the temperature. 2. Average Velocity: No change, as average velocity depends only on the temperature. 3. Frequency of Collisions: As the container volume decreases, there is less space for the gas molecules to move around, which means that they will collide more frequently with the walls of the container. Therefore, the frequency of collisions will increase.

d. The number of moles of neon is doubled

When the number of moles of neon in the container is doubled, we expect the following changes in the gas properties without changing the volume and the temperature: 1. Average Kinetic Energy: No change, as average kinetic energy depends only on the temperature. 2. Average Velocity: No change, as average velocity depends only on the temperature. 3. Frequency of Collisions: As the number of moles increases, there will be more gas molecules in the container. Consequently, there will be more collisions with the walls of the container. Therefore, the frequency of collisions will increase.

Key concepts

Average Velocity in Gases

Understanding the average velocity of gas molecules is crucial as it provides insights into the behavior of gas particles under varying conditions. In simplest terms, the average velocity refers to the speed at which gas particles travel within a container. It's important not to confuse average velocity with average speed; velocity is a vector quantity and includes directional information.

According to the kinetic theory of gases, the average velocity of particles increases with temperature. This is due to the increase in energy provided to the particles which makes them move more rapidly. For example, heating a gas will cause its molecules to pick up speed, increasing the average velocity. Conversely, cooling the gas will have the opposite effect, slowing the molecules down. This relationship is pivotal when examining gas behaviors according to the kinetic molecular theory. In the context of the given exercise, increasing the temperature to 100°C results in an increased average velocity, while decreasing it to -50°C causes the velocity to drop. If the volume or number of moles changes but the temperature remains constant, the average velocity remains unaffected.

Collision Frequency in Gases

Collision frequency in gases is another fundamental aspect of gas behavior. It indicates how often gas molecules collide with each other or with the walls of their container. The concept is significant as it influences properties like pressure and rate of reaction in confined gas systems.

The frequency of these collisions changes with alteration in any of the gas parameters like volume, temperature, or amount of gas present. At higher temperatures, molecules have more kinetic energy and move faster, leading to an increase in collision frequency. When the temperature is decreased, as in the exercise with a drop to -50°C, molecules have less kinetic energy and the collision frequency goes down.

Moreover, decreasing the volume of a container, as seen in the exercise from 1.0L to 0.5L, logically increases the collision frequency since molecules have less space to travel before a collision occurs. When the number of gas molecules is doubled, the presence of more particles leads to an increased number of collisions, assuming the size of the container and temperature remain constant.

Gas Laws and Temperature

The connection between gas laws and temperature is a key principle in understanding gas behavior. The gas laws are a series of individual laws that describe the relationship between the pressure, volume, temperature, and the quantity of gas. Temperature, measured on an absolute scale (Kelvin), is vital in these relationships because it reflects the average kinetic energy of the gas molecules.

For instance, Charles's Law states that the volume of a gas is directly proportional to its temperature when pressure is held constant. This means that if we raise the temperature, the gas expands to occupy a larger volume. Similarly, Gay-Lussac's Law describes how pressure of a gas is directly proportional to its temperature at a constant volume. In the exercise, increasing the temperature to 100°C accordingly results in higher pressure and volume if their counterparts are not fixed.

These gas laws also explain why changing the number of molecules, as in doubling the moles of neon gas, does not affect the temperature or average kinetic energy if the pressure and volume are maintained. The temperature is an intrinsic indicator of the energy within the system, and it is this energy that dictates how gas laws come into play during alterations of a gas system's conditions.