Question

Explain in molecular terms why the pressure of a gas increases with increasing temperature when the volume is held constant.

Short answer

In molecular terms, the pressure of a gas increases with increasing temperature (at constant volume) because temperature is a measure of the average kinetic energy of gas molecules, and higher temperatures mean the molecules move faster. This increased molecular speed leads to more frequent and forceful collisions with the walls of the container, which is what we perceive as an increase in pressure.

Step-by-step solution

Understanding the Basics of Gas Pressure

At the microscopic level, a gas is composed of a large number of molecules moving in random directions. These molecules collide with each other and with the walls of their container. Each time a molecule hits the wall, it exerts a small force on it. The cumulative effect of these collisions is what we perceive as pressure. So pressure is effectively the result of the many molecular collisions with the walls of a container per unit area.

Understanding the Effect of Temperature on Molecular Speed

Temperature is a measure of the average kinetic energy of molecules in a substance. Higher temperatures mean increased kinetic energy. In the case of gases, this added kinetic energy is manifested as an increase in the speed of gas molecules. As temperature rises, the gas molecules move faster.

Connecting Temperature, Molecular Speed and Pressure

When the gas molecules move faster due to the increase in temperature, they collide with the container’s walls more frequently and with more force. As established in step 1, pressure is directly related to the number and force of these collisions. Therefore, given the same volume, if the gas molecules are moving faster (due to increased temperature), they will collide more often and with more force with the walls of the container, leading to a higher pressure.

Key concepts

Kinetic Molecular Theory of Gases

The kinetic molecular theory of gases provides a framework for understanding the behavior of gas molecules at the molecular level. According to this theory, gas molecules are in constant, random motion and the temperature of the gas corresponds to the average kinetic energy of these molecules.

Here's the concept broken down:

• Gas consists of many small particles, or molecules, that are in constant motion.
• These molecules are very far apart compared to their size. This means that the volume of the molecules themselves is negligible relative to the volume of the container.
• Gas molecules collide with each other and with the walls of the container without losing energy in perfectly elastic collisions.
• The speed at which these molecules move is directly related to the temperature; the higher the temperature, the faster the molecules move.

When you consider a gas under these assumptions, it becomes clear why changes in temperature affect gas pressure, a concept we will explore further in the following sections.

Temperature's Effect on Gas Molecules

Temperature plays a pivotal role in determining the behavior of gas molecules. At the molecular level, an increase in temperature translates into an increase in kinetic energy of the gas molecules. This is because temperature is, in essence, a measure of this energy.

Here’s how temperature affects the molecules:

• At low temperatures, gas molecules move more slowly because their kinetic energy is lower.
• As you increase the temperature, the kinetic energy of the molecules increases too, and they start to move more quickly.
• This increase in molecular speed is significant because it changes the way the molecules interact with their environment, particularly with the walls of the container in which they are held.

Understanding this concept allows us to predict how a gas will behave when subjected to changes in temperature, assuming other factors like volume and the amount of gas remain constant.

Relationship Between Gas Pressure and Temperature

The relationship between gas pressure and temperature is more than just a theoretical concept; it has practical implications in everyday phenomena. According to Gay-Lussac's law, which describes this relationship, the pressure of a gas at constant volume is directly proportional to its absolute temperature.

Exploring the relationship in detail:

• As the temperature increases, so does the average kinetic energy of the gas molecules. This means the molecules are moving faster.
• At a faster speed, molecules collide with the walls of a container more frequently and with more force.
• Since pressure is the result of these molecular collisions with the container's walls, an increase in frequency and force of collisions leads to an increase in pressure.

The law is mathematically presented as \( P \thicksim T \text{ (at constant volume)} \text{,} \) where P stands for pressure and T stands for temperature. This proportional relationship allows us to predict how pressure will change as temperature changes, as long as the volume of the gas does not change.