Question

At constant volume for a fixed number of a moles of gas, the pressure of the gas increases with the rise in temperature due to (a) increase in average molecular speed (b) increase in rate of collisions (c) increase in molecular attraction (d) increase in mean free path

Short answer

(a) Increase in average molecular speed and (b) increase in rate of collisions.

Step-by-step solution

Understanding the Question

We need to determine why the pressure of a gas increases with a rise in temperature at constant volume and a fixed number of moles.

Relation between Pressure and Temperature

According to Gay-Lussac's law, for a given amount of gas at constant volume, the pressure of a gas is directly proportional to the temperature in Kelvin. This can be expressed as \( P \propto T \) or \( \frac{P}{T} = \text{constant} \).

Analyzing the Effect of Temperature

As temperature increases, the kinetic energy of the gas molecules increases. This increases their speed, causing them to hit the walls of the container more frequently and with greater force.

Identifying Causes of Increased Pressure

Increased molecular speed leads to more frequent and forceful collisions with the container walls, thereby increasing pressure. This corresponds to option (a): increase in average molecular speed, and to option (b): increase in rate of collisions.

Eliminating Other Options

Since pressure is related to collisions and speed, we can rule out (c) increase in molecular attraction and (d) increase in mean free path, as these do not directly relate to increased collisions or force on the container walls due to temperature rise.

Key concepts

Gay-Lussac's Law

Gay-Lussac's Law is a cornerstone principle in the study of gases. It describes the direct relationship between the pressure and temperature of a gas, keeping volume constant.

This law can be mathematically represented by the equation \( \frac{P}{T} = k \), where \( P \) is the pressure of the gas, \( T \) is the absolute temperature in Kelvin, and \( k \) is a constant for a given sample of gas at a fixed volume. This means that if the temperature of a gas increases, so does its pressure as long as the volume doesn't change.

In practical terms, if you heat a sealed rigid container filled with gas, the gas molecules gain energy and move faster. This increased activity causes them to strike the vessel's walls more forcefully and frequently, raising the pressure inside the container. Gay-Lussac's Law underpins many safety guidelines in industries dealing with gases, emphasizing the need for pressure-resistant containers capable of withstanding temperature changes.

Kinetic Molecular Theory

The Kinetic Molecular Theory provides a framework for understanding how gases behave at the molecular level. It assumes that gases are composed of a large number of tiny particles, which are in constant, random motion.

This theory helps explain key gas laws, like Gay-Lussac's Law, by linking them to molecular movement. According to this theory, temperature is directly proportional to the average kinetic energy of gas particles. As the temperature increases, so does the kinetic energy of the gas molecules. This is because when heated, these particles move more rapidly, which results in increased pressure when they collide with the walls of their container.

Some important aspects of the Kinetic Molecular Theory include:

• Particles move in straight lines until they collide with something.
• Most of the volume of a gas is empty space.
• Collisions between gas molecules and the walls of the container are perfectly elastic.

This theory not only helps in understanding the behavior of gases under varying conditions but also in predicting how changes in conditions like temperature will affect gas properties.

Temperature and Pressure Relationship

The relationship between temperature and pressure in gases is fundamental to understanding how gases behave in different conditions.

This relationship is crucial in various applications, from inflating tires to understanding atmospheric phenomena. At a basic level, when the temperature of a gas rises, the kinetic energy of its particles increases. Consequently, these energetic particles collide more often and with greater force against the walls of their container, leading to an increase in pressure. This is because pressure is essentially the force exerted by gas particles per unit area of the container walls.

Understanding this relationship helps predict how gases will react to changes in their environment. For instance, if you leave a basketball in the sun, its pressure might increase significantly due to the rise in temperature. This is the same principle that governs the behavior of gas in a sealed container or how weather balloons expand as they rise through the atmosphere.

To summarize, the pressure of a gas is directly proportional to its temperature when volume remains constant, evidencing the delicate balance and interconnectedness of physical variables in gas dynamics.