Problem 52 - A Complete Resource Book in Chemistry for JEE Main

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

Step-by-Step Solution

Answer

(a) Increase in average molecular speed and (b) increase in rate of collisions lead to higher pressure.

Step 1: Read the Question Carefully

The exercise asks us to determine why the pressure of a gas increases with a rise in temperature at constant volume and a fixed number of moles. We have four options to consider: increase in average molecular speed, rate of collisions, molecular attraction, and mean free path.

Step 2: Understand the Relationship Between Pressure and Temperature

When a gas is heated, its molecules move faster. According to the kinetic theory of gases, the pressure of a gas is related to the kinetic energy of the molecules, which depends on their speed. Therefore, as temperature increases, the average speed of the molecules increases, impacting pressure.

Step 3: Analyze Each Option

(a) Increase in average molecular speed: This directly affects pressure as increased speed means more forceful impacts with the container walls. (b) Increase in rate of collisions: As temperature rises, molecules move faster, leading to more collisions, which contributes to increased pressure. (c) Increase in molecular attraction: Not significant for ideal gases as these are considered negligible in the ideal gas law, and would typically decrease pressure due to cohesion. (d) Increase in mean free path: A higher temperature doesn't increase mean free path; it would remain constant at a fixed volume.

Step 4: Select the Correct Answer

The main factors leading to increased pressure when temperature increases at constant volume are options (a) an increase in average molecular speed and (b) an increase in rate of collisions. These are closely related and both contribute to the increase in pressure.

Key Concepts

Average Molecular Speed

In the realm of kinetic theory, the average molecular speed is a measure of how quickly molecules move within a gas. This speed tends to increase when the temperature rises and the volume stays constant.
When the temperature of a gas increases, it means that the thermal energy supplied to the gas also rises. This energy is translated into kinetic energy, causing gas molecules to move faster.
Consequently, with faster speeds, molecules will collide more forcefully with the walls of the container. These frequent and vigorous collisions are pivotal for understanding changes in pressure.

Pressure-Temperature Relationship

The pressure-temperature relationship in gases is fundamentally linked through the kinetic energy of particles. When the temperature of a gas is increased, its particles gain kinetic energy and move with greater speed.
This enhanced movement results in an increased rate of collision of gas molecules against the container walls, leading to higher pressure.
The kinetic theory of gases explains this concept:

• The pressure of a gas is caused by collisions of particles with the walls of its container.
• As the temperature increases, the frequency and force of these collisions increase, thereby increasing pressure.

This relationship is crucial in understanding why gases behave the way they do under various temperatures. The more we heat a gas at constant volume, the more it wants to "expand" by increasing pressure.

Ideal Gas Law

The ideal gas law is an equation of state that describes how pressure, volume, and temperature interact for an ideal gas. It is succinctly expressed as: \[ PV = nRT \] where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature.
This equation can help us see how pressure is directly proportional to temperature when volume and moles of gas are constant.

• A rise in temperature means more kinetic energy for the particles.
• In a fixed volume, this increase in kinetic energy results in higher pressure.

The ideal gas law simplifies the complex dynamics of gas behavior into a more manageable form, helping us predict how changes in temperature affect gas pressure.