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) and (b) increase pressure with temperature rise.

Step 1: Understand the Relationship

According to the ideal gas law, when gas is kept at constant volume and the number of moles is fixed, its pressure is directly proportional to its temperature. This means that if the temperature increases, the pressure also increases.

Step 2: Analyze Options

We must consider the physical reasons why pressure increases with temperature at constant volume. When gas molecules have higher temperature, their kinetic energy and speed increase.

Step 3: Determine the Consequence of Increased Speed

As the speed of gas molecules increases, they collide with each other and the walls of the container more frequently and with greater force. This increases the rate of collisions, which, in turn, increases the pressure of the gas.

Step 4: Evaluate Other Options

An increase in molecular attraction actually reduces the pressure, as molecules tend to stay closer rather than colliding. The mean free path (average distance traveled by a molecule between collisions) doesn't explain the increase in pressure due to temperature rise at constant volume.

Step 5: Conclude With Findings

Both the increase in average molecular speed and the increase in rate of collisions lead to higher pressure when temperature rises. Therefore, options (a) and (b) correctly explain the observation.

Key Concepts

Pressure-Temperature Relationship

In the realm of gases, a fundamental principle called the Ideal Gas Law plays a significant role in explaining how pressure relates to temperature. Specifically, at constant volume and a fixed amount of gas, the pressure of the gas is directly proportional to its temperature. This means if you increase the temperature, the pressure will also increase because higher temperatures provide more energy to gas molecules.
The formula expressing this relationship can be written as \( P \propto T \) where \( P \) is pressure and \( T \) is temperature (both in Kelvin). Simply put, when you heat a gas at a constant volume, its molecules speed up and exert more force on the container's walls, thus increasing the pressure.

• Pressure increases: This results from the acceleration of the gas molecules.
• Temperature rises: Encourages more vigorous molecular motion.

Understanding this dynamic helps explain why, for example, a can of soda may bulge or even explode if left in a hot car.

Kinetic Molecular Theory

The Kinetic Molecular Theory (KMT) provides an insightful explanation of the behavior of gas particles at the microscopic level. Fundamentally, KMT posits that gas comprises a large number of tiny particles, usually molecules, that are in constant random motion.
The temperature of the gas is directly related to the average kinetic energy of these molecules. Therefore, a rise in temperature results in an increase in the average molecular speed and kinetic energy.

• Gases consist of many small particles.
• These particles are in random, constant motion.
• Their kinetic energy dictates the temperature of the gas.

By understanding these principles, one can infer that higher temperatures cause molecules to move more aggressively, thus influencing pressure and other properties significantly.

Gas Collisions

Gas collisions are a crucial factor in understanding pressure within a gas. These collisions occur between gas molecules and the container's walls. When temperature increases, the kinetic energy of the gas molecules goes up, causing them to move faster and collide more frequently and forcefully.

• Collisions with container walls lead to pressure exerted by the gas.
• Higher temperatures accelerate these collisions.

These frequent collisions explain the increase in pressure at constant volume as changes in molecular speed directly affect how often and how hard the molecules hit the walls.
Thus, the dynamic relationship between temperature and pressure in gases is largely defined by these continuous and vigorous collisions.

Molecular Speed

Molecular speed in gases is not only a fascinating concept but a pivotal aspect in predicting gas behavior under various conditions. At a microscopic level, the temperature of the gas determines the speed distribution of its molecules. An increase in temperature generally speeds up these molecules.
The speed at which these molecules move is linked closely to the temperature; higher speeds result in higher temperatures and vice versa.

• Molecular speed indicates how fast gas molecules are moving.
• Temperature affects this speed directly.

As molecules speed up, they collide more frequently and with greater energy, thus increasing the pressure in a constant volume scenario. Understanding this process showcases the intrinsic relationship between molecular speed and other physical properties of gases.