Problem 14 - Arihant AIEEE Physics

Question

When without change in temperature, a gas is forced in a smaller volume, its pressure increases because its molecules : (a) strike the unit area of the container wall more often (b) strike the unit area of the container wall at higher speed (c) strike the unit area of container wall with greater force id) have more energy

Step-by-Step Solution

Answer

Molecules strike the unit area more often (option a).

Step 1: Understanding the Scenario

The situation described involves a gas being compressed into a smaller volume with no change in temperature. Under these circumstances, we need to analyze how the behavior of gas molecules changes, particularly in relation to the pressure exerted on the walls of the container.

Step 2: Pressure and Molecular Collisions

Pressure in a gas is caused by molecules colliding with the walls of the container. When the volume of the container decreases without a change in temperature, the molecules have less space to move, leading to more frequent collisions with the walls as they travel shorter distances before hitting the wall again.

Step 3: Evaluating the Options

We need to evaluate each option based on the scenario: - (a) More frequent collisions occur as molecules have less distance to travel. - (b) Speed doesn't change with constant temperature. - (c) Force per collision remains constant without an increase in speed or mass. - (d) Energy remains constant since temperature is unchanged.

Step 4: Selecting the Correct Answer

From the evaluation, only option (a) correctly explains why pressure increases: the molecules strike the unit area of the container wall more often due to reduced volume, leading to more frequent collisions and thus increased pressure.

Key Concepts

Molecular Collisions

In any gas, molecules are constantly moving and bumping into each other and the walls of their container. These encounters, known as molecular collisions, play a crucial role in generating gas pressure. As gas molecules move, they have kinetic energy, which relates to how fast they are moving. Each time a molecule hits the container wall, it exerts a force, which contributes to the overall pressure.

• Collision frequency refers to how often gas molecules hit the container wall.
• Factors affecting collision frequency include the speed of the molecules and the volume of the container.

When the volume is reduced, without changing temperature, molecules have less space to move. Thus, they collide with the walls more frequently. This increased collision frequency is what raises the gas pressure in the container.

Pressure and Volume Relationship

The relationship between pressure and volume in gases is a fundamental concept in physics and chemistry. Known as Boyle's Law, it describes how pressure and volume are inversely related when temperature is held constant. This means that if the volume of a gas decreases, the pressure increases, assuming no change in temperature.

• Boyle's Law can be expressed as: \[ P_1V_1 = P_2V_2 \] where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume.
• This illustrates that as volume decreases, pressure must increase to maintain the constant product.

This principle is why compressing a gas in a sealed container without changing its temperature causes the pressure inside to increase. Molecules collide with the walls more frequently, increasing the force per area, and thus the pressure.

Ideal Gas Law

The Ideal Gas Law provides a comprehensive equation that relates a gas's pressure, volume, temperature, and number of moles. It is expressed as:\[ PV = nRT \] Here, \( P \) stands for pressure, \( V \) is volume, \( n \) represents the number of moles, \( R \) is the ideal gas constant, and \( T \) denotes temperature. This law combines several earlier gas laws, including Boyle’s Law, Charles’s Law, and Avogadro's Law, into a single equation.

• This equation assumes an ideal gas, meaning it follows the gas laws at all conditions of temperature and pressure.
• For a constant temperature, as in our scenario, if the volume \( V \) decreases, the pressure \( P \) must increase to maintain the equality, in accordance with Boyle’s Law.

Understanding the Ideal Gas Law helps us quantify and predict changes in gas behavior due to changes in pressure, volume, and temperature.