Problem 178 - Master Resource Book in JEE Main Physics

Question

Pressure of an ideal gas is increased by keeping temperature constant. What is the effect on kinetic energy of molecules? [UP SEE 2006] (a) Increase (b) Decrease (c) No change (d) Can't be determined

Step-by-Step Solution

Answer

No change in kinetic energy (Option c).

Step 1: Understand the Ideal Gas Law

The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature. The problem states that the temperature is constant.

Step 2: Relate Pressure and Volume

Given that temperature \( T \) is constant, we can use the relation \( PV = nRT \). If pressure \( P \) increases with a constant temperature, to maintain equality in the equation, the volume \( V \) must decrease.

Step 3: Analyze the Effect on Kinetic Energy

The kinetic energy of molecules for an ideal gas is associated with its temperature and is given by \( KE = \frac{3}{2} k T \), where \( k \) is Boltzmann's constant. Since temperature \( T \) is constant, the kinetic energy, which depends on temperature, also remains constant.

Key Concepts

Pressure and Volume Relationship

When dealing with ideal gases, an important relationship to understand is how pressure and volume interact. This principle is highlighted by Boyle's Law, which is part of the overarching ideal gas law. The ideal gas law is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles of gas, \( R \) is the universal gas constant, and \( T \) is the temperature. Under the conditions of constant temperature (isothermal conditions), Boyle's Law provides us with a simple insight: if you increase the pressure exerted by a gas, its volume decreases, and vice versa.

• This is because, at a constant temperature, the energy of the gas doesn't change, so a higher pressure must be compensated by a decrease in volume.
• This inverse relationship is crucial when performing calculations involving gases and predicting how a gas will behave when subjected to different pressures.

This relationship helps us understand changes in a gas’s behavior, such as when a bicycle tire is pumped with more air, thereby increasing the pressure and subsequently reducing the volume occupied by the gas molecules inside the tire.

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental scientific theory that explains the behavior of gases. It is based on the idea that gas molecules are in a constant state of random motion. Here are some key points of the kinetic theory:

• Gases consist of many small particles (atoms or molecules) that are in constant, random motion.
• The pressure of a gas results from collisions between gas molecules and the walls of their container.
• The average kinetic energy of the gas molecules is proportional to the temperature of the gas in Kelvin.

This theory helps to provide a molecular-level interpretation of the macroscopic properties such as pressure and temperature. According to this theory, the ideal gas behavior arises when gases are at a high temperature and low pressure, conditions under which the interactions between molecules are negligible.

Temperature and Kinetic Energy

In the context of ideal gases, temperature is a measure of the average kinetic energy of gas molecules. The relationship is simple but fundamental: higher temperatures mean higher average kinetic energy, and vice versa. The kinetic energy for an ideal gas is derived from the equation \( KE = \frac{3}{2} k T \), where \( k \) is Boltzmann's constant, and \( T \) is the absolute temperature.

• In this equation, temperature plays a pivotal role because it directly influences the kinetic energy of the gas molecules.
• If the temperature remains constant, as in the problem described, the kinetic energy of the gas molecules does not change.
• This explains why, when pressure is increased through a decrease in volume, the kinetic energy of the gas does not change, as it depends solely on temperature.

Consequently, understanding this relationship helps to connect the dots between macroscopic observations (like an increase in pressure) and microscopic behavior (kinetic energy) in gases.