Problem 44 - A Complete Resource Book in Chemistry for JEE Main

Question

If a gas expands at constant temperature (1) the pressure decreases (2) the kinetic energy of the molecules remains the same (3) the kinetic energy of the molecules decreases (4) the number of molecules of the gas increase (a) 1,2 (b) \(1,2,3\) (c) \(1,2,4\) (d) 2,3

Step-by-Step Solution

Answer

The correct answer is (a) 1,2.

Step 1: Understanding Constant Temperature in Gases

When a gas expands at a constant temperature, it is undergoing an isothermal process. In such a process, the temperature of the gas remains unchanged. According to the ideal gas law \( PV = nRT \), if temperature \( T \) and the number of moles \( n \) are constant, an increase in volume \( V \) will result in a decrease in pressure \( P \).

Step 2: Evaluating Kinetic Energy of Molecules

In an isothermal process, since the temperature remains constant, the average kinetic energy of the gas molecules stays the same. The kinetic energy of a gas is directly related to its temperature.

Step 3: Reviewing the Options

Let's examine the options: 1. The pressure indeed decreases as the gas expands at constant temperature, which is true. 2. The kinetic energy remains the same due to the constant temperature, which is also true. 3. The kinetic energy of the molecules decreases is false because the temperature is constant. 4. The number of molecules increases is incorrect because we have no indication that additional gas is being added.

Step 4: Choosing the Correct Answer

Comparing the true statements to the answer choices: (a) 1,2 is correct because both 1 and 2 are true. (b) 1,2,3 is incorrect because 3 is false. (c) 1,2,4 is incorrect because 4 is false. (d) 2,3 is incorrect because 3 is false.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics that describes the behavior of an ideal gas. It is expressed by the equation \( PV = nRT \), where:

• \( P \) represents pressure,
• \( V \) stands for volume,
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant, and
• \( T \) is the temperature in Kelvin.

In an isothermal process, as described in the exercise, the temperature \( T \) remains constant. This means that any change in either volume or pressure must be compensated by an inverse change in the other variable. So, when a gas expands (meaning volume increases), the pressure must decrease if the temperature is constant. This helps explain why choices 1 and 2 in the exercise are true.

Kinetic Energy in Gases

The kinetic energy of gas molecules is associated with their motion, and it is directly proportional to the temperature of the gas. The relationship can be expressed with the equation for the average kinetic energy, which is given by:\[ KE_{avg} = \frac{3}{2} kT \]where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature. Since the kinetic energy is derived from motion, it makes sense that any change in the temperature of the gas would affect the kinetic energy of its molecules.

However, in an isothermal process, as described in your exercise, the temperature does not change. Therefore, the average kinetic energy of the gas molecules remains constant. This explains why option 2 in the original exercise is true while option 3 is false. The misunderstanding often arises from assuming more complex factors are involved when actually, constant temperature simplifies many considerations.

Thermodynamics Concepts

Thermodynamics is the study of heat, energy, and the work done by them in a system, such as an ideal gas. It includes concepts like the four laws of thermodynamics and various process types, including isothermal, adiabatic, isobaric, and isochoric processes.

An isothermal process specifically involves changes in a system, like a gas, without a change in temperature. In such processes, while the internal energy of the system remains constant, pressure and volume can change as dictated by the Ideal Gas Law.

• Work done in an isothermal process can be calculated, but it requires the use of logarithmic functions due to the constant temperature consideration.
• During expansion, the gas does work on its surroundings, leading to a decrease in pressure, as seen in the original exercise.

This aligns with thermodynamic principles, where energy conservation and conversion are central themes. Understanding the relationships presented by the Ideal Gas Law, kinetic energy, and thermodynamic processes helps demystify gas behaviors under constant temperature conditions.