Question

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): With the increase in the pressure of an ideal gas, the volume falls off more rapidly in an isothermal process in comparison to the adiabatic process.
Reason (R): In an isothermal process, \( PV = \text{constant} \), while in an adiabatic process \( PV^\gamma = \text{constant} \). Here, \( \gamma \) is the ratio of specific heats, \( P \) is the pressure and \( V \) is the volume of the ideal gas.
In the light of the above statements, choose the correct answer from the options given below:

• A. Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
• B. (A) is true but (R) is false
• C. Both (A) and (R) are true and (R) is the correct explanation of (A)
• D. (A) is false but (R) is true

Hint:

In an isothermal process, \( P \) and \( V \) are inversely proportional, while in an adiabatic process, the relationship between \( P \) and \( V \) follows \( PV^\gamma = \text{constant} \), and the decrease in volume is more rapid.

Answer 1

The Correct Option is D

- In an isothermal process, the temperature remains constant, and \( PV = \text{constant} \). As the pressure increases, the volume decreases in a manner such that the product remains constant.
- In an adiabatic process, \( PV^\gamma = \text{constant} \), where \( \gamma \) is the ratio of specific heats. Here, the volume decreases more rapidly than in the isothermal process.
Thus, the assertion (A) is false because in fact, the volume decreases more slowly in the isothermal process than in the adiabatic process. The reason (R) is true and correctly describes the nature of the isothermal and adiabatic processes.

Answer 2

Step 1: Understanding the statements.
Assertion (A): With the increase in pressure of an ideal gas, the volume falls off more rapidly in an isothermal process in comparison to the adiabatic process.
Reason (R): In an isothermal process, \( PV = \text{constant} \), while in an adiabatic process \( PV^\gamma = \text{constant} \), where \( \gamma \) is the ratio of specific heats.

Step 2: Analyzing the processes.
For an isothermal process:
\[ PV = \text{constant} \Rightarrow V \propto \frac{1}{P} \] For an adiabatic process:
\[ PV^\gamma = \text{constant} \Rightarrow V \propto P^{-\frac{1}{\gamma}} \] Since \( \gamma > 1 \), the exponent \( -\frac{1}{\gamma} \) is smaller in magnitude than \(-1\).
Therefore, in an adiabatic process, the volume decreases more rapidly with increasing pressure compared to an isothermal process.

Step 3: Evaluating the assertion.
The Assertion claims the opposite — that the volume decreases more rapidly in an isothermal process than in an adiabatic one. This is incorrect.
Hence, (A) is false.

Step 4: Evaluating the reason.
The Reason correctly states the relationships for isothermal and adiabatic processes:
- \( PV = \text{constant} \) for isothermal
- \( PV^\gamma = \text{constant} \) for adiabatic
Hence, (R) is true.

Final Answer:
(A) is false but (R) is true