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 comparision to the

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Solution

Q.
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 isothermal process, PV = constant, while in adiabatic process $P V^{γ}$ = constant. Here $γ$ 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: [2025]

1 Both (A) and (R) are true but (R) is NOT the correct explanation of (A)

2 (A) is true but (R) is false

3 Both (A) and (R) are true and (R) is the correct explanation of (A)

4 (A) is false but (R) is true

Ans.
(4)

For isothermal process

PV = constant

$\Rightarrow P d V + V d P = 0$

$\Rightarrow \frac{d V}{V} = \frac{- d P}{P}$

$\Rightarrow d V = - (\frac{V}{P}) d P$

For adiabatic $P V^{γ}$ = constant

$\Rightarrow P_{γ} V^{γ - 1} d V + V^{γ} d P = 0$

$\Rightarrow d V = \frac{- V}{γ^{P}} (d P)$

Magnitude of $| \frac{V}{P} |$ is greater than $| \frac{V}{γ^{P}} |$

So, volume falls more rapidly in adiabatic.

2nd statement is the process description of the isothermal and adiabatic.

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