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 PV^\gamma = 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:

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

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

Heat and Thermodynamics PYQ — JEE Main 2025 · 29 Jan · Shift 2 · Q55 with Solution

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Correct answer: D

Given relations
- For an isothermal process of an ideal gas: $PV = \text{constant}$ so, $V \propto \frac{1}{P}$
- For an adiabatic process: $PV^{\gamma} = \text{constant}$ so, $V \propto P^{-1/\gamma}$
where for an ideal gas, $\gamma > 1$ .
Compare how volume changes with pressure
- In isothermal process: $V = kP^{-1}$
- In adiabatic process: $V = k'P^{-1/\gamma}$
Since $\gamma > 1$ , we have $\frac{1}{\gamma} < 1$ Hence, $P^{-1}$ decreases faster with increasing $P$ than $P^{-1/\gamma}$ .

Therefore, as pressure increases, volume decreases more rapidly in the isothermal process than in the adiabatic process.

So, Assertion (A) is true.
Check the Reason (R)

The reason states:
- Isothermal: $PV = \text{constant}$
- Adiabatic: $PV^{\gamma} = \text{constant}$
These are correct standard equations for ideal-gas isothermal and adiabatic processes.

So, Reason (R) is true.
Does the Reason explain the Assertion?

Yes. Using these two equations directly gives: $V \propto P^{-1} \quad \text{(isothermal)}$ $V \propto P^{-1/\gamma} \quad \text{(adiabatic)}$ and since $\gamma > 1$ , the isothermal volume falls faster with pressure.

Thus, (R) is the correct explanation of (A).
Correct option

$\boxed{\text{D}}$

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

View written solutionFree

Correct answer: D

Given relations
- For an isothermal process of an ideal gas: $PV = \text{constant}$ so, $V \propto \frac{1}{P}$
- For an adiabatic process: $PV^{\gamma} = \text{constant}$ so, $V \propto P^{-1/\gamma}$
where for an ideal gas, $\gamma > 1$ .
Compare how volume changes with pressure
- In isothermal process: $V = kP^{-1}$
- In adiabatic process: $V = k'P^{-1/\gamma}$
Since $\gamma > 1$ , we have $\frac{1}{\gamma} < 1$ Hence, $P^{-1}$ decreases faster with increasing $P$ than $P^{-1/\gamma}$ .

Therefore, as pressure increases, volume decreases more rapidly in the isothermal process than in the adiabatic process.

So, Assertion (A) is true.
Check the Reason (R)

The reason states:
- Isothermal: $PV = \text{constant}$
- Adiabatic: $PV^{\gamma} = \text{constant}$
These are correct standard equations for ideal-gas isothermal and adiabatic processes.

So, Reason (R) is true.
Does the Reason explain the Assertion?

Yes. Using these two equations directly gives: $V \propto P^{-1} \quad \text{(isothermal)}$ $V \propto P^{-1/\gamma} \quad \text{(adiabatic)}$ and since $\gamma > 1$ , the isothermal volume falls faster with pressure.

Thus, (R) is the correct explanation of (A).
Correct option

$\boxed{\text{D}}$

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