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 ${\mathrm{PV}}^{\gamma }=$ 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 :

The assertion (A) states that with an increase in pressure, the volume of an ideal gas decreases more rapidly in an isothermal process compared to an adiabatic process. The reason (R) given is that in an isothermal process, $(PV=\text{constant)}$, while in an adiabatic process, $(P{V}^{\gamma }=\text{constant)}$, where ( $\gamma$) is the ratio of specific heats.

To analyze this, consider the equations governing each process. For an isothermal process, Boyle's law gives$(PV=\text{ constant )}$, leading to $V\propto \frac{1}{P}$. For an adiabatic process, the relation is $P{V}^{\gamma }=\text{ constant }$, leading to $V\propto {\left(\frac{1}{P}\right)}^{1/\gamma }$.

Since $\gamma >1,$ the exponent $\frac{1}{\gamma }$ is less than 1. When pressure P increases, the volume V decreases as $\left(\frac{1}{P}\right)$ for isothermal and ${\left(\frac{1}{P}\right)}^{1/\gamma }$ for adiabatic. Because \( $\frac{1}{\gamma }<1,$ the term ${\left(\frac{1}{P}\right)}^{1/\gamma }$ decreases more slowly than $\left(\frac{1}{P}\right)$. Therefore, for the same increase in pressure, the volume decreases more rapidly in an isothermal process compared to an adiabatic process.

The reason (R) correctly explains the assertion (A) by providing the respective equations for each process, which show that the volume decreases more rapidly with pressure in an isothermal process due to the different exponents in the equations.

Thus, both assertion (A) and reason (R) are true, and reason (R) is the correct explanation for assertion (A).

${\left(\frac{\mathrm{dP}}{\mathrm{dV}}\right)}_{\text{Adiabatic}}>{\left(\frac{\mathrm{dP}}{\mathrm{dV}}\right)}_{\text{Isothermal}}$