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(B) For an adiabatic process,the relation between pressure $P$ and temperature $T$ is given by $P^{1-\gamma} T^{\gamma} = 	ext{constant}$.Given that

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$\frac{R}{2}$

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(B) For an adiabatic process,the relation between pressure $P$ and temperature $T$ is given by $P^{1-\gamma} T^{\gamma} = 	ext{constant}$.Given that $P^2 \propto T^3$,we can write $P \propto T^{3/2}$,which implies $P T^{-3/2} = 	ext{constant}$.Comparing $P T^{-3/2} = 	ext{constant}$ with $P^{1-\gamma} T^{\gamma} = 	ext{constant}$,we rewrite the given relation as $P T^{-3/2} = 	ext{constant}$,or $P^{1} T^{-3/2} = 	ext{constant}$.Raising to the power of $\frac{1}{1-\gamma}$,we get $P T^{\frac{-3/2}{1-\gamma}} = 	ext{constant}$.Comparing the exponents of $T$,we have $\frac{\gamma}{1-\gamma} = -\frac{3}{2}$.$2\gamma = -3(1-\gamma) \implies 2\gamma = -3 + 3\gamma \implies \gamma = 3$.For an ideal gas,the molar specific heat at constant volume is $C_V = \frac{R}{\gamma - 1}$.Substituting $\gamma = 3$,we get $C_V = \frac{R}{3-1} = \frac{R}{2}$.

For $1 \text{ mole}$ of an ideal gas,during an adiabatic process,the square of the pressure of a gas is found to be proportional to the cube of its absolute temperature. The specific heat of the gas at constant volume is ($R$ is universal gas constant)

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