Question

The initial pressure and volume of a given mass of an ideal gas $\left(with\frac{C_{p}}{C_{v}} =\gamma\right),$ taken in a cylinder fitted with a piston, are P$_0$ and V$_0$ respectively. At this stage the gas has the same temperature as that of the surrounding medium which is T$_0$. It is adiabatically compressed to a volume equal to $\frac{v_{0}}{2}.$ Subsequently the gas is allowed to come to thermal equilibrium with the surroundings. What is the heat released to the surroundings ?

• A. 0
• B. $\left(2^{\gamma-1}-1\right)\frac{P_{0} V_{0}}{\gamma-1}$
• C. $\gamma P_{0}V_{0} in 2$
• D. $\frac{P_{0}V_{0}}{2\left(\gamma-1\right)}$

Answer 1

Answer: (B)

$\left(2^{\gamma-1}-1\right)\frac{P_{0} V_{0}}{\gamma-1}$

Answer 2

$T_{0}V_{0}^{\gamma-1} =T\left(\frac{V_{0}}{2}\right)^{\gamma-1} \Rightarrow T =T_{0}2^{\gamma-1}$
$\therefore After compression, we assume the piston to be fixed.$
$\therefore\Delta Q =nC_{v}\Delta T =n\frac{R}{\gamma-1}\left(T_{0}-T_{0}2^{\gamma-1}\right) =\frac{P_{0}V_{0}}{\gamma-1}\left(1-2^{\gamma-1}\right)$
$\therefore heat released =\frac{P_{0}V_{0}}{\gamma-1} \left(2^{\gamma-1}-1\right)$