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. $\gamma P_0{V}_{0}in2$
• D. $\frac{P_0{V}_0}{2\left(\gamma -1\right)}$

Answer 1

Answer: (B)

$T_0{V}_0^{\gamma -1}=T{\left(\frac{V_0}{2}\right)}^{\gamma -1}\Rightarrow T=T_0{2}^{\gamma -1}$
$\therefore Aftercompression,weassumethepistontobefixed.$
$\therefore \Delta Q=n{C}_v\Delta T=n\frac{R}{\gamma -1}\left(T_0-T_0{2}^{\gamma -1}\right)=\frac{P_0{V}_0}{\gamma -1}\left(1-2^{\gamma -1}\right)$
$\therefore heatreleased=\frac{P_0{V}_0}{\gamma -1}\left(2^{\gamma -1}-1\right)$