Question

An ideal gas $(C_P/C_V=\gamma )$ is taken through a process in which the pressure and the volume vary as $P=a{V}^b$. Find the value of b for which the specific heat capacity in the process is zero.

Answer 1

For the specific heat capacity to be 0, heat supplied must be 0. That is $Q=0$.

Thus $W=\Delta U$

$W=\int -PdV=\int a{V}^{b}dV=-\frac{a}{b+1}(V_2^{b+1}-V_1^{b+1})$

Since $P=a{V}^b$

$W=-\frac{P_2{V}_2-P_1{V}_1}{b+1}$

Also,

$\Delta U=\frac{nR}{\gamma -1}(T_2-T_1)=\frac{P_2{V}_2-P_1{V}_1}{\gamma -1}$

Equating the two, we get

$b+1=1-\gamma$

Thus $b=-\gamma$

Answer is $-\gamma$