An ideal gas $(C_{P} / C_{V} = γ)$ 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.

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Solution

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

Thus $W = Δ U$

$W = \int - P d V = \int a V^{b} d V = - \frac{a}{b + 1} (V_{2}^{b + 1} - V_{1}^{b + 1})$

Since $P = a V^{b}$

$W = - \frac{P_{2} V_{2} - P_{1} V_{1}}{b + 1}$

Also,

$Δ U = \frac{n R}{γ - 1} (T_{2} - T_{1}) = \frac{P_{2} V_{2} - P_{1} V_{1}}{γ - 1}$

Equating the two, we get

$b + 1 = 1 - γ$

Thus $b = - γ$

Answer is $- γ$

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