The relation between U, P and V for an ideal gas in an adiabatic process is given by relation U = a + bPV. Find the value of adiabatic exponent ($\gamma$) of this gas 

(1) $U=a+bPV ......\left(1\right)$

In adiabatic change, $$\begin{gathered} dU=-dW=\frac{nR}{\gamma -1}\left(T_2-T_1\right)=\frac{nR}{\gamma -1}\left(dT\right) \\ \Rightarrow U=\int dU=\frac{nR}{\gamma -1}\int dT \\ or U=\left(\frac{nR}{\gamma -1}\right)T+a=\frac{PV}{\gamma -1}+a .....\left(2\right) \end{gathered}$$

where a is the constant of integration. Comparing (1) and (2), we get 

$b=\frac{1}{\gamma -1}\Rightarrow \gamma =\frac{b+1}{b}$