Explanation

(b) For adiabatic expansion, we have the formula
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ pV^{\gamma} = constant$ …(i)
Gas equation is ,
pV=RT
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p = \frac{RT}{V}$ …(ii)
From Eqs. (i) and (ii), we obtain
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \frac{RT}{V} \right)V^{\gamma} = constant$
$\Rightarrow \ \ \ \ \ TV^{\gamma - 1} = constant$ …(iii)
But $\ \ \ \ \ \ \ \ T \propto \frac{1}{\sqrt{V}}$ (given)
as $\ \ \ \ \ \ \ \ \ \ \ \ TV^{1/2} = constant$ …(iv)
Thus, using Eqs. (iii) and (iv) togther, we get
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \gamma - 1 = \frac{1}{2}$
$or\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \gamma = \frac{3}{2} = 1.5$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{C_{p}}{C_{v}} = 1.5$