$P=p \frac{R T}{M}$ (Ideal gas equation)

$\Rightarrow \frac{p_{2}}{\rho}=\frac{P}{\rho}\left(\frac{\rho R T}{M}\right)=P T \frac{R}{M}=$ constant

$\therefore$ The graph of the above process on the $P-T$ diagram is hyperbola. For the above process

$\left(\frac{P_{2}}{\rho}\right)_{1}=\left(\frac{P^{2}}{\rho}\right)_{2} \Rightarrow \frac{P_{2}}{\rho}=\frac{P_{2}^{2}}{\rho / 2} \Rightarrow P_{2}=\frac{P}{\sqrt{2}} \ldots .(i)$

and

$P_{1} T_{1}=P_{2} T_{2} \Rightarrow P T=\frac{P}{\sqrt{2}} T_{2} \Rightarrow T_{2}=\sqrt{2} T \ldots .(ii)$