During an experiment, an ideal gas is found to obey a condition $\frac{{{P^2}}}{\rho }$ = constant [$\rho =$ density of the gas]. The gas is initially at temperature $T,$ pressure $P$ and density $\rho$ . The gas expands such that density changes to $\rho/2$
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$\frac{P^{2}}{\rho}=$ constant
$P=p \frac{R T}{M}$ (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
$P_{1} T_{1}=P_{2} T_{2} \Rightarrow P T=\frac{P}{\sqrt{2}} T_{2} \Rightarrow T_{2}=\sqrt{2} T \ldots .(ii)$
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