The pressure $P_{1}$ and density $d_{1}$ of diatomic gas $\left(\gamma=\frac{7}{5}\right)$ changes suddenly to $P _{2}\left(> P _{1}\right)$ and $d _{2}$ respectively during an adiabatic process. The temperature of the gas increases and becomes $......$ times of its initial temperature.$\left(\right.$ given $\left.\frac{ d _{2}}{ d _{1}}=32\right)$
JEE MAIN 2022, Medium
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$PV ^{\gamma}=$ const $\quad d =\frac{ m }{ v }$
$p \left(\frac{ m }{ d }\right)^{\gamma}=$ const
$\frac{ p }{ d ^{\gamma}}=$ const $\quad \frac{ d _{2}}{ d _{1}}=32$
$\frac{ p _{1}}{ p _{2}}=\left(\frac{ d _{1}}{ d _{2}}\right)^{\gamma}=\left(\frac{1}{32}\right)^{7 / 5}=\frac{1}{128}$
$\frac{ T _{1}}{ T _{2}}=\frac{ P _{1} V _{1}}{ P _{2} V _{2}}=\frac{1}{128} 32=\frac{1}{4}$
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