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  4. The pressure P1 and density d1 of diatomic gas (γ=(7/5)) changes suddenly to P 2(> P 1) and d 2 respectively during an adiabatic process. The temperature of the gas increases and becomes times of its initial temperature. (given ( d 2/ d 1)=32 )

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Q. 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.
(given $\frac{ d _2}{ d _1}=32$ )

JEE MainJEE Main 2022Kinetic Theory

Solution:

$ PV ^\gamma=\text { const } \quad d =\frac{ m }{ V } $
$ p \left(\frac{ m }{ d }\right)^\gamma=\text { const } $
$ \frac{ p }{ d ^\gamma}=\text { const } $
$ \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 _1}=\frac{ P _1 V _1}{ P _2 V _2}=\frac{1}{128} 32=\frac{1}{4}$