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 \(\left.\frac{ d _{2}}{ d _{1}}=32\right)\)

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Correct Answer

(C) \(4\)

Detailed explanation

\(PV ^{\gamma}=\) 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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