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The pressure and density of a diatomic gas $\gamma = 7/5$ change adiabatically from $(P, d)$ to $(P', d').$ If $\frac{{d'}}{d} = 32,$ then $\frac{{P'}}{P}$should be
$1/128$
$32$
$128$
None of these
Solution
For adiabatic process,
$\mathrm{PV}^{\mathrm{y}}=\mathrm{constant} \ldots \ldots \ldots(1)$
As volume $=[(\text { mass }) /(\text { density })]$ i.e. $\quad \mathrm{V}=(\mathrm{m} / \mathrm{d})$
$\left(\mathrm{V}_{1} / \mathrm{V}_{2}\right)=\left(\mathrm{d}_{2} / \mathrm{d}_{1}\right) \ldots \ldots \ldots(2)$
From $(1),\left(P_{1} / P_{2}\right)=\left(V_{2} / V_{1}\right)^{Y}$
From $(2),\left(P_{1} / P_{2}\right)=\left(d_{1} / d_{2}\right)^{Y}$
Here $P_{1}=P, P_{2}=P^{\prime}, d_{1}=d, d_{2}=d^{\prime}$
Hence $\left(P / P^{1}\right)=\left(d / d^{\prime}\right)^{y}$
$\therefore\left(\mathrm{P} / \mathrm{P}^{1}\right)=(1 / 32)^{7 / 5}$
$\therefore\left(\mathrm{P} / \mathrm{P}^{1}\right)=\left[1 / 2^{\{5 \times(7 / 5)\}}\right]=\left(1 / 2^{7}\right)$
$\therefore P^{\prime}=2^{7} \cdot P$
$\therefore\left(P^{\prime} / P\right)=128$
