A hypothetical gas expands adiabatically such that its volume changes from $08 litres$ to $27 litres$ . If the ratio of final pressure of the gas to initial pressure of the gas is $\frac{16}{81}$ . Then the ratio of $\frac{C_{p}}{C_{v}}$ will be.

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$\frac{4}{3}$

$\frac{1}{2}$

$\frac{3}{2}$

$\frac{3}{1}$

answer is D.

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Detailed Solution

In adiabatic process, ${PV}^{γ} = const$
$\begin{aligned} P_{1} V_{1}^{γ} = P_{2} V_{2}^{γ} \\ \frac{P_{2}}{P_{1}} = {(\frac{V_{1}}{V_{2}})}^{γ} \\ \frac{16}{81} = {(\frac{8}{27})}^{γ} \\ γ = \frac{4}{3} \end{aligned}$

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