A monatomic gas at a pressure P, having a volume V expands isothermally to a volume 2V and then adiabatically to a volume 16V. The final pressure of the gas is (Take ) [2014]

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Solution

Q.
A monatomic gas at a pressure P, having a volume V expands isothermally to a volume 2V and then adiabatically to a volume 16V. The final pressure of the gas is (Take $γ = \frac{5}{3}$ ) [2014]

1 64 P

2 32 P

3 P/64

4 16 P

Ans.
(3)

First, isothermal expansion

$P V = P' (2 V) \Rightarrow P' = \frac{P}{2}$

Then, adiabatic expansion, $P' {(2 V)}^{γ} = P_{f} {(16 V)}^{γ}$

(For adiabatic process, $P V^{γ}$ = constant)

$\frac{P}{2} {(2 V)}^{5 / 3}$ $= P_{f} {(16 V)}^{5 / 3}$

$P_{f} = \frac{P}{2} {(\frac{2 V}{16 V})}^{5 / 3} = \frac{P}{2} {(\frac{1}{8})}^{5 / 3} = \frac{P}{2} {(\frac{1}{2^{3}})}^{5 / 3} = \frac{P}{64}$

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