# Initial pressure and volume of a gas are P and V respectively. First it is expanded isothermally to volume 4V and then compressed adiabatically to volume V. The final pressure of gas will be Option 1) 1P Option 2) 2P Option 3) 4P Option 4) 8P

As we learnt in

Equation of state -

$dQ= 0$

$n\, C_{V}\, dT+PdV= 0$

- wherein

On solving

$\gamma \frac{dV}{V}+\frac{dP}{P}= 0$

$\Rightarrow PV^{\gamma }= constant$

First process

PV= Constant

$PV= P_{1}\: 4V\ \, \, or\ \, \, P_{1}= \frac{P}{4}$

$P_{1}V_{1}\gamma = P_{2}V_{2}\gamma\ \, \, or\ \: \: \frac{P}{4}(4V)^{\gamma}=P_{2}V^{\gamma}$

$P_{2}= \frac{P}{4}. (4^{\gamma})$

$\Rightarrow$    $\gamma$ is not given.

Option 1)

1P

Incorrect

Option 2)

2P

correct

Option 3)

4P

Incorrect

Option 4)

8P

Incorrect

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