Question

An ideal gas of volume $V$ and pressure $P$ expands isothermally to volume $16V$ and is then compressed adiabatically to volume $V$. The final pressure of the gas is :-
[Take $\gamma =1.5$]

• A. $P$
• B. $3P$
• C. $4P$
• D. $6P$

Answer 1

The correct option is C $4P$

Initially, ideal gas undergoes isothermal expansion.
For an isothermal process, temperature remains constant
i.e $P_1{V}_1=P_2{V}_2$
From the data given in the question,
$PV=16V\times P^{\prime }$
$P^{\prime }=\frac{P}{16}......\left(1\right)$
Later, gas undergoes adibatic compression to come back to initial volume $V$.
Let the final pressure be $P^{\prime \prime }$.
The equation of state in terms of $P$ and $V$ is given by
$P_2{V}_2^{\gamma }=P^{\prime \prime }{V}_1^{\gamma }$
From the data given in the question, we get
$P^{\prime }\times (16V{)}^{1.5}=P^{\prime \prime }\times (V{)}^{1.5}$ $(\because P_2=P^{\prime }\text{and}{V}_2=16V)$
$\frac{P}{16}\times (16V{)}^{1.5}=P^{\prime \prime }\times (V{)}^{1.5}$
$\Rightarrow P^{\prime \prime }=4P$
Thus, option (c) is the correct answer.