Question

An ideal gas initially at pressure $p_0$ undergoes a free expansion (expansion against vaccum under adiabatic conditions) until its volume is $3$ times its initial volume. The gas is next adiabatically compressed back to its original volume. The pressure after compression is $s^{\frac{2}{3}}{P}_0$. The pressure of the gas after the free expression is :

• A. $\frac{P_0}{3}$
• B. ${P_0}^{\frac{1}{3}}$
• C. $P_0$
• D. $3P_0$

Answer 1

Answer: (A)

$\frac{P_0}{3}$

given: from A to B process is a free expansion

A B

$Po,Vo,To$ $V1=3Vo$

$\text{Solution}$:

for process $A\to B$

$\Rightarrow$ Q =0 $---\therefore$adiabatic process

$\Rightarrow$ W = 0 $---\therefore$ free expansion

from 1st Law of thermodynamics

$\Delta U=Q-W$ = 0 (as $Q$and $W$ are 0 ).

we know that $\Delta U=nCv\Delta T$

$\Delta U=nCv\Delta T$ = 0 which means$\Delta T=0$

So $T$ comes out to be constant here

from ideal gas equation

$PV=nRT$ $\therefore n,R,T$are constant here

$PV$= Constant

$PaVa=PbVb$

$\frac{PoVo}{3Vo}=Pb$ = $\frac{Po}{3}$

THE CORRECT OPT: A