Question

An ideal gas having initial pressures $P$, volume $V$ and temperature $R$ is allowed to expand adiabatically until its volume becomes $5.66V$ while its temperature falls to $T/2$.
Obtain the work done by the gas during the expansion as a function of initial pressure $P$ and volume $V$.

Answer 1

We know that the work done by the gas during adiabatic expansion is given by
$w=\frac{1}{(\gamma -1)}(PV-P^{\prime }{V}^{\prime })....\left(1\right)$
where pressure $P^{\prime }$ after expansion is obtained from the relation
$\frac{P^{\prime }{V}^{\prime }}{T^{\prime }}=\frac{PV}{T}$
or, $P^{\prime }=P\times \frac{V}{V^{\prime }}\times \frac{T^{\prime }}{T}=P\times \frac{V}{5.66V}\times \frac{T/2}{T}=\frac{P}{11.32}$
Putting $\gamma =1.4,v^{\prime }=5.66V$ and $P^{\prime }=\frac{P}{11.32}$ in Eq. (1), we have
$w=\frac{1}{(1.4-1)}(PV-\frac{P}{11.32}\times 5.66V)$
$=\frac{1}{0.4}(PV-\frac{1}{2}PV)=1.25PV$.