Question

One mole of helium in a vessel gets heat from outside and starts expanding to make its volume $2$ times the original volume. The heat capacity of the gas in this process is constant and is $\frac{R}{2}$. What is the final temperature (in $K$ ) of the gas, if the initial temperature is $200 \,K$ and initial pressure is $40 \,kPa$ ?
(Here, $R=$ Universal gas constant)

Answer 1

The process is polytropic.
$C=C_{v}-\frac{R}{m-1} $
$\frac{R}{2}=\frac{3}{2} R-\frac{R}{m-1} $
$\Rightarrow m=2 $
$P V^{2}=C$
$40 \times V_{0}^{2}=P\left(2 V_{0}\right)^{2} $
$P=10 \,kPa$
$\frac{P V}{T}=\frac{P_{0} V_{0}}{T_{0}}$
$\frac{10 \times 2 V_{0}}{T}=\frac{40 \times V_{0}}{200} $
$\Rightarrow T=100 \,K$