Q. During an adiabatic process, an ideal gas obeys the relation $PV^{\frac{3}{2}}=$ constant. Calculate the final temperature of such a gas initially at a temperature $T$ , if it is compressed adiabatically to half its initial volume.
NTA AbhyasNTA Abhyas 2020
Solution:
$PV^{\frac{3}{2}}=$ constant;
$PV^{\gamma }=$ constant
$\Rightarrow \gamma =\frac{3}{2}$
Now, $TV^{\gamma - 1}=$ constant
where T is temprature
V is volume.
i denotes initial condition of a system
f denotes final condition of a system.
$T_{i}V_{i}^{\gamma - 1}=T_{f}V_{f}^{\gamma - 1}$
$T_{f}=T_{i}\left(\frac{V_{i}}{V_{f}}\right)^{\gamma - 1}=T_{i}\left(\right.2\left(\left.\right)^{\gamma - 1}$
$T_{f}=T_{i}\left(\right.2\left(\left.\right)^{\frac{3}{2} - 1}=\sqrt{2}T_{i}T_{I}=TT_{f}=\sqrt{2}T$
$PV^{\gamma }=$ constant
$\Rightarrow \gamma =\frac{3}{2}$
Now, $TV^{\gamma - 1}=$ constant
where T is temprature
V is volume.
f denotes final condition of a system.
$T_{i}V_{i}^{\gamma - 1}=T_{f}V_{f}^{\gamma - 1}$
$T_{f}=T_{i}\left(\frac{V_{i}}{V_{f}}\right)^{\gamma - 1}=T_{i}\left(\right.2\left(\left.\right)^{\gamma - 1}$
$T_{f}=T_{i}\left(\right.2\left(\left.\right)^{\frac{3}{2} - 1}=\sqrt{2}T_{i}T_{I}=TT_{f}=\sqrt{2}T$