Starting at temperature $300K,$ one mole of an ideal diatomic gas $\left(\gamma =1.4\right)$ is first compressed adiabatically from volume $V_1$ to $V_2=\frac{V_1}{16}.$ It is then allowed to expand isobarically to volume $2V_2$ . If all the processes are the quasi-static then the final temperature of the gas (in ${}^{o}K$ ) is (to the nearest integer) ___________.

Solution

$P{V}^y=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$
$T{V}^{-1}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$
$300{\left(V_1\right)}^{1.4-1}=T_B{\left(\frac{V_1}{16}\right)}^{2/5}$
$T_B=300\times 2^{8/5}$
Now for BC process
$\frac{V_B}{T_B}=\frac{V_c}{T_c}$
$T_c=\frac{V_c{T}_B}{v_B}=2\times 300\times 2^{8/5}$
$T_c=1818.859$
$T_c=1819K$

Hence, the correct answer is 1819.