Question

An ideal diatomic gas $(\gamma =\frac{7}{5})$ undergoes a process in which its internal energy relates to the volume as $U=\alpha \sqrt{V}$, where $\alpha$ is a constant. If the work (in $J$) performed by the gas to increase its internal energy by $100J$ is 16k.Then what is the value of k?

Answer 1

$U=\alpha \sqrt{(}V)$ Eq(1)

So, $\delta U=\alpha (\sqrt{(}{V}_f-\sqrt{V_i})=100$

U = $5nRT/2=5pV/2$ Eq(2)

From Eq (1) and (2) $p=2\alpha /\left(5\sqrt{V}\right)$

$W=\int p\delta V=\frac{2\alpha }{5}{\int }_{V_i}^{V_f}\frac{\delta V}{\sqrt{V}}=0.8\alpha (\sqrt{V_f}-\sqrt{V_i})=80$

Answer. 80 J