Question

Suppose a diatomic gas gets ionised to a certain extent without any expenditure of heat energy. If the fractional change in the number of moles of the gas be $\eta$, then ignoring any internal energy loss, the fractional change in the temperature of the gas will be $\frac{\eta }{\eta +x}$. Then $x$ is

Answer 1

Initially, $E=n_0\frac{5}{2}RT$ -- (i)
Given $\frac{\Delta n}{n_0}=\eta$ or $\Delta n=\eta n_0$ Number of moles of diatomic gas left $=(1-n)n_0$ Number of moles of ions formed
$=2\Delta n=2\eta n_0$
$\therefore$ Finally, $E=(1-\eta )n_0\frac{5RT}{2}+2\eta n_0\frac{3R{T}^{\prime }}{2}$ --(ii)
Equating equations (i) and (ii), $T^{\prime }=\frac{5T}{5+\eta }$ Fractional change i temperature, $1-\frac{T^{\prime }}{T}=\frac{\eta }{\eta +5}$