For an ideal gas the instantaneous change in pressure ' $p$ ' with volume ' $v$ ' is given by the
equation $\frac{\mathrm{dp}}{\mathrm{dv}}=-\mathrm{ap}$. If $\mathrm{p}=\mathrm{p}_{0}$ at $\mathrm{v}=0$ is the given
boundary condition, then the maximum temperature one mole of gas can attain is :
(Here $R$ is the gas constant)
Correct Option: 1
$\int_{p_{0}}^{p} \frac{d p}{p}=-a \int_{0}^{v} d v$
$\ln \left(\frac{\mathrm{p}}{\mathrm{p}_{0}}\right)=-\mathrm{av}$
$\mathrm{p}=\mathrm{p}_{0} \mathrm{e}^{-\mathrm{av}}$
For temperature maximum p-v product should be maximum
$\mathrm{T}=\frac{\mathrm{pv}}{\mathrm{nR}}=\frac{\mathrm{p}_{0} \mathrm{ve}^{-\mathrm{av}}}{\mathrm{R}}$
$\frac{\mathrm{dT}}{\mathrm{dv}}=0 \Rightarrow \frac{\mathrm{p}_{0}}{\mathrm{R}}\left\{\mathrm{e}^{-\mathrm{av}}+\mathrm{ve}^{-\mathrm{av}}(-\mathrm{a})\right\}$
$\frac{\mathrm{p}_{0} \mathrm{e}^{-\mathrm{av}}}{\mathrm{R}}\{1-\mathrm{av}\}=0$
$\mathrm{v}=\frac{1}{\mathrm{a}}, \infty$
$\mathrm{T}=\frac{\mathrm{p}_{0} 1}{\mathrm{Rae}}=\frac{\mathrm{p}_{0}}{\mathrm{Rae}}$
at $v=\infty$
$\mathrm{T}=0$
Option (1)