Question

For an ideal gas the instantaneous change in pressure ' $p$ ' with volume ' $v$ ' is given by the equation $\frac{dp}{dv}=-$ ap. If $p=p_0$ at $v=0$ is the given boundary condition, then the maximum temperature one mole of gas can attain is :
(Here $R$ is the gas constant)

• A. $\frac{p_0}{aeR}$
• B. $\frac{a{p}_0}{eR}$
• C. infinity
• D. $0^{\circ }C$

Answer 1

Answer: (A)

$\underset{p_0}{\overset{p}{\int }}\frac{dp}{P}=-a{\int }_0^{v}dv$
$\ell n\left(\frac{p}{p_0}\right)=-av$
$p=p_0{e}^{-av}$
For temperature maximum $p-v$ product should be maximum
$T=\frac{pv}{nR}=\frac{p_{0}v{e}^{-av}}{R}$
$\frac{dT}{dv}=0$
$\Rightarrow \frac{p_0}{R}\left\{e^{-av}+v{e}^{-av}(-a)\right\}$
$\frac{p_0{e}^{-av}}{R}\{1-av\}=0$
$v=\frac{1}{a},\infty$
$T=\frac{p_{0}1}{Rae}=\frac{p_0}{Rae}$
at $v=\infty$
$T=0$