Question

For an ideal gas the instantaneous change in pressure ' $p$ ' with volume ' $v$ ' is given by the equation $\frac{ d p }{ dv }=-$ 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. $\frac{ p _{0}}{ ae R }$
• B. $\frac{ ap _{0}}{ e R }$
• C. infinity
• D. $0^{\circ} C$

Answer 1

Answer: (A)

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