Question

3 moles of a gas mixture having $8.3m^3$ volume and temperature of $90.5K$ is compressed to $1/5^{th}$ of the initial volume. The change in its adiabatic compressibility, if the gas obeys $P{V}^{19/13}=$ constant is $-x\times {10}^{-3}P{a}^{-1}$ [$R=8.3$ J/mol-K and $\frac{1}{5^{\gamma }}-1=-0.905$], then $x$ is

Answer 1

Bulk modulus $B=\gamma P$
Compressibility $C=\frac{1}{B}=\frac{1}{\gamma P}$
and $\Delta C=C^{\prime }-C$
or $\Delta C=\frac{1}{\gamma }.[\frac{1}{P^{\prime }}-\frac{1}{P}]$

We know,
$P{V}^{\gamma }=P^{\prime }(\frac{V}{5}{)}^{\gamma }$
With, $\gamma =19/13$ and $P^{\prime }=5^{\gamma }P$

$⟹\Delta C=\frac{1}{\gamma P}.[\frac{1}{5^{\gamma }}-\frac{1}{1}]=-\frac{13\times 0.905}{19P}$

$⟹\Delta C=-\frac{13\times 0.905\times 8.3}{19\times 3\times 8.3\times 90.5}=-0.0022P{a}^{-1}$