Question

Column I Contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubble of the $4\times 4$ matrix given in the ORS.

Answer 1

i. In case of free expansion under adiabatic condition, change in internal energy $△U=0$.
Therefore, internal energy and temperature will remain constant.
ii. $p\propto \frac{1}{V^2}$
$p{V}^2=constant...\left(i\right)$
$\left(\frac{nRT}{V}\right)V^2=constant$
$T\propto \frac{1}{V}....\left(ii\right)$
If volume is doubled, temperature will decrease as per Eq. (ii)
Further, molar heat capacity in process $p{V}^x=constant$.
$C=C_v=\frac{R}{1-x}$
From Eq. (i), $x=2$
$C=\frac{3}{2}R+\frac{R}{1-2}=+\frac{R}{2}$
Since molar heat capacity is positive, according to $Q=nC△T,Q$ will be negative if $△T$ is negative, or gas loses heat if temperature is decreasing.
iii. $P\propto \frac{1}{V^{4/3}}$
$p{V}^{4/3}=constant$
${\left(\frac{nRT}{V}\right)}^{V^{4/3}}=constant$
$T\propto \frac{1}{V^{1/3}}$
Further, with increase in volume, temperature will decrease.
Here, $T\propto \frac{1}{V^{1/3}}$
$C=\frac{3}{2}R+\frac{R}{1-\frac{4}{3}}=-15R$
As molar heat capacity is negative, $Q$ will be positive if $triangleT$ is negative. The gas gains with decrease in temperature.
iv. $T\propto pV$
In expansion from $V_1$ to $2V_2$, product of $pV$ is increasing.
Therefore, temperature will increase or $△U=positive$.
Further, in expansion, work done is also positive.
Hence, $Q=W+△U=$ positive, gas gains heat.