Question

An ideal momoatomic gas is subjected to the processes mentioned in Column-I. Column-II mentions some values of molar specific heat capacity. Match the process in column-I to its corresponding molar specific heat capacity in Column-II.

Answer 1

In polytropic process,
$P{V}^n=constant$
(A):
$\frac{V}{T^2}=constant$
Substituting $T=\frac{PV}{R}$
we get
$P{V}^{\frac{1}{2}}=constant$
Hence, polytropic exponent $\eta =\frac{1}{2}$
$C\Delta T=\Delta U+\Delta W=C_v\Delta T+\frac{-R\Delta T}{\eta -1}=\frac{R}{\gamma -1}\Delta t+\frac{-R\Delta T}{\eta -1}$
$\therefore C=\frac{R}{\gamma -1}-\frac{R}{\eta -1}$
Substituting the values $\gamma -1=\frac{5}{3}-1=\frac{2}{3}$ and $\eta =\frac{1}{2}$
$C=\frac{3R}{2}+2R=\frac{7}{2}R$
(B):
$\frac{V}{T^3}=constant$
Substituting $T=\frac{PV}{R}$
we get
$P{V}^{\frac{2}{3}}=constant$
Hence, polytropic exponent $\eta =\frac{2}{3}$
$\therefore C=\frac{R}{\gamma -1}-\frac{R}{\eta -1}=\frac{3R}{2}+3R=\frac{9}{2}R$
(C):
$P{T}^2=constant$
Substituting $T=\frac{PV}{R}$ we get
$P{V}^{\frac{2}{3}}=constant$
Hence, polytropic exponent $\eta =\frac{2}{3}$
$\therefore C=\frac{9}{2}R$ same as in (B)
(D):
$P{T}^3=constant$
Substituting $T=\frac{PV}{R}$ we get
$P{V}^{\frac{3}{4}}=constant$
Hence, polytropic exponent $\eta =\frac{3}{4}$
$\therefore C=\frac{3}{2}R+4R=\frac{11}{2}R$