Question

A monatomic ideal gas is following the cyclic process $ABCA$. Then choose the incorrect option.

• A. Molar heat capacity for the process $AB$ is $\frac{R}{2}$
• B. Heat is rejected by the system in path $BC$
• C. Molar heat capacity for the process $BC$ is $\frac{2}{3} R$
• D. Work done by the system in the process $CA$ is $\frac{2U_0}{3} ln \,4$

Answer 1

Answer: (C)

For path $AB$
$TV =$ constant $\Rightarrow PV^2 =$ constant $\quad ...(i)$
For polytropic process $PV^m =$ constant $\quad ...(ii)$
On comparing eqn. $(i)$ and $(ii)$ $m = 2$ The molar heat capacity is given as
$C= \frac{R}{\gamma-1} - \frac{R}{ m-1} , \gamma = \frac{5}{3}$
$ C = \frac{3R}{2} - R = \frac{R}{2} $
$C = \frac{R}{\frac{5}{3}-1} - \frac{R}{2-1}$
$ = \frac{3}{2}R -R = \frac{R}{2}$
So option $(a)$ is correct.
For $BC$ it is an isochoric process and temperature is
decreasing so option $(b)$ is correct.
$C _V = \frac{3}{2}R ;$ so option $(c)$ is wrong.
' For process $CA$ (isothermal)
$W = nRT ln \left(\frac{V_{A}}{V_{C}}\right)$
$ W= \frac{2}{3} \left(\frac{3}{2}nRT\right)ln \left(4\right)\quad$ (AS density at $C$ is $4\rho_0)$
$=\frac{2}{3}U_{0} ln \left(4\right)$
So, option $(d)$ is correct.