Question

One mole of an ideal monoatomic gas undergoes a process described by the equation $P{V}^3=\text{constant}$. The molar heat capacity of this gas during this process is

• A. $\frac{3}{2}R$
• B. $\frac{5}{2}R$
• C. $2R$
• D. $R$

Answer 1

The correct option is D $R$

We know that, for polytropic process of index $x$,
$P{V}^x=\text{constant}$
Molar heat capacity is given as,
$C=C_V+\frac{R}{1-x}$
$\Rightarrow C=\frac{fR}{2}+\frac{R}{1-x}...\left(1\right)$
$f=3$ [monoatomic gas]
$x=3$ [$P{V}^3=\text{constant},x=3$]
On putting these values in Eq.$\left(1\right)$
$\Rightarrow C=\frac{3\times R}{2}+\frac{R}{1-3}$
$\Rightarrow C=\frac{3R}{2}-\frac{R}{2}$
$\Rightarrow C=R$
The molar heat capacity of the gas during the given process is $R$