Question
A mono-atomic ideal gas undergoes a process in which the ratio of $$P$$ to $$V$$ at any instant is constant and equals to 1. What is the molar heat capacity of the gas
A.
$$\frac{{3R}}{2}$$
B.
$${2R}$$
C.
$$0$$
D.
$$\frac{{5R}}{2}$$
Answer :
$${2R}$$
Solution :
In general, the molar heat capacity for any process is given by
$$\eqalign{
& C = {C_v} + \frac{R}{{1 - n}},\,{\text{when}}\,P{V^n} = {\text{constant}} \cr
& {\text{Here}}\,\,\frac{P}{V} = 1,\,\,\,i.e.\,\,P{V^{ - 1}} = {\text{constant}} \cr
& {\text{For}}\,{\text{monoatomic}}\,{\text{gas,}}\,\,{C_v} = \frac{3}{2}R \cr
& \therefore \,\,C = \frac{3}{2}R + \frac{R}{{1 - \left( { - 1} \right)}} = \frac{3}{2}R + \frac{R}{2} = \frac{{4R}}{2} = 2R. \cr} $$