Question

A gas has molar heat capacity $C=37.55$J $mol{e}^{-1}{K}^{-1}$, in the process PT$=$constant. The number of degrees of freedom of the molecules of the gas.

• A. $6$
• B. $3$
• C. $1$
• D. $5$

Answer 1

The correct option is D $5$

Here, $C=37.55$J $mol{e}^{-1}{K}^{-1}$; and

PT$=$K(constant) .(i)

According to standard gas equation

PV$=$RT or P$=$RT/V

From (i), $\frac{RT}{V}\times T=K$ or $V=\frac{R{T}^2}{K}$

$\therefore \frac{dV}{dT}=\frac{2RT}{K}$

But $\frac{T}{K}=\frac{1}{P}$ from eqn.(i), therefore, $\frac{dV}{dT}=\frac{2R}{P}$ (ii)

As, $C=C_v+P\frac{dV}{dT}$, therefore, using (ii)

$C=C_v+P\times \frac{2R}{P}=C_v+2R$ or $C_v-C=2R$

As, $C_v=\frac{n}{2}R$

$\therefore \frac{n}{2}R=C-2R$ or $n=\frac{2(C-2R)}{R}=\frac{2(37.55-2\times 8.3)}{8.3}=5$