If one mole of the polyatomic gas is having two vibrational modes and $\beta$ is the ratio of molar specific heats for polyatomic gas $\left(\beta=\frac{C_{P}}{C_{V}}\right)$ then the value of $\beta$ is:

JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Heat & Thermodynamics

If one mole of the polyatomic gas is having two vibrational modes and

β

is the ratio of molar specific heats for polyatomic gas

(β = \frac{C _{P}}{C _{V}})

then the value of

β

is:

(A)1.02
(B)1.25
(C)1.35
(D)1.2✓ Correct

Answer: (D)

Solution

By equipartition each translational and each rotational degree of freedom contributes

\frac{1}{2} R

to the molar heat capacity, while each vibrational mode contributes a full

R

, since a vibration carries both kinetic and potential energy. A polyatomic molecule has 3 translational and 3 rotational degrees of freedom, so with two vibrational modes

C_{V} = \frac{3}{2} R + \frac{3}{2} R + 2 R = 5 R

C_{P} = C_{V} + R = 6 R

The ratio is

β = \frac{6 R}{5 R} = 1.2

JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Single correct option

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