According to the law of equipartition of energy, the number of vibrational modes of a polyatomic gas of constant (where and are the specific heat capacities of the gas at constant pressure and constant volume, respectively) [2024]

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Solution

Q.
According to the law of equipartition of energy, the number of vibrational modes of a polyatomic gas of constant $γ = \frac{C_{P}}{C_{V}}$

(where $C_{P}$ and $C_{V}$ are the specific heat capacities of the gas at constant pressure and constant volume, respectively) [2024]

1 $\frac{4 + 3 γ}{γ - 1}$

2 $\frac{3 + 4 γ}{γ - 1}$

3 $\frac{4 - 3 γ}{γ - 1}$

4 $\frac{3 - 4 γ}{γ - 1}$

Ans.
(3)

Polyatomic molecule has 3 translational, 3 rotational and vibrational modes.
According to law of equipartition of energy,

$U = (\frac{3 R T}{2} + \frac{3 R T}{2} + f R T)$

$U = 3 R T + f R T$

$C_{V} = \frac{d u}{d t} = 3 R + f R \Rightarrow f = \frac{C_{V} - 3 R}{R} = \frac{C_{V}}{R} - 3$

As, $C_{V} = \frac{R}{γ - 1}$

$∴$ $f = \frac{R}{(γ - 1) R} - 3 \Rightarrow \frac{1}{γ - 1} - 3 = \frac{4 - 3 γ}{γ - 1}$

So, the correct option is 3.

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