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# If $\gamma$ be the ratio of specific heat of a perfect gas, the number of degrees of freedom of a molecule is(A) $\dfrac{{25}}{2}(\gamma - 1)$ (B) $\dfrac{{3\gamma - 1}}{{2\gamma - 1}}$ (C) $\dfrac{2}{{\gamma - 1}}$  (D) $\dfrac{9}{2}(\gamma - 1)$

Last updated date: 14th Apr 2024
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Hint In this question we need to find the specific heat of a gas at constant volume and pressure in terms of degrees of freedom. Dividing those 2 quantities we will get $\gamma$ which can be manipulated to find degrees of freedom.

Complete step by step solution
As we know that the vibrational degree of freedom of a diatomic gas molecule is 3 and the rotational degree of freedom is 2. This makes the total degree of freedom as 5. Let's consider this in a more general sense, let the total degree of freedom of a body be n, then its internal energy will be
$U\, = \,\dfrac{n}{2}RT$
This internal energy when taken at constant pressure will become the molar heat capacity at a constant volume which is :
${C_v}\, = \,\dfrac{n}{2}RT$
${C_p} - {C_v}\, = \,RT$
Substituting ${C_v}$ in this relation we get,
${C_p}\, = \,R + {C_v} \\ {C_p}\, = \,RT(1 + \dfrac{n}{2}) \\$
Where n is the number of degrees of freedom. Dividing ${C_p}$ by ${C_v}$ we get:
$\dfrac{{{C_p}}}{{{C_v}}}{\text{ }} = {\text{ }}\dfrac{{RT(1 + \dfrac{n}{2})}}{{\dfrac{n}{2}RT}} \\ \gamma \, = \,\dfrac{{2 + n}}{n} \\ n\gamma {\text{ }} = {\text{ }}2 + n \\ n = \dfrac{2}{{(\gamma - 1)}} \\$
Note For a single molecule, the energy of the system is expressed as $\dfrac{n}{2}{k_B}T$ where n the degree of freedom of the molecule. When this number is multiplied by Avogadro's number we get the energy as $\dfrac{n}{2}RT$