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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 :

\[{C_v}\, = \,\dfrac{n}{2}RT\]

We already know the relation:

\[{C_p} - {C_v}\, = \,RT\]

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)}} \\

\]

**Therefore the option with the correct answer is option C.**

**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 \[\dfrac{n}{2}RT\]

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