Question

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) $\dfrac{9}{2}(\gamma - 1)$

Answer 1

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 $\dfrac{n}{2}RT$