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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) \[\dfrac{9}{2}(\gamma - 1)\]

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Last updated date: 14th Apr 2024
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MVSAT 2024
Answer
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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 \[\dfrac{n}{2}RT\]