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