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

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Last updated date: 13th Jun 2024
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Answer
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Hint: The ratio of specific heat at constant pressure and the specific heat at constant volume is denoted by \[\gamma \]. Difference between specific heat at constant pressure and that at constant volume is the same as universal gas constant. Also, degrees of freedom of a molecule of the gas is proportional to its specific heat at constant volume.

Formula Used:
Definition of Heat capacity ratio:
\[\gamma = \dfrac{{{C_P}}}{{{C_V}}}\] (1)
Where,
\[\gamma \] is the heat capacity ratio,
\[{C_P}\] is the specific heat capacity at constant pressure,
\[{C_V}\] is the specific heat capacity at constant volume.

Relation between specific heat capacities and universal gas constant is given as:
\[{C_P} - {C_V} = R\] (2)
Where,
R is the universal gas constant.

The relationship between degrees of freedom and specific heat capacity at constant volume is known as:
\[{C_V} = \dfrac{{nR}}{2}\] (3)
Where,
n is the no. of degrees of freedom.

Complete step by step answer:
Step 1
First, rewrite the expression of eq.(2) to get an expression for \[{C_P}\]:
\[
  {C_P} - {C_V} = R \\
  \therefore {C_P} = {C_V} + R \\
 \] (4)

Step 2
Now, use the eq.(3) in eq.(4) to get the form of \[{C_P}\] as:
 \[{C_P} = \dfrac{{nR}}{2} + R = \dfrac{{nR + 2R}}{2} = \left( {\dfrac{{n + 2}}{2}} \right)R\] (5)

Step 3
Substitute the value of \[{C_P}\] from eq.(5) and value of \[{C_V}\]from eq.(3) in eq.(1) to get the value of n as:
\[
  \gamma = \dfrac{{\left( {\tfrac{{n + 2}}{2}} \right)R}}{{\tfrac{{nR}}{2}}} \\
   \Rightarrow \gamma = \dfrac{{n + 2}}{n} \\
   \Rightarrow \gamma = 1 + \dfrac{2}{n} \\
   \Rightarrow \dfrac{2}{n} = \gamma - 1 \\
  \therefore n = \dfrac{2}{{\gamma - 1}} \\
 \]

Hence, you will get the relationship between n and \[\gamma \].

Final answer:
The number of degrees of freedom of a molecule of the gas is (c) \[\dfrac{2}{{\gamma - 1}}\].

Note: This problem can be done in a tricky manner. If you just follow the values of \[\gamma \]and degrees of freedom then you will notice that as the number of atoms increases in a molecule degrees of freedom keeps increasing and \[\gamma \] keeps decreasing. So, clearly they are inversely related. Hence, in this question only possible inverse relation is given by option (c) which is the correct answer.