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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 isA) $\dfrac{{25}}{2}(\gamma - 1)$B) $\dfrac{{3\gamma - 1}}{{2\gamma - 1}}$C) $\dfrac{2}{{\gamma - 1}}$D) $\dfrac{9}{2}\left( {\gamma - 1} \right)$

Last updated date: 13th Jun 2024
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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.

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 $\gamma$.

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.