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Kelvi

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The ratio of the specific heat $\frac{C_P}{C_V} = \gamma$ in terms of degrees of freedom $(n)$ is given by

medium

Kinetic Theory

2015

physics

$1 + \frac{n}{3}$

$1 + \frac{2}{n}$

Explanation

To solve this problem, we need to find the ratio of specific heats $\frac{C_P}{C_V} = \gamma$

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$1 + \frac{n}{2}$

$1 + \frac{1}{n}$

in terms of degrees of freedom

(n). \\

For an ideal gas, the specific heat capacities at constant volume

(C_V)

and constant pressure

(C_P)

are related to the degrees of freedom

(n)

as follows:

\\ C_V = \frac{n}{2} R \\ C_P = C_V + R = \frac{n}{2} R + R = \left(\frac{n}{2} + 1\right) R \\

The ratio of specific heats

\gamma

is given by:

\\ \gamma = \frac{C_P}{C_V} = \frac{\left(\frac{n}{2} + 1\right) R}{\frac{n}{2} R} \\

Simplifying the expression:

\\ \gamma = \frac{\frac{n}{2} + 1}{\frac{n}{2}} \\ \gamma = \frac{n + 2}{n} \\

This can be rewritten as:

\\ \gamma = 1 + \frac{2}{n} \\

Therefore, the correct option is Option 2:

1 + \frac{2}{n} \\

This corresponds to the given correct option.