Question

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

• A. $\left(1 + \frac{2}{n} \right)$
• B. $\left(1 + \frac{n}{2} \right)$
• C. $\left(1 + \frac{1}{n} \right)$
• D. $\left(1 + \frac{n}{3} \right)$

Answer 1

The Correct Option is A

To solve this problem, we need to determine how the ratio of specific heats $\gamma = \frac{C_P}{C_V}$ is related to the degrees of freedom (n) of a gas.

The specific heats at constant pressure ( $C_P$) and volume ( $C_V$) are related to the degrees of freedom by the following equations:

• For an ideal gas, the specific heat at constant volume is given by: $C_V = \frac{n}{2}R$
• The specific heat at constant pressure is given by: $C_P = C_V + R = \frac{n}{2}R + R = \left(\frac{n}{2} + 1\right)R$

Substituting these expressions into the formula for $\gamma$ gives:

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

Thus, the correct expression for the ratio of specific heats, in terms of the degrees of freedom, is $\left(1 + \frac{2}{n}\right)$.

This matches the given correct option: $\left(1 + \frac{2}{n} \right)$.

Let's rule out other options:

• $\left(1 + \frac{n}{2}\right)$ does not fit the derived form based on degrees of freedom.
• $\left(1 + \frac{1}{n}\right)$ is also incorrect based on the formula derivation.
• $\left(1 + \frac{n}{3}\right)$ doesn't match the established relation.

Therefore, the correct answer is $\left(1 + \frac{2}{n}\right)$.