A gas has $$n$$ degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be

We need to find the ratio of specific heat at constant volume ($$C_v$$) to the specific heat at constant pressure ($$C_p$$) for a gas with $$n$$ degrees of freedom. According to the equipartition theorem, each degree of freedom contributes $$\dfrac{1}{2}kT$$ of energy per molecule; therefore, for one mole of an ideal gas with $$n$$ degrees of freedom, the internal energy is $$U = \dfrac{n}{2}RT$$, where $$R$$ is the universal gas constant and $$T$$ is the temperature.

Since $$C_v = \dfrac{dU}{dT}$$, it follows that $$C_v = \dfrac{n}{2}R$$. Substituting into Mayer's relation $$C_p - C_v = R$$ gives $$C_p = C_v + R = \dfrac{n}{2}R + R = \left(\dfrac{n}{2} + 1\right)R = \dfrac{n + 2}{2}R$$.

From this, $$\dfrac{C_v}{C_p} = \dfrac{\dfrac{n}{2}R}{\dfrac{n+2}{2}R} = \dfrac{n}{n+2}$$.

Verification: For a monoatomic gas ($$n = 3$$): $$\dfrac{C_v}{C_p} = \dfrac{3}{5}$$, which matches the known value $$\dfrac{1}{\gamma} = \dfrac{3}{5}$$.

The correct answer is Option A: $$\dfrac{n}{n+2}$$.