Question

Each molecule of a gas has \(\mathrm{f}\) degrees of freedom. The radio \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)=\gamma\) for the gas is (A) \(1+(\mathrm{f} / 2)\) (B) \(1+(1 / \mathrm{f})\) (C) \(1+(2 / \mathrm{f})\) (D) \(1+\\{(\mathrm{f}-1) / 3\\}\)

Short answer

The correct option for the ratio \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)=\gamma\) for the gas is (C) \(1+(2 / \mathrm{f})\).

Step-by-step solution

Relationship between specific heat capacities and degrees of freedom

To find the relationship between the specific heat capacities and the degrees of freedom (\(f\)), we can recall that: \(C_p=C_v+nR\) Where \(C_p\) = specific heat capacity at constant pressure, \(C_v\) = specific heat capacity at constant volume, \(n\) = number of moles, and \(R\) = ideal gas constant.

Equation of specific heat capacity at constant volume

For a gas with \(f\) degrees of freedom, the specific heat capacity at constant volume (\(C_v\)) is given by: \(C_v=\frac{f}{2}nR\) We are going to use this equation to express \(\frac{C_p}{C_v}\) in terms of \(f\).

Derive the expression for gamma (\(\gamma\))

To find the ratio \(\gamma\), we can substitute the expression of \(C_v\) (from step 2) into the equation from step 1: \(C_p - C_v = nR\) Then divide both sides by \(C_v\): \(\frac{C_p}{C_v}-1 = \frac{2}{f}\) Add 1 on both sides of the equation to get the ratio: \(\frac{C_p}{C_v} = \gamma = 1 + \frac{2}{f}\) Comparing our derived expression with the given options, we find that it matches with option (C).

Answer

The correct option for the ratio \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)=\gamma\) for the gas is (C) \(1+(2 / \mathrm{f})\)