Animated Solution for Physics - Thermodynamics: The correct relation between the degrees of freedom $f$ and the ratio of specific heat $\gamma$ is

Have you ever wondered how the microscopic motion of gas molecules dictates the macroscopic properties we measure, like specific heat? The bridge between these two worlds is beautifully captured by a single parameter: the degree of freedom, denoted by $f$.

According to the law of equipartition of energy, every active degree of freedom of a gas molecule contributes $\frac{1}{2}{k}_{B}T$ to its average kinetic energy. For one mole of an ideal gas, this translates to a molar specific heat at constant volume given by:

But what happens when we heat the gas at constant pressure? The gas must do work against the surroundings to expand. This extra energy requirement is given by Mayer's relation:

Substituting our expression for $C_v$, we get:

The ratio of these two specific heats is a fundamental constant for a given gas, denoted by $\gamma$ (gamma):

Let's plug in our expressions in terms of $f$:

Notice how beautifully the universal gas constant $R$ cancels out. We are left with a pure, dimensionless relationship:

The question asks us to express $f$ in terms of $\gamma$. This is a simple algebraic rearrangement. First, we isolate the term containing $f$:

Finally, taking the reciprocal of both sides and multiplying by 2, we arrive at our destination:

This elegant equation is a powerful tool in thermodynamics, allowing us to instantly deduce the microscopic structure of a gas just by measuring its macroscopic heat capacities!