Animated Solution for Physics - Thermodynamics: A polyatomic ideal gas has 24 vibrational modes. What is the value of $\gamma$?

Imagine a complex polyatomic gas molecule. It is not just flying around in a straight line; it is tumbling, spinning, and its constituent atoms are vibrating as if they are connected by tiny, invisible springs. To understand the thermodynamic behavior of such a gas, we must quantify all these independent modes of motion, known as degrees of freedom ($f$).

To find the heat capacity ratio, $\gamma$, we need to know the total degrees of freedom, $f$. The master formula connecting them is elegantly simple:

Let's break down the degrees of freedom for a non-linear polyatomic molecule. The molecule can move in three independent spatial directions ($x$, $y$, and $z$), giving it $3$ translational degrees of freedom ($f_{\text{trans}}=3$). Furthermore, it can rotate around three independent axes, contributing $3$ rotational degrees of freedom ($f_{\text{rot}}=3$).

Now, the question states there are $24$ vibrational modes. Here is the crucial catch! Each vibrational mode is modeled as a harmonic oscillator, which involves both kinetic energy (due to the motion of atoms) and potential energy (due to the restoring force of the bonds). According to the Law of Equipartition of Energy, each of these energy forms contributes $1$ degree of freedom.

Therefore, a single vibrational mode contributes $2$ degrees of freedom. The total vibrational degrees of freedom will be:

Let's add them all up to find the total degrees of freedom:

This gives us a massive total of $54$ degrees of freedom. We now substitute this value back into our master formula for $\gamma$:

Simplifying the fraction, we get:

When we divide this, we get approximately $1.037$. Rounding it to match the given options, we arrive at our final answer:

Notice how close $\gamma$ is to $1$. As a molecule gets more complex and its degrees of freedom increase towards infinity, the value of $\gamma$ asymptotically approaches $1$. This is a beautiful mathematical reflection of how energy is distributed in highly complex molecular systems!