Animated Solution for Physics - Thermodynamics: Consider a gas of triatomic molecules. The molecules are assumed to be triangular and made of massless rigid rods whose vertices are occupied by atoms. The internal energy of a mole of the gas at temperature $T$ is

The journey to understanding the internal energy of a gas begins with a profound realization: temperature is simply a macroscopic manifestation of microscopic chaos. When we heat a gas, we are pumping energy into its molecules, causing them to dance, spin, and vibrate with greater vigor. But how exactly is this energy distributed? This is where the elegant Law of Equipartition of Energy comes into play.

Imagine you are shrinking down to the atomic scale. You are floating in a void, and a single molecule of our gas drifts by. The problem tells us it is a triatomic molecule, meaning it consists of three atoms. Furthermore, it is triangular, not linear like carbon dioxide ($C{O}_2$). The atoms sit at the vertices of a triangle, connected by what the problem describes as "massless rigid rods."

This geometry is crucial. The shape of a molecule dictates how it can move through space, and consequently, how it can store energy. Every independent way a molecule can store kinetic or potential energy is called a degree of freedom.

Before we count these degrees of freedom, we need our master tool: the Equipartition Theorem. This fundamental principle of statistical mechanics states that in thermal equilibrium, every active degree of freedom contributes exactly $\frac{1}{2}kT$ of energy per molecule, where $k$ is the Boltzmann constant and $T$ is the absolute temperature.

However, the question asks for the internal energy of one mole of the gas, not just a single molecule. Since one mole contains Avogadro's number ($N_A$) of molecules, we multiply the energy per molecule by $N_A$. Because $N_A\times k=R$ (the universal gas constant), the energy contribution becomes $\frac{1}{2}RT$ per mole for each degree of freedom.

Therefore, the total internal energy $U$ of one mole of an ideal gas is given by the beautifully simple equation: where $f$ is the total number of degrees of freedom. Our mission is now clear: we must find $f$.

Let's break down the molecule's possible motions into three categories: translation, rotation, and vibration.

1. Translational Degrees of Freedom ($f_{\text{trans}}$) No matter how complex a molecule is, its center of mass can always move freely through three-dimensional space. It can travel left or right (along the X-axis), up or down (along the Y-axis), and forward or backward (along the Z-axis). Each of these independent directions represents a translational degree of freedom.

2. Rotational Degrees of Freedom ($f_{\text{rot}}$) Now, let's watch the molecule spin. A linear molecule (like a dumbbell) only has two meaningful axes of rotation because spinning along its own axis doesn't involve any significant displacement of mass.

But our molecule is a 2D triangle! Because its mass is spread out in a plane, it has a significant moment of inertia about all three perpendicular axes (X, Y, and Z). It can tumble end-over-end in two different ways, and it can spin like a frisbee in the plane of the triangle. This gives us three independent rotational degrees of freedom.

3. Vibrational Degrees of Freedom ($f_{\text{vib}}$) Finally, we must consider vibration. Can the atoms oscillate back and forth like masses on springs? The problem statement gives us a critical constraint: the atoms are connected by "rigid rods."

In physics, "rigid" is a magic word that means the distance between the atoms cannot change. The bonds cannot stretch, compress, or bend. Because the structure is locked in place, the molecule cannot store energy through internal vibrations.

We have successfully analyzed all possible modes of motion. Now, we simply sum them up to find the total degrees of freedom:

Our triangular molecule has exactly 6 degrees of freedom. We are now ready to deploy our master equation. Substituting $f=6$ into the equipartition formula:

Simplifying the fraction, we arrive at our final, elegant result:

The internal energy of one mole of this rigid, triatomic gas is $3RT$. This perfectly matches option (d). By carefully visualizing the geometry of the molecule and applying the fundamental laws of thermodynamics, we've decoded the microscopic behavior of the gas!