Animated Solution for Physics - Thermodynamics: What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature $T$?

Imagine a container filled with a monoatomic gas, like Helium or Argon. Unlike complex molecules that look like dumbbells or intricate webs, a monoatomic gas is beautifully simple. It consists of single, isolated atoms zipping around. Because these atoms are essentially point masses, they don't have any meaningful moment of inertia. This means they can't store energy by spinning (rotational energy) or by vibrating like a spring (vibrational energy).

Their only way to express kinetic energy is by moving straight through space. They can move left or right (along the x-axis), up or down (along the y-axis), and forward or backward (along the z-axis). Because there are exactly three independent directions of motion, we say that a monoatomic gas has $3$ degrees of freedom ($f=3$).

Now, how does temperature relate to this motion? This is where the brilliant Law of Equipartition of Energy comes into play. Formulated by classical thermodynamics, this law states that for a system in thermal equilibrium at an absolute temperature $T$, the total thermal energy is shared equally among all active degrees of freedom.

Specifically, the energy associated with each degree of freedom for a single molecule is exactly:

Here, $k_B$ is the Boltzmann constant, a fundamental physical constant that bridges the macroscopic world of temperature with the microscopic world of particle energy.

To find the total average energy ($E_{av}$) of our monoatomic gas molecule, we simply multiply the energy per degree of freedom by the total number of degrees of freedom it possesses.

Since we established that $f=3$ for a monoatomic gas, we substitute this value into our master equation:

And there we have it! The total average kinetic energy of a single monoatomic gas molecule in thermal equilibrium is $\frac{3}{2}{k}_{B}T$. This elegant result is a cornerstone of the Kinetic Theory of Gases.