Animated Solution for Physics - Kinetic Theory of Gases: An HCl molecule has rotational, translational and vibrational motions. If the rms velocity of HCl molecules in its gaseous phase is $\bar{v}$, $m$ is its mass and $k_B$ is Boltzmann constant, then its temperature will be

Imagine you are observing a single molecule of Hydrogen Chloride ($\text{HCl}$) floating in a gaseous state. It is not just sitting still; it is performing a complex, energetic dance.

The problem explicitly tells us that this molecule is exhibiting translational, rotational, and vibrational motions.

This is a crucial piece of information because it dictates how the molecule stores its thermal energy.

To understand the energy of the molecule, we first need to count its degrees of freedom ($f$). Think of degrees of freedom as the independent ways a molecule can move and store energy.

For a diatomic molecule like $\text{HCl}$: 1. It can move in 3-dimensional space (x, y, z axes), giving it 3 translational degrees of freedom. 2. It can rotate around two independent axes perpendicular to the bond connecting the atoms, giving it 2 rotational degrees of freedom. 3. Because the problem specifies vibrational motion, the atoms can oscillate back and forth along the bond. This contributes 2 vibrational degrees of freedom (one for kinetic energy and one for potential energy).

Adding these up, we get the total degrees of freedom:

Now, we bring in a powerful principle from statistical mechanics: the Law of Equipartition of Energy.

This law states that in thermal equilibrium, every single degree of freedom contributes an equal amount of average energy to the molecule, specifically $\frac{1}{2}{k}_{B}T$, where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.

Since our $\text{HCl}$ molecule has 7 degrees of freedom, its total average thermal energy ($E$) is:

We also know from the kinetic theory of gases that the total kinetic energy of a molecule can be expressed macroscopically using its root mean square velocity ($v_{\text{rms}}$).

The formula for this kinetic energy is:

The problem states that the rms velocity is $\bar{v}$. So, we can write:

We now have two different expressions for the same total energy of the molecule. Let's equate them to find the temperature $T$:

Notice how beautifully the $\frac{1}{2}$ on both sides cancels out!

Finally, we isolate $T$ by dividing both sides by $7k_B$:

And there we have it! The temperature of the gas is directly related to the mass of the molecule, the square of its rms velocity, and inversely proportional to its degrees of freedom. This perfectly matches option (b).