$5.6.1.$ The average energy of a single gas molecule over a wide temperature range is determined quite accurately by the formula $\bar{\varepsilon} = \frac{i}{2}kT$, where $i$ is the number of degrees of freedom of the molecule, which is equal to the number of coordinates determining its position. Using this formula, find the average energy of $H_2$,$N_2$,$H_2O$, and $CH_4$ molecules at a temperature $T$.
Theoretical Background: Degrees of Freedom & Molecular Models
In thermodynamics, a **degree of freedom ($i$)** is an independent coordinate required to completely specify the position and orientation of a molecule in space. It represents the number of independent ways a molecule can move and store kinetic energy.
At normal temperatures, we consider translational motion (the center of mass moving along coordinate axes) and rotational motion (tumbling around axes). Let's visualize all three fundamental models:
**1. Monatomic Gas (e.g., $He$,$Ne$,$Ar$)** A single atom is treated as a point mass. It can move in 3D space, but its moment of inertia is practically zero, so rotation does not store thermodynamic energy.
* **Translational ($i_{\text{trans}}$):** 3 (motion along the $X$,$Y$, and $Z$ axes, as shown by the pink arrows). * **Rotational ($i_{\text{rot}}$):** 0 * **Total $i = 3$**
**2. Diatomic Gas (e.g., $H_2$,$N_2$)** Two atoms connected by a rigid bond (a "dumbbell" model). Its motion is divided into two parts.
First, the entire center of mass can move in 3D space. * **Translational ($i_{\text{trans}}$):** 3 (motion along the $X$,$Y$, and $Z$ axes).
Second, it can rotate end-over-end. As shown in the diagram above, if the molecule lies along the $Y$-axis, it can rotate independently around the $X$ and $Z$ axes (indicated by the green arrows). Rotation strictly around its own bond axis ($Y$-axis in this specific orientation) has a negligible moment of inertia and is therefore ignored. * **Rotational ($i_{\text{rot}}$):** 2 * **Total $i = 5$**
**3. Polyatomic Non-linear Gas (e.g., $H_2O$,$CH_4$)** These molecules possess a complex 3D structure with substantial mass distribution around all three spatial axes.
* **Translational ($i_{\text{trans}}$):** 3 (motion along $X$,$Y$, and $Z$ axes, always present). * **Rotational ($i_{\text{rot}}$):** 3 (as shown by the green arrows, the molecule can tumble around the $X$,$Y$, and $Z$ axes independently). * **Total $i = 6$**
Solution
To find the average energy of a single gas molecule using the formula: $$\bar{\varepsilon} = \frac{i}{2}kT$$ we determine the number of degrees of freedom $i$ based on the geometric structures analyzed above.
For the diatomic gases $H_2$ and $N_2$, the number of degrees of freedom is $i = 5$. Therefore, the average energy of a single molecule of these gases is: $$\bar{\varepsilon}_{H_2} = \bar{\varepsilon}_{N_2} = \frac{5}{2}kT$$
For the non-linear polyatomic molecules $H_2O$ and $CH_4$, the number of degrees of freedom is $i = 6$. Therefore, the average energy of a single molecule is: $$\bar{\varepsilon}_{H_2O} = \bar{\varepsilon}_{CH_4} = \frac{6}{2}kT = 3kT$$
