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+### Statement
+
+$5.6.1.$ [Insert the problem statement]
+
+### Solution
+
+### Statement
+
+$5.6.1.$ 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.
+
+![Monatomic Translation](Degree-of-freedom-1.png)
+* **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.
+
+![Diatomic Translation](Degree-of-freedom-3.png)
+First, the entire center of mass can move in 3D space.
+* **Translational ($i_{\text{trans}}$):** 3 (motion along the $X$, $Y$, and $Z$ axes).
+
+![Diatomic Rotation](Degree-of-freedom-4.png)
+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.
+
+![Polyatomic Rotation](Degree-of-freedom-2.png)
+* **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:
+$$\bar{\varepsilon}_{H_2O} = \bar{\varepsilon}_{CH_4} = \frac{6}{2}kT = 3kT$$
+
+#### Answer
+$$\bar{\varepsilon}_{H_2} = \bar{\varepsilon}_{N_2} = \frac{5}{2}kT$$
+$$\bar{\varepsilon}_{H_2O} = \bar{\varepsilon}_{CH_4} = 3kT$$
+
+#### Answer
+
+[Insert a concise answer or boxed result]