## Related questions with answers

In one model of a solid, the material is assumed to consist of a regular array of atoms in which each atom has a fixed equilibrium position and is connected by springs to its neighbors. Each atom can vibrate in the $x, y$, and $z$ directions. The total energy of an atom in this model is

$E=\frac{1}{2} m v_x^2+\frac{1}{2} m v_y^2+\frac{1}{2} m v_z^2+\frac{1}{2} k x^2+\frac{1}{2} k y^2+k z^2$

What is the average energy of an atom in the solid when the temperature is $T$ ? What is the total energy of $1 \mathrm{~mol}$ of such a solid?

Solution

Verified**Conceptual answer.**

From the **equipartition theorem** we know two important facts for a system in equilibrium:

- By each degree of freedom the average energy is $\frac{1}{2}kT$ per molecule.
- The total energy is $\frac{1}{2}RT$ per mole by each degree of freedom.

Thus, since in this model the total energy of the solid atom depends on $v_x$,$v_y$,$v_z$,$x$,$y$, and $z$, the system has six degrees of freedom.

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