FIGURE 20.9 reminds you of our “bedspring model” of a solid with particle-like atoms connected by a lattice of spring-like molecular bonds. How many degrees of freedom does a solid have? Three degrees of freedom are associated with the kinetic energy, just as in a monatomic gas. In addition, the molecular bonds can be compressed or stretched independently along the x-, y-, and z-axes. Three additional degrees of freedom are associated with these three modes of potential energy. Altogether, a solid has six degrees of freedom.

FIGURE 20.9 A simple model of a solid.

The energy stored in each of these six degrees of freedom is $\frac{1}{2}N{k}_{\text{B}}T$. The thermal energy of a solid is the total energy stored in all six modes, or

We can use this result to predict the molar specific heat of a solid. If the temperature changes by $\text{Δ}T$, then the thermal energy changes by

In Chapter 19 we defined the molar specific heat of a solid such that

By comparing Equations 20.34 and 20.35 we can predict that the molar specific heat of a solid is

Not bad. The five elemental solids in Table 19.2 had molar specific heats clustered right around $25\text{J}\text{/}\text{mol}\text{K}$. They ranged from $24.3\text{J}\text{/}\text{mol}\text{K}$ for aluminum to $26.5\text{J}\text{/}\text{mol}\text{K}$ for lead. There are two reasons the agreement between theory and experiment isn’t quite as perfect as it was for monatomic gases. First, our simple bedspring model of a solid isn’t quite as accurate as our model of a monatomic gas. Second, quantum effects are beginning to make their appearance. More on this shortly. Nonetheless, our ability to predict C to within a few percent from a simple model of a solid is further evidence for the atomic structure of matter.