FIGURE 20.8 shows a monatomic gas such as helium or neon. The atoms in an ideal gas have no molecular bonds with their neighbors; hence $U_{\text{micro}}=0$. Furthermore, the kinetic energy of a monatomic gas particle is entirely translational kinetic energy $ϵ$. Thus the thermal energy of a monatomic gas of N atoms is

where ${ϵ}_i$ is the translational kinetic energy of atom i. We found that ${ϵ}_{\text{avg}}=\frac{3}{2}{k}_{\text{B}}T\text{;}$ hence the thermal energy is

where we used $N=n{N}_{\text{A}}$ and the definition of Boltzmann’s constant, $k_{\text{B}}=R\text{/}{N}_{\text{A}}$.

FIGURE 20.8 The atoms in a monatomic gas have only translational kinetic energy.

We’ve noted for the last two chapters that thermal energy is associated with temperature. Now we have an explicit result for a monatomic gas: $E_{\text{th}}$ is directly proportional to the temperature. Notice that $E_{\text{th}}$ is independent of the atomic mass. Any two monatomic gases will have the same thermal energy if they have the same temperature and the same number of atoms (or moles).

If the temperature of a monatomic gas changes by $\text{Δ}T$, its thermal energy changes by

In Chapter 19 we found that the change in thermal energy for any ideal-gas process is related to the molar specific heat at constant volume by

Equation 20.29 is a microscopic result that we obtained by relating the temperature to the average translational kinetic energy of the atoms. Equation 20.30 is a macroscopic result that we arrived at from the first law of thermodynamics. We can make a micro/macro connection by combining these two equations. Doing so gives us a prediction for the molar specific heat:

This was exactly the value of $C_{\text{V}}$ for all three monatomic gases in Table 19.4. The perfect agreement of theory and experiment is strong evidence that gases really do consist of moving, colliding molecules.