The change in height of the potential energy barrier at the surface of a metal with the expansion of the metal due to heating is investigated. Also, the change of the normal maximum energy of an electron in a metal is calculated as a function of the temperature of the metal. These calculations show that the work function of a metal is a linear function of its temperature. These results, when combined with the thermionic emission equation, show that the thermionic emission constant is a characteristic of the metal, and is no longer the same for all pure metals. The calculated results for the thermionic emission constants for several metals show fair agreement with the experimentally determined values.