The relationship between the thermodynamic and Boltzmann entropies: the role of degrees of freedom.

Abstract This paper shows that the statistical and thermodynamic entropies are not generally equivalent. However, they are connected as changes in the internal energy cause changes to the distribution of energy among the effective number of active degrees of freedom in a system. In systems in which the heat capacity is independent of temperature, such as an ideal, classical gas, the number of degrees of freedom is independent of temperature and the two are seen to be equivalent. In more complex systems it is necessary to define an effective number of degrees of freedom. This is done for some solids and it is shown that in such cases that the thermodynamic entropy is not the same as the Boltzmann entropy. The consequences for the chemical potential are explored and the educational implications are discussed.