In thermodynamics, we often inquire about the occupation of a given
state of a system. This terminology derives from the quantum
underpinnings that we have already
discussed. We will want to be able to quickly say what the probability
is of occupying a state of a system, and to be able to give an answer
both relative to the occupation of other states as well as absolute.
To this end, we will need to develop what is known as the Boltzmann
factor, a probabilistic measure of the relative occupation of a given
state. Adding up all of these probabilities yields the ubiquitous
partition function which we use at first to normalize our results
and later to derive numerous other quantities. We will investigate how
the Helmholtz Free Energy relates to the
partition function.
We will apply these concepts to investigate the spectrum of
electromagnetic radiation in a cavity. Such a spectrum is given by the
Planck distribution function. We will learn that the energy density
of this radiation is given by the Stefan-Boltzmann law of radiation.
We will consider the effects of the chemical potential on the
probabilities of occupation of states, and come up with the Gibb's
Sum. We will discuss how all of these tools are sufficient to tackle
some challenging problems, such as the ideal gas.