Boltzmann's seductive connection of statistics to thermodynamics holds (at least) one more
instructive surprise . . . not counting Shannon's "surprisal." If we rearrange the
expression to focus upon the ratio of state populations to the total population, we see the
origin of an important entropy expression remembered from Physical Chemistry. Those ratios
are the probabilities of occupancy of the system's states (wavefunctions) and thus
the mole fraction of the systems occupying those states.Even without invoking the Boltzmann Distribution (the
i haven't been given
their Boltzmann values here), we see that S=k(lnW) is the
"Entropy of Mixing" of the states. Note that this isn't just about ideal fluids
mixing, although the expression is identical; it is a fundamental truth about Entropy:
All Entropy is the Entropy of Mixing. It represents the measure of multitude of
different configurations an ensemble represents and, of even greater value, the direction
in which it may evolve to increase that multitude.So Entropy is not just a temperature-scaled reversible heat or even just a measure of the Enthalpy unavailable for Work; it is, as we see now, a handle on the complexity (W) of the Universe . . . or at least of our system's corner of it.
As fundamental as is Entropy, we'll find that the single most valuable quantity linking the statistical to the thermodynamic world is Z, the partition function. As the "Sum over States," the partition function is adding up the fractional availability of states. Indeed the mole fraction, Xi, of the last derivation is, when we use the Boltzmann Distribution, nothing more than the exponential factor (the availability of a given state, i) in ratio to the partition function (the availability of all states).
One could be forgiven for asking "availability for what?" . . . thermal availability for occupancy. Each exponential factor's ratio of state energy,
i , to kT is a
measure of that fraction of the systems at temperature T which have energy at least as
great as that state requires. (Remember the Arrhenius exponential factor?) Since 100% of
any system has at least the zero point energy (at any temperature), Z, the sum of all the
exponential factors, is always a number no less than one.The greater Z becomes, the more states are within reach of a thermal system, but it would be a mistake to assume that Z=2 means two states can be occupied. Instead, Z=2 means that the ground state is fully available (which accounts for 1), and the sum of the fractional availability of all higher states adds another 1 to that. The practical value of Z is that a species A isoenergetic with a rearrangement of itself, B, but with twice B's partition function (twice B's available states) will come to equilibrium with 2/3 in the A form and 1/3 in the B form. This is the fundamental stuff of chemical equilibrium as we'll see in the next lecture.
There's a temptation to consider those state energies as energy levels instead, and, of course, they are, but it would be a mistake to assume that Z is the "sum over levels." It isn't. It's the "sum over states." The distinction isn't pedantic; instead, it recognizes that some energy levels are degenerate. More than one wavefunction (state) are associated with such a level. If a level is triply degenerate, its exponential factor will appear three times in the "sum over states." And so on.
Often explicit recognition is given to the degeneracies by summing over the levels (instead of the states) and correcting for the undercount with degeneracy factors, like the 2J+1 of diatomic rotation fame. This means that we must be very aware of all the sources and magnitudes of degeneracies, some of which are fiendishly subtle (like the rotation factors of symmetrical molecules which, by diminishing the partition function, run counter to the effect of degeneracy . . . more about that later).