wavelength be much, much smaller than the distance between the particles. If
this is so, then the thermal de Broglie wavelength of any two particles is neg-
ligible compared to their separation, and the individual gas particles can be
considered truly independent of each other. Therefore, conditions of low pres-
sure and high temperature—both of which contribute to an increased inter-
atomic separation—are desirable when comparing theory with experiment. We
will see examples of how naive predictions of statistical thermodynamics do
not agree with experiment at particularly low absolute temperatures in the
next chapter.
17.8 Summary
The mathematics of statistics are applicable to atoms and molecules, as we have
seen by applying statistical math to a monatomic gas. By considering how
many ways we can distribute energy among many gas particles that are dis-
tributed into an ensemble (a canonical ensemble, in particular), it becomes
clear that the overall properties of the gas can be understood if we know the
properties on only one distribution, the most probable distribution. This un-
derstanding leads to an expression for the partition function. Using the statis-
tical mathematics of averages, we can express measurables like energy and en-
tropy in terms of that partition function. The partition function,q,thus
becomes the central focus of our understanding of the statistical nature of
thermodynamics.
Quantum mechanics and calculus allows us to determine an explicit ex-
pression for qfor the three-dimensional motion of the gas particles. Using that
expression, we can determine expressions for E(called Uin phenomenologi-
cal thermo),H, and heat capacities. The true test, however, is S: we know
absolute values ofSexperimentally, so a comparison ofSvalues determined
experimentally with those calculated using statistical thermodynamics is cru-
cial. Table 17.1 shows that the equations derived from a statistical approach to
thermo passed the test.
Boltzmann’s derivations depended on the existence of matter being, ulti-
mately, particulate. This is consistent with modern atomic theory. Boltzmann’s
ideas—including the idea that atoms behave statistically—have been accepted
as a correct understanding of matter.
17.8 Summary 613