Boltzmann found that the absolute entropy Sof the system is proportional to
the natural logarithm of the number of possible combinations:
Sln
To make a proportionality an equality, a proportionality constant is necessary:
Skln (3.24)
where kis known as Boltzmann’s constant.
There are several important ramifications of equation 3.24. First, it intro-
duces the concept that an absoluteentropy can be determined. Entropy thus
stands alone among state functions as the only one whose absolute values can
be determined. Therefore, in large thermodynamic tables ofUand Hvalues,
parallel entries for entropy are for S, not S. It also implies that the entropies
found in tables are not zero for elements under standard conditions, because
we are now tabulating absoluteentropies, not entropies for formation reac-
tions. We can determine changesin entropies,S’s, for processes; up to now we
have dealt exclusively with changes in entropy. But Boltzmann’s equation 3.24
means that we can determine absolute values for entropy.
Second, equation 3.24 brings up an intriguing notion. Consider a system
where all species (atoms or molecules) of the component are in the same state.
One way of illustrating this is to assume that it is in the form of a perfect
crystal, implying perfect order. If this was the case, then (the number of
possible combinations of conditions that would have this arrangement) would
be 1, the logarithm of would be zero, and thus Swould be zero. It seems un-
likely that such a circumstance might exist under normal conditions.
However, science has the ability to dictate the conditions of systems under
study. In the late 1800s and early 1900s the properties of matter at extremely
low temperatures were being investigated. As the thermodynamics of materials
were measured at temperatures approaching absolute zero, the total entropy
of cold, crystalline materials—which could be measured experimentally using
expressions like equation 3.18—began approaching zero. Since entropy is an
obvious function ofTfor all substances, the following mathematical statement
became obvious:Tlim→0KS(T) ^0 for a perfectly crystalline material (3.25)
This is the third law of thermodynamics, which can be stated verbally as
follows:
The third law of thermodynamics: Absolute entropy approaches zero
as the absolute temperature approaches zero.
Thus, this statement provides entropy with an absolute minimum value of zero
and establishes the ability to determine absolute entropies. Equation 3.24,
defining a statistical origin of entropy, is such a fundamental idea in science
that it is carved on Ludwig Boltzmann’s tombstone in Vienna. (See Figure 3.7.)
Boltzmann’s constant is, interestingly enough, related to the ideal gas law
constant R. It can be shown that
RNAk (3.26)
where NAis Avogadro’s number (6.022 1023 ). The constant ktherefore
has a value of 1.381 10 ^23 J/K. Its relative magnitude implies that there are
an enormous number of possible combinations of states that atoms and mol-
ecules of macroscopic samples can adopt, as seen in the following example.80 CHAPTER 3 The Second and Third Laws of ThermodynamicsFigure 3.7 Above the bust of Boltzmann,
you might be able to make out the equation
Skln.Courtesy of Frantisek Zboray, Vienna