Page 79 - Essentials of physical chemistry

The Kinetic Molecular Theory of Gases 41

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‘‘What would the weighting factor be if we are averaging a sample of a mole (6.022 10 ) values
for molecules and the range of values of some molecular property is (nearly) continuous?’’
We believe that Boltzmann probability weighting is one of the most ‘‘essential’’ concepts in this
whole text. We saw above that we can itemize the values of the thing we want to average, but we
also need to weight that value and the weighting can be considered as a probability. Boltzmann may
have considered a number of mathematical relationships before he ‘‘discovered’’ the formula, now
given his name. The basic consideration is that for 3D quantities such as kinetic energy, one needs to
merge the contributions in the various (x, y, z) directions in a way that will represent the total
probability. This follows the common sense understanding of probabilities in that if an event is
1=3in x,1=3in y, and 1=3in z, then the probability of the event simultaneously in x, y, and z is 1=27.
The idea that a probability product must also add energies leads to putting the energy expression in
an exponent where a product will add exponents. The properties of the natural logarithm base ‘‘e’’
were known for over 100 years when Boltzmann wrote his PhD thesis, so he chose that base,
realizing that separate probabilities in different coordinates would add in the exponent of a common
base. We should emphasize that the weighting number n i above is really the probability of the value
being weighted when you divide by the number of values. Boltzmann knew that ‘‘energy runs down
hill’’ so that higher energies are less probable. On the other hand, he also knew that higher
temperatures favor higher energies. Add those factors to your thinking and perhaps you will
come up with the same result as he did:

e
Boltzmann probability ¼ e e=kT ¼ exp :
kT
This amazing relationship came from Boltzmann’s consideration of all the above factors. We do not
actually know how Boltzmann thought of this relationship, and it was sheer genius at the time, but
we need to appreciate the several aspects of this very important mathematical expression. If the
temperature is constant, then a given probability is favored less as the energy e increases and makes
the exponential more negative. If the energy is constant, then a higher temperature makes it more
probable, while a lower temperature makes it less probable. Finally, if we have different forms of
energy such as (e x , e y , e z ), the probabilities will add in the exponent to a common base, namely, e.
We should also appreciate that e is a microscopic energy on the atomic scale and that k ¼ R=N Av
is the ideal gas constant divided by Avogadro’s number N Av so k ¼ R=N Av ¼ 1:38 10 16
erg= K ¼ 1:38 10 23 J= K.

Our goal here is to simplify some topics but still treat the full details of the difficult material.
Instead of a cubical box, we can shift our attention to a spherical system and consider the motion of
atoms=molecules in any direction with a scalar speed v. Velocity is a vector quantity, which requires
both a scalar value and a direction. An automobile speedometer indicates only scalar speed and
needs to be accompanied by a compass reading to determine the direction needed to specify the
velocity of the automobile. Here, we can integrate over all angles (u, f) with the factor of (4p)
shown in the Mathematics Review and only consider the scalar speed v. We can follow the example
above of the average grade to set up the average kinetic energy of atoms=molecules in a spherical
2
system. Here, we clearly see that the thing we are averaging is (mv =2) and the weighting
2
probability factor is the Boltzmann factor with e ¼ (mv =2), assuming the only form of energy
the particles have is kinetic energy with essentially no potential energy. Actually, there should be an
expression of some form of potential energy when the atoms=molecules get very close, but we will
see that most of the time they are very far apart. This is called the ‘‘hard sphere’’ approximation.
Note that the denominator is needed to divide by all the possible values of the probability just as we
divided the weighted grades by the number of students in the class. Later, we will see that the
denominator is always the same in Boltzmann averaging these quantities over a spherical volume
and can be inverted (flipped) as a factor in the numerator. Here, we want to emphasize the idea of the
weighted average compared to the example of the class average grade.