Page 78 - Essentials of physical chemistry

40 Essentials of Physical Chemistry

4.5
4
Number of students 2.5 3 2
3.5

1.5

0.5 1
0
0 20 40 60 80 100 120
Student grade
FIGURE 3.2 Hypothetical grade distribution in a class of 15 students.

In this author’s experience, this concept needs to be identified in every place it occurs to help
students understand what is being averaged. First, let us consider a simple example of weighted
averaging. Suppose, we administer a midterm examination to a class of 15 students and record the
grades on a graph from the data (Figure 3.2). Although the example is a discrete distribution, we
have plotted the data so you can see a ‘‘distribution function’’ line, and we anticipate that if we had
grades in increments of 1 point and a class of 250 freshmen, the graph would be a smoother ‘‘curve’’
but still based on a discrete set of points.
We now come to a key idea, which the student should make sure he or she understands since we
will use it over and over in later applications. We introduce the symbol hi to denote an averaging
process, in this case a ‘‘weighted average’’ as shown in Table 3.1.

P
1(100) þ 2(90) þ 3(80) þ 4(70) þ 3(60) þ 1(50) þ 1(40)
i n i G i
¼ P ¼ 71:33:
1 þ 2 þ 3 þ 4 þ 3 þ 1 þ 1 n i
hGi¼
i
P
Note that the symbol indicates a discrete summation over specific values. Also make sure you
i
note that we are weighting the G i value by the number of times it occurs or is ‘‘weighted’’ in the
summation. To gain perspective, we could look at the graph distribution and use the weighting of a
given grade, say n i =15 to estimate the probability that a student would get a certain grade G i . So far
that is sort of obvious, but the interesting point is that we have to divide ‘‘by the number of students
in the class’’ and in effect this ‘‘normalizes’’ the process to the average grade for just ‘‘one average’’
grade of a hypothetical single student. This ‘‘normalizing’’ process will be a key idea in several
applications in quantum chemistry as well as here for Boltzmann averaging. Next, we need to take
a side trip to the amazing discovery of Boltzmann weighted averaging and ask the question

TABLE 3.1
Weighted Average Grade of Class
No. of Students Grade
1 100
2 90
3 80
4 70
3 60
1 50
1 40
Weighted average (1070=15) ¼ 71.33