Page 134 - Essentials of physical chemistry

96 Essentials of Physical Chemistry

FIGURE 5.5 Twelve numbered white poker chips in a one dozen egg carton.

FIGURE 5.6 Twelve ‘‘indistinguishable’’ white chips in 12 egg wells are perfectly ordered.

The concept of ‘‘indistinguishability’’ is important in molecular science because in many cases
there is no way to tell one H atom from another as well as many other indistinguishable cases.
Now suppose we want a mixture of red and white chips in the egg carton such that the sum of
poker chips is still 12. We could put some red chips in the empty carton in some order as shown in
Figure 5.7. Then we could put some numbered white chips into the empty egg wells. Finally, we
could turn over the red and white chips to hide the numbering.
Suppose we fill in just six red chips in some random order and hide the labels and then fill the
other egg wells with white chips that also have hidden labels, as in Figure 5.6. Now the question is,
what is W for this mixture of red and white chips?

12! 479, 001, 600
¼ 924:
(6!)(6!) 518, 400
W ¼ ¼
The 12! in the numerator comes from the fact that there are a total of 6 þ 6 ¼ 12 chips and we could
have picked any of the 12 as the first, second, third, etc. to place in the egg carton without noting
their color. However, the possible order within the red set of chips could have only been chosen
in 6! ways and only 6! ways would have been possible for the order of choosing the white chips.