Page 135 - Essentials of physical chemistry

The Second and Third Laws of Thermodynamics 97

FIGURE 5.7 Placing only six labeled red chips in the carton shows the many possibilities. Here that
corresponds to hiding the labels but the principle should be clear.

Thus, even though there are still 924 possible ways the (six red, six white) arrangement could have
been achieved, we now have a way to take care of the fact that all the chips are the same. So, for a
binary system of (a, b) species, we have a formula for W(a, b)as

(N a þ N b )! (N a þ N b )!
and so S ¼ k ln :
W(a, b) ¼
(N a !)(N b !) (N a !)(N b !)
Now, we have to worry over the fact that we have to treat a number of particles of the order
of Avogadro’s number in the 10 23 range. Fortunately, two mathematical tricks are available at this
point. First, we should realize that we can use the logarithm of the number of particles, which will
be a much smaller number while still following the same trends as the number itself. Second, we
have available Stirling’s approximation for large values of ln (n!) as found in several handbooks
(Table 5.4).

ffiffiffiffiffiffiffiffiffi
p n n
Stirling s approximation: n! ffi ( 2pn)n e :
0
We can take the natural logarithm of this expression to find a simpler formula:
ffiffiffiffiffiffiffiffiffi
p
ln (n!) ¼ n ln (n) n þ ln ( 2pn):

TABLE 5.4
Selected Values of Stirling’s Approximation for ln (n!)
n n! ln (n!) n ln (n) n Error Error (%)
10 3.6288E6 15.1044 13.0258 2.0786 13.7616
20 2.4329E18 42.3356 39.9146 2.4210 5.7186
30 2.6525E32 74.6582 72.0359 2.6223 3.5124
40 8.1592E47 110.3206 107.5552 2.7654 2.4428
50 3.0414E64 148.4777 145.6012 2.8765 1.9373
60 8.3210E81 188.6282 185.6607 2.9675 1.5732