Page 136 - Essentials of physical chemistry

98 Essentials of Physical Chemistry

The last term of this expression becomes negligible for very large values of ‘‘n,’’ so we arrive at a
very useful approximation for the natural logarithm of n!as

ln (n!) ffi n ln (n) n:

We can see that n! soon exceeds the range of even a 10-place calculator, but we can also see that the
23
percent error decreases as the number gets larger and that when n ¼ 10 , the approximation will be
very good.
Now consider an egg carton with only red poker chips and another with only white chips. Let red
chips be the ‘‘A’’ chips and ‘‘B’’ the white chips. Then in each perfectly ordered carton box we
would have (the numbers need not be six and six but could be eight and four etc.)

N A ! N B !
S A ¼ k ln ¼ 0 for the red chips and S B ¼ k ln ¼ 0 for the white chips:
N A ! N B !
Then after mixing the red and white chips in a larger ‘‘two dozen’’ crate as a simulation of pouring
two liquids together, we suppose that A and B are something like n-heptane and n-octane, which are
so similar in structure and nonpolar that there is very little energy interaction between them so we
make the approximation that DH mix ¼ 0. Thus, whatever happens in this mixing is a result of only
an entropy effect. Now let us calculate DS mix ¼ S after S before using the Boltzmann formula:

N! N A ! N B !
DS mix ¼ k B ln k B ln k B ln
(N A !)(N B !) N A ! N B !
and then apply Stirling’s approximation:

DS mix ¼ k B [N ln N N N A ln N A þ N A N B ln N B þ N B ] 0 0,
where

N ¼ N A þ N B :

So,

DS mix ¼ k[N A ln N A þ N B ln N B N ln N]:

Since N A þ N B cancels N inside the bracket

DS mix ¼ k B [N A ln N A þ N B ln N B (N A þ N B )ln N]:

N A N B N N A N A N B N B
DS mix ¼ k B N A ln þ N B ln ¼ Nk B ln þ ln , where
N N N N N N N
we have multiplied the bracket by (N=N) and dragged N under both terms. What if N is equal to
Avogadro’s number? That will convert the expression to a molar basis. Note that N A =N is a mole
fraction.
N A N B
are the respective mole fractions of the A and B species, and we obtain
A
B
x ¼ and x ¼
N N
the final expression in terms of mole fractions and if Nk B ¼ n tot R the final result is
DS mix ¼ nR[x ln x þ x ln x ],
B
B
A
A