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518 14MOLECULAR INTERACTIONS
Solutions to problems
P14E.2 �e equilibriumconstant for the formation ofmicelles containingNmonomers,
MN , is given by [14E.6b–627] (the factors of 1/c−○ are omitted for clarity)
K = [MN]
([M]total − N[MN])N
In the text the fraction of surfactant molecules present as micelles is de�ned
as f = [MN]/[M]total; it could be argued that a better de�nition would be f =
N × [MN]/[M]total on the grounds that each micelle contains N monomers.
Using the �rst de�nition for f the equilibrium constant is rewritten
K = f
[M]N−1total(1 − N f )N
�is equation is solved numerically to �nd f as a function of [M]total for given
values of K and N .�e results for three such situations are shown in Fig. 14.14.
For N = 30 there is a sudden increase in f once [M]total/c−○ exceeds a certain
value; this corresponds to the criticalmicelle concentration.�is concentration
depends on both N and K.
0 2 4 6 8 10
0.0
0.1
0.2
0.3
0.4
[M]total/c−○
f
f (K = 1 N = 3)
10 f (K = 1 N = 30)
10 f (K = 10−6 N = 30)
Figure 14.14
Solutions to integrated activities
I14.2 (a) �e Lennard-Jones potential is given by [14B.12–600],VLJ(r) = 4ε{(r0/r)12−
(r0/r)6}. As shown in Fig. 14B.8 on page 600, the depth of the potential
well is given by ε and the position of the minimum is given by re = 21/6r0.
�e results for He2 give the depth of the well as hcD̃e = 1.51×10−23 J and
the position of the minimum as R = 297 pm so it follows that
ε = 1.51 × 10−23 J and r0 =
re
21/6
= 297 pm
21/6
= 265 pm
A plot of the potential with these values is shown in Fig. 14.15.