wt,;L
4a
,
:
Questions
on this exam concern dilute amphiphilic molecules
in (and at the
surface of) aqueous solutions. One such molecule (denoted "A"
for
"amphiphile"),
consisting of a polar head and a nonpola,r
tail, is sketched below.
A
=
?*0""
n"':
.,
)-nonRolar
tail
1. These amphiphilic solutes can shield their hydrophobic tails from solvent by forming
micelles,
roughly spherical clusters of solutes whose head groups point outward
toward solvent.
micelle
(cross
section)
Each micelle
(denoted A,) is made up of n amphiphile molecules. We will not
specify a value for n,.except to say that it is much greater than one.
Let
p,1be the chemical
potential of an individual amphiphile A that is well
separated
from other solutes. Let p,n be the chemical
potential of an entire micelle
A,o. Similarly, we will denote the number of unassociated amphiphiles
per unit
volume # pr, and the number of'micelles
per unit volume 6
Pn.
Finally, let N be
the total number of amphiphiles
(including those
in micelles), so that c
:
NIV
:
h
* npn is their total concentration'
Temperature ? and pressure p will be fixed throughout this problem.
(i) What is the change AG,, in Gibbs free energy when ri, amphiphiles a,re brought
iogether to form a micelle, i.e., resulting from a single instance of the reaction
ni
--+ Arr. Your answer should involve some (but not necessarily all) of the
following
quantitles: the chemical
potentials p1 and
1.rn,
the size n of a micelle, ?,
p, N, c, and fundamental constants.