292 Chapter 8. Chemical forces and self-assembly[[Student version, January 17, 2003]]
In an ideal gas or other collection of independent particles, for example a dilute solution, we
haveμ=kBTln(c/c 0 )+μ^0 (Equation 8.3). Herecis the number density andc 0 is a con-
ventional reference concentration.μ^0 depends on temperature and on the choice of reference
concentration, but not onc.For a charged ion in an external electric potential, addqV(x)to
μ^0 ,toget the “electrochemical potential.”- Grand ensemble: The probability of finding a small subsystem in the microstatei,ifit’s in
contact with a reservoir at temperatureTand chemical potentialsμ 1 ,...,is(Equation 8.5)
Z−^1 e−(Ei−μ^1 N^1 ,i−μ^2 N^2 ,i···)/kBT.HereZis a normalization factor (the partition function) andEi,N 1 ,i,...are the energy and
populations for stateiof the subsystem.- Mass Action: Consider a reaction in whichν 1 molecules of species X 1 ,... react in dilute
solution to formνk+1 molecules of species Xk+1 and so on. Let ∆G^0 =−ν 1 μ^01 −···+
νk+1μ^0 k+1+···,and let ∆Gbethe similar quantity defined using theμ’s. Then the equilibrium
concentrations obey (Equation 8.16)
∆G=0 ,or [Xk+1]νk+1···[Xm]νm
[X 1 ]ν^1 ···[Xk]νk
=Keq.where [X]≡cX/(1M)andKeq=e−∆G(^0) /kBT
. Note that ∆G^0 andKeqboth depend on the
reference concentrations chosen when defining them. The above formula corresponds to taking
the reference concentrations all equal to 1M.Often it’s convenient to define pK=−log 10 Keq.
If the “reaction quotient” above differs fromKeq,the system is not in equilibrium and the
reaction proceeds in the net direction needed to move closer to equilibrium.
- Acids and bases: The pH of an aqueous solution is−log 10 [H+]. The pH of pure water
reflects the degree to which H 2 Odissociates spontaneously. It’s almost entirelyundissociated:
[H+]=10−^7 ,whereas there are 55 mole/Lof H 2 Omolecules. - Titration: Each residueαof a protein has its own pKvalue for dissociation. The probability
to be protonated,Pα,equals 1/2 when the surrounding solution’s pH matches the residue’s
pK.Otherwise we have (Equation 8.28)
Pα=(1+10xα)−^1 ,wherexα=pH−pKα.- Critical micelle concentration: In our model the total concentration of amphiphile molecules
ctot is related to the concentrationc 1 of those remaining unaggregated byctot =c 1 (1 +
(2c 1 /c∗)N−^1 )(Equation 8.32). The critical micelle concentrationc∗is the concentration at
which half of the amphiphile molecules are in the form of micelles; its value reflects the
equilibrium constant for self-assembly.
Further reading
Semipopular:
On the physics and chemistry of food: McGee, 1984.
Intermediate:
Biophysical chemistry: Atkins, 2001; Dill & Bromberg, 2002; van Holde et al., 1998; Tinoco Jr.