- Problems[[Student version, January 17, 2003]] 297
form the ion HPO^24 −.Inreality, all four possible protonation states, from three H’s to none, exist
in equilibrium. The three successive proton-removing reactions have the following approximate pK
values:
H 3 PO 4
pK 1 =2
H 2 PO− 4pK 2 =7
HPO^24 −pK 3 =12
PO^34 −.Find the relative populations of all four protonation states at the pH of human blood, around 7.4.
8.7Electrophoresis
In this problem you will make a crude estimate of a typical value for the electrophoretic mobility
of a protein.
a. Let us model the protein as a sphere of radius 3nm,carrying a net electric charge of 10e,in
pure water. If we apply an electric field ofE=2volt cm−^1 ,the protein will feel a forceqE.Write a
formula for the resulting drift velocity and evaluate it numerically.^7
b. In the experiment discussed in Section 8.3.4, Pauling and coauthors used an electric field of
4. 7 volt cm−^1 ,applied for up to 20 hours. In order for a mixture of normal and defective hemoglobin
to separate into two distinguishable bands, they must travel different distances under these condi-
tions. Estimate the separation of these bands for two species whose charges differ by just one unit,
and comment on the feasibility of the experiment.
8.8 T 2 Grand partition function
Review Section 8.1.2 on page 262.
a. Show that the distribution you found in Your Turn 8b is the one that minimizes thegrand
potentialof system “a” atT,μ,defined by analogy to Equation 6.32 as
Ψa=〈Ea−μNa〉−TSa. (8.37)b. Show that the minimal value of Ψ thus obtained equalskBTlnZ.
c.Optional: Forthe real gluttons, generalize the above results to systems exchanging particles
and energy,andchanging volume as well (see Section 6.5.1).
(^7) T 2 Actually, one uses a salt solution (buffer) instead of pure water. A more careful treatment would account for
the screening of the particle’s charge (Section 7.4.3′on page 250); the result contains an extra factor of (3/2)(λD/a)
relative to your answer.