Chem 4595
Fall 1998
Homework Set #7
Due Friday, November 13
1. The physics/theory community tends to go for number density , rather than the
concentration as grams/mL, which is c. With concentration units c, we write the osmotic
pressure relationship as:  = cRT(M-1 + A2c + ...). Using a symbol A2, for the 2nd virial
coefficient and number density, we instead have  = kT(1 + A2, + ...). Obtain the
relationship between A2 and A2,.
2. Below are some data for a simple protein. Solutions were prepared by weighing
aqueous solvent and pure, dried protein at 20 oC. Determine the molecular weight of the
protein and also make your best analysis of its radius of gyration. Can you say anything
about shape? This is the first time we deal with real numbers in osmometry. Watch your
units!
Solution
1
2
3
4
5
weights & densities
(mm H2O)
o
o
g2 / mg
g1 / gm density,
26 C
25 C
22 oC
21 oC
(yes,
(yes,
/g-mL-1
miligm!) grams! @ 26 oC
)
1.523 2.1056
1.00018 2.826
2.82
2.81
2.79
3.056 2.0009
1.00038 5.978
5.97
5.95
5.89
9.063 2.1512
1.00105 16.59
16.55
16.50
16.19
12.152 2.2003
1.00137 21.81
21.76
21.70
21.18
15.821 1.9984
1.00197 31.42
31.36
31.17
30.06
3. This problem deals with controlling polydispersity and enhancing molecular weight in
condensation polymers. If one is willing to have a star polymer, rather than a linear one,
then condensation polymerization can actually produce fairly high molecular weights and
reasonably sharp distributions. Consider the polymerization of AB (e.g., a hydroxyacid
like HOOC-(CH2)6-OH in the presence of “star” functional center--something like this:
CH2OH
HOCH2
HOCH2
CH2OH
CH2OH
CH2OH
The mole fraction expression we would derive by following the method shown in class
would be xi = (1-p)6pi-1. This has to be modified to include a degeneracy factor
representing that there are now six unreacted -OH sites for each molecule. The new mole
fraction expression is xi = (1-p)6pi-1(5+i)!/5!i!.
a) Why?
b) Derive expressions for n , w and wn in terms of the extent of reaction, p.
c) What are n , w and wn for p = 0.99?
d) Compare to the linear case.
4. Is the viscosity average for a semiflexible polymer, whose Mark-Houwink "a value" is
1.35, larger than or smaller than Mw?
5. This problem treats atoms in macromolecular terms. Compute Mn, Mw, Mz and Mz+1
in amu for the element Xenon, using the following natural abundance data from a
website: http://isotopes.lbl.gov/isotopes/toi.html. Which average corresponds most
closely with the mass value given in a periodic table? You may assume the mass
number (superscripted integer to the left of the symbol) represents the true mass in
amu.
ISOTOPE
%Natural Abundance
124
0.10
0.09
1.91
26.4
4.1
21.2
26.9
10.4
8.9
Xe 54
Xe 54
128
Xe 54
129
Xe 54
130
Xe 54
131
Xe 54
132
Xe 54
134
Xe 54
136
Xe 54
126
6. What assumptions do we make by assigning the true mass of the isotope in amu to the
mass number?