Chemistry 4595
Problem Set #3
Fall Semester, 1998
Due: Thursday, September 17
1.
Suppose you kick over a mound of fire ants. Disoriented, the little pests come streaming
out from the center and wander about aimlessly. For the ensemble of all ants, compute the
average position <r> and mean square position <r2> from the center of the mound as a function
of time t. Assume that each step requires t seconds and the step length is l. Also assume that all
ant motions are on a 8-point compass (i.e., step i + 1 deviates from step i by  = 45o).
2.
Evaluate Rg = < s2>1/2 for a solid cube of uniform density. Express Rg in terms of the
cube length, L. To do this problem, divide the cube into little segments and convert the
summation expression we used in class to an integral.
3.
In the study of polymer-colloid interactions, it is sometimes found that a linear polymer
(we'll model it as a stiff rod) can form a "bridge" between two colloidal particles (we'll model
them as spheres). Compute the radius of gyration, Rg, for the assembly, assuming that the linear
polymer connecting the two spheres is thin and essentially massless, while the massive spheres
have uniform density. DO NOT assume that the sphere radius, r, is much less than d, the length
of the connecting polymer, because that makes the problem too simple. Note: d is the length
between the surfaces of the spheres, not between their centers.
2r
d
Chem 4595 Homework Set #1 -- Page 1
4.
Consider a chain with 2 bonds per repeat unit as shown below. Derive an expression for
the RMS end to end length in the free rotation limit for n repeat units (2n bonds) where n is a
large number. Your answer can include bond lengths, like a and b, the number of bonds, and the
bond angle 
b O
O
a
a
b
5.
Here are some coordinates for a two-dimensional polymer made up of 20 identical beads.
0,0; 1,2; 0,4; -2,2; -4,0; -2,-2; -4,-4; -2,-6; 0,-4; 2,-6; 4,-4; 2,-2; 0,-2; 2,0; 4,-2; 6,0; 4,2; 6,4; 4,6;
2,4.
Compute the contour length (sum of absolute value of all bond lengths), the end-to-end distance,
and the radius of gyration. Now repeat the computation for a shorter polymer containing just the
first ten beads in the same locations. You could do this by hand, but it would be more fun to
write a little program to it. Show your work.
6.
Brain teaser (not directly related to polymers, just suggested by one of my graduate
students for fun). Four people are standing on one side of a bridge. They must all attend a
meeting on the other side, and the meeting begins in 17 minutes. The bridge can only support
two people at a time. It is a dark night, and each party crossing the bridge--either one or two
people--must carry a flashlight. Each person walks at a different rate, and any pair must walk
together at the rate of the slower person. The flashlight cannot be thrown back across the bridge,
but must be carried all the way back after each crossing. Find the correct sequence that enables
the four people to cross the bridge so that all arrive in 17 minutes, given these crossing times:
Amanda (1 minute); Bobby (2 minutes); Thomas (5 minutes); Zack (10 minutes). Evidently,
Microsoft asks this question to prospective employees. To become a Microserf, you must answer
in 5 minutes. There are no tricks; there are two solutions. It is said that 50 Motorola engineers
failed to solve the problem. One guy solved it by a computer program, requiring 37 minutes to
write. About half the students in my lab have solved it, but all required more than 5 minutes. A
visiting professor from Turkey has solved it fastest. No one has written the computer program to
solve it yet, let alone write it in just 37 minutes. Have fun!
Chem 4595 Homework Set #1 -- Page 2