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Cell & Molecular Biophysics Assignment 3 Fall 2015

1. A simple model of a polymer has a 1-D freely jointed chain where Nlinks of length lcan forma 1-D random walk path. Suppose that nof the links point left and mlinks point right. The end-to-end length of the polymer isL= (n-m) landN= n+ m.a) Working by analogy with the entropy for particles in a box, and writing nand min terms ofN,l and L as in the 1-D random walk derivation, show (using Stirlings approximation) that theentropy, S, of a general configuration for the chain in terms ofN, landLis given by:

1 ln 1 1 ln 1 2ln 22

B

N L L L LS k

Nl Nl Nl Nl

b) If the polymer is completely stretched out soL=Nl, what is the entropy? What is the entropyifL= 0?c) Show that the entropy is a maximum whenL= 0 (i.e. when m= n).d) Suppose now, that the end-to-end length is much less than the total contour length the polymerwould have if all of its Nlinks were stretched out into a line (i.e. if L/Nl

7/23/2019 Assignment 3 2015

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Cell & Molecular Biophysics Assignment 3 Fall 2015

and plotted the net displacement in that time interval. He collected 500 data points in this wayand calculated

1/2for the displacement to be 7.84 m.

a) Use this information to obtain an estimate for the Boltzmann constant, kB.b) Use this kBestimate to find an estimate of Avogadros Number.

4. [Required for Grad students/Optional for Undergrads] Brownian motion and gravitationalsedimentation can be combined naturally by considering that a particles position higher up inthe suspension has energy Egrav= mgz,where m is the difference between the mass of theparticles and the mass of the water they displace, g is the acceleration of gravity and z is theheight above the bottom of the tube. One can then write a Boltzmann factor to find equilibriumparticle concentration as a function of height.a)Nparticles are initially mixed into volume V, to achieve concentration co=N/V. The density

of the particles is p. They are suspended in a fluid which has a density f. If the container is aflat-bottomed cylindrical tube of height h = 0.1 m write the equation for the equilibriumconcentration profile as a function of heightz. Note that the earth's gravity acts in the z-direction,with acceleration g= 9.8 m/sec

2. Hint: Use a Boltzmann factor for the relative probability wrt the

probability at the bottom of the container. Find the proportionality constant (it is not just co) from

the fact that integrating ( )c z dV over the volume should give the total number of particles,N.b) Plot the relative concentration c(z)/c0 as a function of height for z = 0 to h (= 0.1 m) for

mg/kBT= 2 m-1

and mg/kBT= 2 x 106m

-1.

c) One of the other experiments that Jean Perrin did to calculate Avogadros number involvedlooking at the equilibrium concentration of emulsions of rubber particles suspended in water (seethe website http://web.lemoyne.edu/~giunta/perrin.htmlfor more information). Perrin found therelative concentration of granules of the vegetable latex gamboge (density = 1.2067 g/cm

3;

radius = 0.213 m) by counting 13,000 granules in total. This was done at room temperature(20C). If we define the relative concentration at h = 5 m from the bottom of the viewingchamber to be 1.00, and then we get the following data for relative concentration as a function ofincreasing height above the bottom of the chamber:

Height Relative Density(m) [c/c(0)]

5 1.00035 0.47065 0.22695 0.121

(the suspending medium is water). From this data, and the relation found in part a [which should

have been of the form / (0) exp( [ 0 ] )Bc z c mg z z k T ] find another estimate for

Boltzmanns constant kBand hence Avogadros number (the answer is, again, a little high).