Physics 400, Fall 2011: Problem Set 7
Due Thursday Nov. 17, 11:59 pm
1. Diffusion
In this chapter, we began with counting arguments. One of the ways we will use counting
arguments is in thinking about diffusive trajectories. Consider eight particles, four are black
and four are white. Four particles can fit left of a permeable membrane and four can fit right
of the membrane. Imagine that due to random motion of the particles every arrangement
of the eight particles is equally likely. Some possible arrangements are: BBBB-WWWW,
BBBW-BWWW, WBWB-WBWB; the membrane position is denoted by -.
(a) How many different arrangements are there?
(b) Calculate the probability of having all four black particles on the left of the permeable
membrane. What is the probability of having one white particle and three black particles
on the left of the membrane. Finally, calculate the probability that two white and two
black particles are left of the membrane. Compare these three probabilities. Which
arrangement is most likely?
(c) Imagine that in one time instant a random particle from the left-hand side exchanges
places with a random particle on the right-hand side. Starting with three black particles
and one white particle on the left of the membrane, compute the probability that after
one time instant there are four black particles on the left. What is the probability that
there are two black and two white particles on the left, after that same time instant?
Which is the more likely scenario of the two?
2. Elasticity of polymers
The thermodynamic identity for a one-dimensional system is
TdS
dU
Jdl
(1)
where J is the external force exerted on the line and dl is the extension of the line. (The
direction of the force is opposite to the conventional direction of the pressure.)
(a) Find an expression relating
J to a derivative of the entropy.
(b) Now consider a freely-jointed chain of N links each of length p, with each link equally
likely to be directed to the right and to the left. How many arrangements give a head­
to-tail length of l = 2lslp? You can write the result in terms of sand N.
(c) Write the entropy of the chain as a function of l for lsi <t:: N. Your result should be of
the form S(l) = Cl + c2l2 where q are constants.
(d) Calculate the force at extension l.
1
You'll see that the force is proportional to temperature. The force arises because the polymer
wants to curl up: entropy is higher in a random coil than in an uncoiled configuration.
Wanning rubber makes it contract.
3. Carbon monoxide poisoning
In carbon monoxide poisoning the CO replaces oxygen adsorbed on hemoglobin (Hb) molecules
in the blood. To show the effect, consider a model for which each adsorption site on a heme
may be vacant, occupied by oxygen with energy fA, or occupied by CO with energy fB. Let
N fixed heme sites be in equilibrium with oxygen and CO in the gas phases at concentrations
such that the absolute activities are ,\ = 1 X 10- 5 for oxygen and ,\ = 1 X 10- 5 for CO.
(a) Assuming the presence of oxygen alone, evaluate fA such that 90% of the Hb sites are
occupied by oxygen. Express the answer in eV per oxygen molecule.
(b) Now admit CO under the specified conditions. Find fB such that only 10% of the
Hb sites are occupied by oxygen. What fraction is occupied by CO? What fraction is
unoccupied?
4. Nordlund 14.9
5. Nordlund 14.11
6. Nordlund 14.16
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Prohl.m 14.9 Temperalute depeu.cIeot Equilibria
SoJutioos:
(1) : :
where X
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=euthalpy c:baDae ia multiples of 10.21 J.
Typeofcbange -Euthalpy change
(ua.its of Jeri J)
d (HI)
dT HA _It
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(K.l)
Covalent hood
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(2) At300K., HI =exp( n.46X)=7.13XIO.... : NB amoun1:stoO.0713%Oftbetotal
N"
T
SioceA(Ha)=..!.(H8)
AT = (1.7 XlO-'1C-1)(101C) = 1.7 X10-5 • the%ofB increases by
HA
dT NA lOOK
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amount ofNa--tlOt very much..)
Prohl.m 14.11 Mixing ED1ropy
Solution.:
(I)
Dilutioa. oflhe water aCC01lDts for oaly +0. lOR oftb.1s-about 1%.
AG.. =-TAS.... = -136RT = -3.4tJ I mole at 300It
AS.. =-k
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SS5
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(2)
• Awt
5S.5 ]
SS.SOl
Water accooats f« O.OOIR.« 4%.
Proble JD 14.1 is UnimolecuJar Equilibria
Solution:
AG=M{-TAS=lL7kJ/mo1e-TbSs-l0..9kJ/moIe
(1) -TbS=-22.6kJ/mo1e=entropic frI:e eDer8Y (See Table S.6.)
22. fl.X)J/mo1e
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(3) N. .
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