Physical Sciences 2
Nov. 19 – Dec. 1
Physical Sciences 2: Assignments for Nov. 19 – Dec. 1
Homework #10: Statistical Mechanics, Brownian Motion, and Diffusion
Due Friday, Dec. 4 at 5:00 pm
This assignment is officially due at 5:00PM on Friday, December 4, however extensions until
Tuesday, December 8 are available without penalty. Please write your answers to these
questions on a separate sheet of paper with your name and your section TF’s name written at
the top. Turn in your homework to the basket in the front of lecture with your section TF’s name.
You are encouraged to work with your classmates on these assignments, but please write the
names of all your study group members on your homework.
After completing this homework, you should be able to…
• Interpret the meaning of the Boltzmann distribution and describe its applicability
• Describe how temperature affects the Boltzmann distribution
• Explain the connection between Brownian motion and random walks
• Explain how to use a random walk to model Brownian motion
• Show that the mean square displacement is linear in time and directly proportional to the
diffusion coefficient
• Understand how the diffusion coefficient indicates how long it takes a molecule or
particle in solution to move a given distance
• Use the Stokes-Einstein equation to predict the diffusion coefficient
• Explain how random microscopic motion leads to macroscopic flux (via Fick’s law)
• Understand and be able to use Fick’s first law to describe how flux relates to a
concentration gradient
• Understand and be able to use Fick’s second law to describe how concentration changes
with respect to both time and space
• Explain why and how concentration gradients relax over time
Physical Sciences 2
Nov. 19 – Dec. 1
Physical Sciences 2
Nov. 19 – Dec. 1
Physical Sciences 2
Nov. 19 – Dec. 1
0. Reflections on Last Assignment (1 pt)
Pick one question from Homework 9 that you found particularly difficult and
a) describe any mistakes or misunderstandings you made
b) describe the best strategies to ensure you learn from your mistakes and won’t have the
same misunderstanding again
1. Two-state molecule (3 pts). A certain molecule can exist in two states: the ground state,
which has U = 0, and the excited state, which has U = Ue, a constant. At room temperature (~
300 K), 90% of the molecules in a given sample are observed to be in the ground state.
a) What is the value of the constant Ue?
b) At what temperature would 99% of the molecules be in the ground state?
c) At what temperature would 51% of the molecules be in the ground state?
2. Disk-shaped cells (2 pts). Inside a cell, glycerol molecules have a diffusion coefficient of
approximately D = 2.1 × 10–10 m2/s. Consider a cell shaped like a thin disk, with a diameter of
5 µm and a thickness of 1 µm. Consider glycerol molecules
released at the center of the cell. How long, on average, will
it take these molecules to reach …
a) … the outer edge of the cell (e.g. point A)?
b) … the top or bottom of the cell (e.g. point B)?
3. Long axons (5 pts). The longest cells in the human body are the nerve cells whose cell bodies
are located in the base of your spinal cord and whose axons run down to the ends of your toes:
these cells are approximately L = 1 meter long. Treat the axon as a cylinder with a radius of r = 1
µm.
a) Consider a neurotransmitter packaged in a spherical vesicle with a radius of 10 nm. Estimate
its diffusion constant. (Assume that the viscosity inside the axon is roughly η = 6 × 10–3
kg/m·s.)
b) Under passive diffusion, how long would it take on average for this package to move from
the cell body to the end of the axon? Is diffusion a reasonable method for transport in this
case?
Suppose that the cell body manufactures these neurotransmitters such that the concentration at
the top of the axon is kept constant at c0 = 1 mM (1 millimolar, i.e. 10–3 moles per liter). At the
bottom of the axon, the neurotransmitter is consumed at a steady rate, so that its concentration is
zero.
Physical Sciences 2
Nov. 19 – Dec. 1
c) Assuming that the concentration decreases linearly from the cell body (y = L) to the bottom
of the axon (y = 0), write an expression for c(y), the concentration of the neurotransmitter as
a function of position.
d) Calculate the flux (in molecules per unit area per unit time) of neurotransmitter at the bottom
of the axon.
e) To send a signal to a muscle cell, the bottom of the axon must release about 3 × 106
molecules of the neurotransmitter. If the axon relied on diffusion alone, how often could the
neuron send a signal?
concentration (c)
4. The diffusion equation (1 pts). Suppose that the concentration of a solute as a function of
position at a particular time is given by the graph below. Re-draw that graph, and superimpose
on it a graph of the concentration a short time later.
position (x)
5. Take a deep breath (2 pts). At rest, your metabolism requires the consumption of about
1020 molecules of oxygen per second. Atmospheric air has an oxygen concentration of about
5 × 1021 molecules per liter; venous blood (i.e. blood that is depleted of oxygen) has an oxygen
concentration of about 1 × 1021 molecules per liter.
a) In the alveoli, oxygen must diffuse a distance of about 10 µm through the membrane and
fluid that separates the gas from the red blood cells. If the diffusion coefficient of O2 is D ≈ 1
× 10–10 m2/s through this membrane, calculate the flux (in molecules per unit area per unit
time) of oxygen from air into venous blood through this membrane.
b) What total area of alveoli is required to provide enough oxygen to support your resting
metabolism? The actual total area is quite generous (approximately 80 m2).
6. Vacuoles (3 pts). We can model a vacuole as a sphere of radius r surrounded by a very thin
semi-permeable membrane of thickness w. A particular enzyme can diffuse across the membrane
with diffusion constant D. At t = 0, the concentration of the enzyme in the vacuole is c0. Assume
Physical Sciences 2
Nov. 19 – Dec. 1
that the enzymes are distributed uniformly throughout the interior of the vacuole, and that the
outside world is large enough that the enzyme concentration outside the vacuole is always
essentially zero.
a) Find an expression for J, the flux of enzyme molecules across the vacuole membrane, in
terms of D, w, and c. Use the sign convention that positive values of J correspond to a net
outflux of enzyme molecules.
b) Let N(t) be the total number of enzyme molecules inside the vacuole as a function of time.
Derive an expression for the derivative dN/dt in terms of r and J. What is the physical
interpretation of dN/dt?
c) At any given instant, the concentration and number of enzymes inside the vacuole are related
by
N(t)
c(t) = 4 3 .
πr
3
Using this fact, and your answer to part b), show that the flux and concentration are related
by
r dc
J =−
.
€
3 dt
d) Combining the results of parts a) and c) yields a differential equation for c(t). Show that
c(t) = c 0e−t / τ
€
is a solution to this equation, where τ is a constant. Determine the value of τ in terms of D, r,
and w.
7. Diffusion and Fick’s Laws (3€pts).
a) For each of the following, choose the scenario in which the diffusion constant is greatest.
i. Three beads are placed in the same liquid. Bead 1 has a radius of length R; Bead 2 has a
radius of length 2R; and Bead 3 has a radius of length 3R. For which bead will the diffusion
constant be the greatest and why?
ii. Bead 1 (with radius R) is now placed in fluids of different viscosity and the diffusion
constant is measured. Fluid 1 has a viscosity of η; Fluid 2 has a viscosity of 2η; and Fluid 3
has a viscosity of 3η. In which fluid does Bead 1 have the greatest diffusion constant and
why?
iii. Bead 1 (with radius R) is now placed in Fluid 1 (with viscosity η) and the temperature is
varied. At Temperature 1, the temperature is T; at Temperature 2, the temperature is 2T; and
at Temperature 3, the temperature is 3T. At what temperature is the diffusion constant the
greatest and why?
b) Suppose Bead 1 has a radius of R = 1 µm and is placed in Fluid 1 which has a viscosity of 10-3
kg/m2. The temperature is kept at 200K. How long will it take for Bead 1 to diffuse a
Physical Sciences 2
Nov. 19 – Dec. 1
distance of 1 m, if it is traveling in a straight line? Comment on what this tells you about how
molecules, enzymes, etc. are transported in a reasonable amount of time.
c) You have seen in previous questions and in lecture that systems such as long axons, alveoli,
and the eye are not closed systems: at either end, the concentration of “stuff” is kept
constant. For example, in the alveoli problem (Question 5), the concentration of oxygen
molecules was on the order of 1021 molecules per liter in the air and 1021 molecules per liter
in venous blood. In these systems, the concentration on either end does not change.
i.
ii.
iii.
In such a system, does the flux increase, decrease, or stays constant with time?
!"
What does this tell you about the change in concentration in space, !"? Explain.
What does this tell you about the concavity, or the second derivative of concentration
!!!
as a function of position !! ! ? Does Fick’s 2nd law apply? Explain your answer.
Now consider a system like the vacuole in Question 6. The initial concentration of enzymes
in the vacuole is c0 while the concentration outside the membrane is always zero. The initial
concentration is shown in the figure below. As time goes on, the concentration of the vacuole
is depleted as the enzymes diffuse across a membrane of thickness w.
iv.
Can the concentration of enzymes inside the vacuole be replenished, given this
information? Justify your answer.
Hint: In problem 5 there is a constant supply of 02 to the Alveoli. In the description of the vacuole in
problem 6, is there a constant supply of enzymes to the vacuole?
v.
!"
In such a system, does the magnitude of the slope !" at the center of the membrane
increase, decrease, or stays constant over all time? Explain
!"
Hint: Is
at the center of the membrane the same when the concentration inside the vacuole has
!"
dropped to, say, 10% of its initial value?
vi.
!!!
What is the initial concavity !! ! at the center of the membrane?
Note: In order to avoid discontinuities in the second derivative we rounded the corners of the
concentration, as seen in the figure above.