Homework #5 Stat. Mech. CHAP E4120x Prof. Ben O?Shaughnessy

Homework #5
Consider a colloidal suspension of latex particles confined between two
plates. The plates are charged, so there is an electric potential difference V
between them (as shown in the figure below). The number of particles is N and
the temperature is T. The system is isolated from the environment and
gravitational effects are negligible.
Now each particle carries a positive charge +q, so particles are more likely
to be near the plate at the top in the figure, which is negatively charged, than the
bottom one, which is positively charged. The system is very dilute, so particles
are independent of each other and the electrostatic interaction between particles
can be neglected. For simplicity, assume that in the z direction each particle
may only be at one of the following two locations: either attached to the top plate,
or attached to the bottom one, as shown in the figure.
(i) Calculate the fraction of particles attached to the top plate. Sketch this as a
function of: (a) T, (b) V. Discuss the physical meaning of your results.
Hint for (i): The total energy of the system, E, depends on how the particles are
distributed between the two plates. Find out the number of configurations
associated with a certain E and express the entropy of the system, S, as a function
of E, V, and N. Then use
S V N
E T
? ,
? = and solve for E.
V
z
Electric
field
– – – – – – – – – – – – – – – –
+ + + + + + + + + + + + + + +
Homework #5 Stat. Mech. CHAP E4120x
Prof. Ben O?Shaughnessy
2
(ii) Consider the case at room temperature (300?K) and q = 10 e, where e is the
magnitude of the charge carried by an electron (1.6?10-19 C). (This is due to the
ionization of 10 charged groups chemically attached to the particle surface.)
How large does V need to be so that the particles have a significant preference to
reside at the top plate? (In reality, there is salt in the solution, and the salt ions
interact with the particles in a complex way and partially neutralize them. So q
here should be regarded as the effective charge carried by a particle after
considering these effects.) 