** Next:** Phase Transitions
**Up:** Thermodynamics of Phase Transitions
** Previous:** Thermodynamics of Phase Transitions

September 23

- 1.
- Draw a plot of the single particle distribution function
as a function of
*z*for the liquid monolayer, adsorbed on the flat surface.*z*is the distance perpendicular to the adsorbing surface. Consider a liquid of hard-spheres of diameter*d*.**Solution:**- There are molecules at the distance
*d*/2 from the wall only (we have a*mono*layer), so has a sharp peak:

- 2.
- How is normalized?
**Solution:**- The expression is the average
number of particles in the volume around the point if one particle is in the point . Integrating this,
we obtain
*N*-1 (one particle is at the point !).

- 3.
- Does at large distances in the perfect crystal?
**Solution:**- The function does not. Crystal has
*long range order:*

When we average integrate over , we obtain*many*peaks, but they do not tend to 1.

- 4.
- Is the derivation of the equations (3-9)
valid for the anisotropic liquid crystals, or for crystalline
solids?
**Solution:**- As long as we don't introduce spherical symmetry
and integrate over
*dr*, our equations are valid

September 30

- 1.
- Equipartition theorem works
- (a)
- For high temperatures
- (b)
- For low temperatures
- (c)
- Always
- (d)
- Never

**Solution:**- Since classical limit is for high temperatures (difference between levels ), the answer (1a) is right

- 2.
- Consider a cubic crystal with atoms,
*N*<*M*<*P*. The minimal value of*q*is- (a)
- (b)
- (c)
- (d)
- (e)
- (f)
- (g)
- (h)

**Solution:**- The values for are
where is the unit vector along the corresponding
axis. Then
This is minimal at
*n*_{x}=*n*_{y}=0,*n*_{z}=1. The correct answer is (2c).

October 2

- 1.
- Sketch the phase diagram for two phase equilibrium in
*P*-*T*and*V*-*T*coordinates.**Solution:**- Since
*P*is an intensive parameter, and*V*is an extensive one, we have:

and in

*V*-*T*space:

- 2.
- Same near triple point.
**Solution:**- See picture:

- 3.
- Consider a one-component system in an external electric field
. It can have dipole moment parallel to
. How many phases can coexist in this system?
**Solution:**- We add a scalar parameter |
*E*|. Therefore the number of phases is 3+1=4.

- 4.
- Same for binary blend in electric field.
**Solution:**- 5 (see above)

© 1997

Tue Oct 7 22:16:36 EDT 1997