Subsections

• September 23
• September 30
• October 2

Quiz Answers

 
September 23

1.

Draw a plot of the single particle distribution function $\rho(z)$ 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 monolayer), so $\rho(z)$ has a sharp peak:

2.

How is $\rho g(r)$ normalized?

Solution:

The expression $\rho g(\mathbf{r})\,d\mathbf{r}$ is the average number of particles in the volume $d\mathbf{r}$ around the point $\mathbf{r}$ if one particle is in the point $\mathbf{0}$. Integrating this, we obtain N-1 (one particle is at the point $\mathbf{0}$!).

3.

Does $g(r)\to 1$ at large distances in the perfect crystal?

Solution:

The function $g(\mathbf{r})$ does not. Crystal has long range order:

When we average integrate over $\mathbf{r}$, 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 $\Delta E \ll kT$), the answer (1a) is right

2.

Consider a cubic crystal with $N\times M\times P$ atoms, N<M<P. The minimal value of q is

(a)

$\pi/Na$

(b)

$\pi/Ma$

(c)

$\pi/Pa$  

(d)

$\pi/Na$

(e)

$\pi a/P$

(f)

$\pi a/M$

(g)

$\pi P/a^2$

(h)

$\pi/a(N^2+M^2+P^2)$

Solution:

The values for $\mathbf{q}$ are

$$\mathbf{q}= \frac{\pi}{Na}n_x\mathbf{e}_x + \frac{\pi}{Ma}n_y\mathbf{e}_z + \frac{\pi}{Pa}n_z\mathbf{e}_z$$

where $\mathbf{e}_i$ is the unit vector along the corresponding axis. Then

$$q^2 = \frac{\pi^2}{N^2a^2}n_x^2 + \frac{\pi^2}{M^2a^2}n_y^2 + \frac{\pi^2}{P^2a^2}n_z^2$$

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 $\mathbf{E}$. It can have dipole moment $\mathbf{p}$ parallel to $\mathbf{E}$. 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)