Statistical Mechanics: Phase Diagrams and Effusion

[ecastro]

Homework Statement

There are two problems:
The first problem is to calculate the slope of a phase-equilibrium line of a substance given its properties (Helium, Water, etc.) as the temperature approaches a certain value (in this case, T approaches zero).
The second problem is the effusion of a gas into a slit to produce an atomic beam. The intensity of the beam is calculated from the number of molecules that leave the beam.

Homework Equations

For the first problem:
$\frac{dp}{dt} = \frac{\Delta S}{\Delta V}$
S is entropy and V is volume.

Second Problem:
$\frac{dN}{dt} = -\frac{1}{4}n\bar{v}A$

The Attempt at a Solution

I do not know how to move on for the first equation, considering that the density of the solid is greater than that of the liquid.
As for the second problem, I'm not sure if this is already I (the given equation). Differentiating it with respect to T, and
$n = \frac{pV}{kT}$
Considering no change in volume as T changes. Also, the velocities of the particles changes with temperature, I'm also not sure if this assumption is correct. Could you please guide me?

 

[mark726]

For the first problem, you can use the Clausius-Clapeyron equation to calculate the slope of the phase-equilibrium line as the temperature approaches zero. This equation relates the rate of change of pressure (dp/dT) to the change in entropy (ΔS) and volume (ΔV) as the temperature changes. You can use the values for entropy and volume at the given temperature to solve for the slope.

For the second problem, you can use the given equation to calculate the intensity of the atomic beam. However, you will also need to consider the change in velocity of the particles as the temperature changes. This can be done by using the ideal gas law to relate the number of molecules (N) to the pressure (p), volume (V), and temperature (T). Then, you can differentiate this equation with respect to temperature to find the change in velocity (v). Finally, you can substitute this value for velocity into the given equation to solve for the intensity of the beam.