Subsections

# Phase Transitions

## Heat of Phase Transition

Take a sample of phase I and slowly transform it into phase II. Some heat Q will be absorbed.

• Since this is reversible, (1)
• Since in equilibrium P1=P2, the process occurs at constant pressure
• Since T1=T2, it occurs at constant temperature
We have or The heat of phase transition is equal to the change of enthalpy.

## Clapeyron-Clausius Formula

### General Law

On the P-T diagram phase transition occur along a line P(T). Let's calculate the tangent to this line dP/dT.

Chemical potentials are equal: Differentiating: (2)
We know from previous lectures that and therefore with s and v being quantities per molecule

From (2): or Since we obtained (3)
This is Clapeyron-Clausius equation

### Consequences

• For liquid-gas phase transitions q>0. Since v2>v1, dT/dP= (dP/dT)-1>0. Boiling point temperature increases with pressure
• For solid-liquid phase transitions q>0 (exception: ). Usually v2>v1, and melting point increases with pressure. For water v2<v1, and it decreases--that is why skating & skiing is possible!

### Example: Liquid-Gas Phase Transition

Assumptions:

1.
Phase transition heat q does not depend on T and P: 2.
Gas is ideal (PV=NkT), and its volume is much grater than of liquid: We obtain: or ## Order of Phase Transition

If we move from phase I to phase II, thermodynamic potential G is continuous (why?), but its first derivative S jumps. Another possibility: S is continuous, but jumps.

Definition:
if the n-th derivative of free energy is discontinuous, and all lower derivatives are not, the transition is of n-th order.

Phase equilibrium:
first order phase transition!

Second order phase transitions:
Curie point, some liquid crystal phase transitions...

Practical guide:
if q=0, the transition is of the second order (or higher!).

First-order-close-to-second-order transitions:
q small (liquid crystals)   Next: Phase Transitions in Van Up: Thermodynamics of Phase Transitions Previous: Quiz Answers

© 1997 Boris Veytsman and Michael Kotelyanskii
Tue Oct 7 22:16:36 EDT 1997