Lecture 20: Phase transition

Note

Phase diagrams are the beginning of wisdom-not the end of it. – William Hume-Rothery

Warning

This lecture corresponds to Chapter 28 of the textbook.

Summary

Attention

In the previous lecture (Lecture 19: Cooling real gas), we introduced the concept of the possibility of the existence of two phases, predicted by the realistic description of a gas, beyond the ideal gas laws. We found that if we obey the principle that a system will always try to minimize Gibbs energy, there is a transition between a phase of high compressibility (gas) and a phase of small compressibility (liquid). This only happens for a point (known as the critical point in the phase diagram. In this lecture, we generalize those findings to general phase transitions.

In general, during a phase transition, the Gibbs energy is the same between the two phases. Furthermore, at the transition the chemical potentials of each phase are identical. The domain of points (in the phase diagram) separating two phases is thus an iso-Gibbs energy function. It is described by the Clausius-Clapeyron relationship (for first-order transitions):

$\frac{ d p}{ d T}=\frac{L}{T\left(V_{2}-V_{1}\right)}$

where $L$ is the latent heat and the denominator is the change in volume between the two phases.

For example, this equation becomes (with the approximation that the latent heat is temperature independent):

$p=p_{0}+\frac{L}{\Delta V} \ln \left(\frac{T}{T_{0}}\right)$

We see that the slope of the curve is described by the latent heat and by the change in volume. We therefore expect the boundary between a solid and a liquid to show a very steep curve in the $p$ versus $V$ phase diagram since the change of volume is very small. In general the slope is positive since the phase transition from a solid to a liquid usually involves a increase in volume. One noteworthy exception is water, where the specific hydrogen bonding in the solid (ice) phase leads to an decrease in volume when melted.

Finally, one notes that all these results are predicated on the validity of the main assumption we’ve made in this course: the fact we work at the thermodynamic limit (that is: a regime where properties can be described by ensemble averages). It appears that some systems can momentarily be found in phases that do not correspond to the lowest Gibbs energy, such as the superheated liquid and the supercooled gas (see definition in the box below).

Note

Categories of phase transition

Ehrenfest proposed to classify phase transitions by their order. A first-order phase transition is categorized by a discontinuity in the first derivatives of Gibbs energy at the critical temperature. For example, such derivatives include the volume and the entropy. A transition from a liquid to gas is an example of first-order phase transition. First-order transition involves a finite latent heat. The summary above is focused on such transitions.

A second-order phase transition is a general phase transition whose second derivative is discontinuous. These transitions include discontinuity in heat capacity. An example of second order phase transition is the paramagnetic to ferromagnetic phase transition. Second-order phase transitions (as well as higher-order phase transitions) do not involve latent heat.
Symmetry breaking: another way to describe a phase transition is by examining symmetry breaking. For instance, the transition between a liquid and a solid breaks (lowers) the symmetry of the system (that’s why there is no critical point for this transition). On the other hand, some phase transition do not involve a break in symmetry. This includes transition between a liquid and a gas does not involve a break in symmetry

Note

The Ising model

Consider a lattice of magnetic moments $S_i$ (in 1D or 2D). The energy of the system can be described by the simple Ising Hamiltonian:

$\hat{H}=-J \sum_{\langle i, j\rangle} \hat{S}_{i} \hat{S}_{j},$

where $J$ is positive for ferrogmatic ordering and negative for an anti-ferromagnetic ordering. The bracket indicates that only first-neighbor interactions are taken into account.

This description departs from the scenario we studied when we examined Curie’s law: here the magnetic do interact with one another.

The Ising model is a very popular model to understand phase transitions. The question students often ask is: I understand that at low temperature the Ising model would predict a fully aligned set of magnetic dipoles (in the case of ferromagnetic coupling). But, why is there a stable configuration at higher temperature that’s not ordered? The answer to this question is that at higher temperature, entropy starts to have a more important role. We can formalize this by looking at the Helmholtz energy:

$F=U-TS.$

At high enough temperature, increases in entropy will eventually take over minimization of $U$ . This is why we can have a phase transition in the 2D Ising model. Interestingly, there is no phase transition in the 1D Ising model because long range order is impossible! The reason is simple: the flip of a single magnetic moment corresponds to many microstates (choice of where the flips takes place) and if the 1D system is long enough the entropy will always be larger than the internal energy at all but strictly zero temperature (which cannot be achieved, due to the third law of thermodynamic).

Key Definitions

Note

Supercooled vapor:

When a gas is cooled below the boiling point, it is possible to continue momentarily on the curve corresponding to $\mu_{\rm gas}$ and to form supercooled vapor, which is a metastable state.

Supeheated liquid:

When a liquid is warmed through its boiling point, it is possible to continue momentarily on the curve corresponding to $\mu_{rm liq}$ and to form superheated liquid, which is a metastable state.

Trouton’s rule:

Empirical rule that predicts that the latent heat involved in the liquefaction of a gas is given approximately by

$L \approx 10 R T_{ b },$

where $T_b$ is the boiling temperature.

Triple point:

Point in the phase diagram where the gas, liquid, and solid phases coexist.

The Metropolis Algorithm:

Computational technique used to sample configuration space with probability amplitude governed by Boltzmann factor. Random moves that decrease the total energy are always accepted. However, moves that increase energy are also accepted with a probability proportional to the change in Boltzmann factor.

A full list of terms, including the ones provided here, can be found in the Index.

Learning Material

Copy of Slides

The slides for Lecture 20 are available in pdf format here: pdf

Screencast

Test your knowledge

When a phase transition requires a positive latent heat to proceed, what can we conclude?
1. The internal energy decreases during the process.
2. The process is always isochoric.
3. There is a decrease in entropy.
4. There is an increase in entropy.
5. There is not enough information to conclude anything.
All phase transitions are associated with a discontinuous change in entropic content.
1. True
2. False
During a phase transition between two phases, what can you conclude about the chemical potentials of the two phases?
1. The chemical potential of the second substance is necessarily lower
2. The chemical potentials of the two phases are identical
3. The chemical potential of the second substance is necessarily higher
A second-order phase transition is defined as
1. a phase transition with a discontinuous change in entropy and heat capacity.
2. a phase transition with a discontinuous change in volume and heat capacity.
3. a phase transition with a discontinuous change in volume and entropy.
4. none of the other claims is correct.
The Clausius-Clapeyron equation…
1. describes how the chemical potential changes during a phase transition.
2. describes the domain, in the $p-T$ phase space, of equilibrium between two different phases, e.g., a liquid and a gas.
3. describes the amount of energy (e.g., latent heat) needed for the phase transition to take place.
4. describes how the ideal gas law can explain some phase transitions.
5. None of the other answers is correct.
To really understand a phase transition, we have to remember that
1. only thermodynamics matters and the system will always lead to the phase with lowest chemical potential.
2. only thermodynamics matters and the system will always lead to the phase with largest chemical potential.
3. thermodynamics only applies at the thermodynamic limit and it may happen that metastable states are observed during the transition.
4. all phase transitions are first-order transitions since additional heat is needed to vaporize a liquid.
5. none of the other claims is correct.
In general, if you increase the volume of a container full of gas at constant temperature, what do you expect will happen to the entropy?
1. It will increase.
2. It will decrease.
3. It will remain the same.
4. More information is needed to answer this question.

Hint

Find the answer keys on this page: Answers to selected test your knowledge questions. Don’t cheat! Try solving the problems on your own first!

Homework Assignment

Solve the following problems from the textbook: