Phase Equilibria and Phase Diagrams

Phases of Matter

A phase of matter, also called a state of matter, is one of the distinct forms in which matter can exist. A phase has a uniform composition (above atomic lengthscales) and is characterized by specific properties and characteristics. The phase that a particular substance exhibits at any one time is determined by its physical conditions such as temperature and pressure, and also the interactions between the particles and how they are arranged. The primary phases of matter are: solid, liquid, gas, plasma.

Solid

A solid is a form of matter that has a fixed shape and volume. The atoms, molecules, or ions that make up a solid are closely packed and do not move or flow freely.

Ordered solids

Have a regular, repeating structure called a crystal lattice.

Examples: diamond, silicon, sodium chloride (NaCl).

Disordered solids

Lack long-range order; particles are randomly arranged.

Examples: glass, rubber, and plastics.

Liquid

Liquids are a state of matter consisting of particles in constant motion that are bound together by intermolecular forces. Liquids have a definite volume, but no fixed shape, and they are nearly incompressible. Liquids are able to flow and take the shape of their container.

Gas

Gases are a state of matter where, like liquids, the particles are in constant motion. However, gases have no definite volume and will expand and fill any container they are placed in.

Plasma

A plasma is similar to a gas but it is composed of charged particles, e.g., ions and free electrons. Because the particles are charged, plasmas exhibit a variety of phenomena and complexity that do not occur in neutral gases.

Gibbs Free Energy

The detail is given by Thermodynamic potentials.

We introduced a state function, the Gibbs Free Energy , whose natural variables are temperature and pressure:

The equilibrium phase is the one with the lowest Gibbs Free Energy.

Gibbs Free Energy is the useful form of potential when the temperature and pressure are given in conditions. The phase with least Gibbs Free Energy is the one most stable.

Why nature needs Gibbs Free Energy to be least

The reason is from the Second Law of Thermodynamics which states that the total entropy of the universe will increase ().

Thus consider a system exchanging heat and work with one reservoir which possesses temperature and pressure . With the Second Law we can say

While the change in entropy can be given by

And the work done on the system is

Thus from the First Law of Thermodynamics:

If we define a quantity called Availability, and recall the Gibbs Free Energy:

Therefore, when temperature and pressure for the process, we can say the change of availability is equal to the change of Gibbs Free Energy. So for any isobaric and isothermal process the Gibbs Free Energy decreases. The system will finally stay in the state of least .

Helmholtz Free Energy (Isothermal and Isovolumic)

The Helmholtz Free Energy is given by

and

Therefore, when the process is isovolumic (), the change in Availability can also equal the change in Helmholtz Free Energy.

Slope of the Gibbs free energy with temperature

Imagine water at temperature : water and ice will coexist because they have the same Gibbs Free Energy. But we can see from the graph of against that the gradients for liquid and solid are not the same. How do we express this?

The Gibbs Free Energy can be expressed in differential form:

We can simplify this by and . With the First Law of Thermodynamics , we get the general differential form of Gibbs Free Energy:

If we assume the process of phase change is under the condition of isobaric, therefore we get

This is the slope of the curves. When ice transitions to water, as ice possesses entropy less than water (implied in the slope), the entropy will have a jump of . To achieve this we must input energy into the system. This energy is called Latent Heat.

Phase Transitions and Latent Heat

Latent Heat

When a substance transfers from solid with low entropy to liquid with high entropy, its entropy has a discontinuity that requires energy input into the system. This energy is called Latent Heat. This kind of discontinuity in the first-order derivative (entropy) is called a first-order phase transition in theoretical physics. The latent heat is given by

Supercooling and Nucleation

Sometimes, when you cool water below it does not freeze and remains in the liquid phase (supercooling). This appears to violate the Gibbs curve, but why?

Because of the nucleation mechanism. In the disordered liquid, molecules want an ordered structure but it is difficult for the molecules to assemble together to form the solid structure, unless they have a substance to adjoin. This is called nucleation at impurity.

Transition Temperature

The reason why ice and water can coexist is that their Gibbs values are equal. Now imagine the pressure changes: this will change their Gibbs values. To let them coexist, nature must adjust the temperature . This is why the boundary of the phase diagram is a line consisting of many points , not a single point.

The Clapeyron Equation

We know that the Gibbs values on the boundary line are equal, so their differentials are equal:

We also know the total differential form of Gibbs:

Therefore

From the latent heat formula we know , then

This is the Clapeyron Equation, which connects the macroscopic phase diagram slope to the microscopic thermodynamical properties.

A classic example is the ice skater: their weight is concentrated on the ice blade, which produces large pressure (). From the Clapeyron Equation, the slope is negative since the change in volume is negative from ice to water, thus . In other words, the melting point of ice is lowered under increased pressure. The molten water (quasi-liquid) would act as a lubricant for the ice blade, allowing the skater to slide smoothly. However, this explanation is actually incorrect: the pressure required to lower the melting point enough is far greater than what a skater’s blade can produce. The real reason ice is slippery is due to a thin quasi-liquid layer that exists on the ice surface.

The Clausius-Clapeyron Equation

In real calculations, Clausius made a correction to the Clapeyron Equation to make it easier to apply, especially for liquid-to-gas or solid-to-gas transitions. In these processes, the change of volume can be given by:

For example, 1 L of water will evaporate to 1700 L of steam. This is a huge difference, so we can neglect the liquid volume and consider the gas to be ideal:

Back to the equation:

This is the differential form of the Clausius-Clapeyron Equation. To aid calculation, we integrate it: