8.1 Phase rule and phase diagrams

Phases, Components, and Degrees of Freedom

Defining Phases, Components, and Degrees of Freedom

A phase is a homogeneous, physically distinct portion of a system separated from other phases by a definite boundary. Ice floating in water is two phases; the ice and water each have uniform properties throughout, but they're clearly distinct from one another.

The number of components ($C$) is the minimum number of independently variable chemical species needed to describe the composition of every phase in the system. This isn't always the same as the number of chemical species present. For example, if you have a chemical equilibrium like $\text{CaCO}_3 \rightleftharpoons \text{CaO} + \text{CO}_2$, there are three species but only two components, because the equilibrium condition provides a constraint that reduces the independent count by one.

Degrees of freedom ($F$), also called the variance, represent the number of intensive variables (temperature, pressure, or composition) you can independently change without altering the number of phases in equilibrium. If $F = 0$, the system is locked in place: you can't change anything without losing a phase.

Equilibrium and Chemical Potential

At equilibrium, the chemical potential of each component must be the same in every phase. Chemical potential ($\mu$) is defined as the partial molar Gibbs energy:

$\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T, P, n_{j \neq i}}$

This equality is what drives the phase rule. If the chemical potential of component $i$ were higher in phase $\alpha$ than in phase $\beta$, there would be a thermodynamic driving force for $i$ to transfer from $\alpha$ to $\beta$. Equilibrium means no net transfer, which requires $\mu_i^\alpha = \mu_i^\beta$ for every component across all phases.

Applying the Gibbs Phase Rule

The Gibbs Phase Rule Equation

The Gibbs phase rule is:

$F = C - P + 2$

where $F$ is the degrees of freedom, $C$ is the number of components, and $P$ is the number of phases in equilibrium. The "+2" accounts for temperature and pressure as the two external intensive variables.

This rule applies universally to any system at thermodynamic equilibrium, whether you're dealing with a pure substance, a binary alloy, or a ternary mixture.

Where does it come from? For $C$ components distributed among $P$ phases, you need $(C - 1)$ mole fractions to specify the composition of each phase (since they must sum to 1), giving $P(C - 1)$ composition variables, plus $T$ and $P$, for a total of $P(C - 1) + 2$ variables. The equilibrium conditions ($\mu_i^\alpha = \mu_i^\beta = \cdots$) provide $C(P - 1)$ constraints. Subtracting constraints from variables gives $F = C - P + 2$.

Degrees of Freedom and Invariant Systems

The maximum degrees of freedom occur when only one phase is present ($P = 1$), giving $F = C + 1$. For a single-component system, that's $F = 2$ (you can vary both $T$ and $P$ freely within a single-phase region).

Here's how it plays out for a single-component system ($C = 1$):

An invariant system ($F = 0$) occurs when $P = C + 2$. For a pure substance, that means three phases coexisting at the triple point. At water's triple point (273.16 K, 611.657 Pa), solid, liquid, and vapor all coexist, and neither $T$ nor $P$ can be changed without destroying the three-phase equilibrium.

Note that $P = C + 2$ also sets the maximum number of phases that can coexist in equilibrium, since $F$ cannot be negative.

Interpreting Phase Diagrams

Single-Component Phase Diagrams

A single-component phase diagram plots pressure vs. temperature. Each region on the diagram corresponds to a single stable phase (solid, liquid, or gas), and the boundaries between regions are coexistence curves where two phases are in equilibrium.

The three coexistence curves are:

• Melting/freezing curve (solid–liquid boundary): For most substances, this has a positive slope because the liquid is less dense than the solid. Water is a famous exception, with a negative slope because ice is less dense than liquid water.
• Vaporization/condensation curve (liquid–gas boundary): This curve runs from the triple point up to the critical point.
• Sublimation/deposition curve (solid–gas boundary): This curve runs from the triple point down toward lower temperatures.

The triple point is the unique $(T, P)$ at which all three phases coexist. With $F = 0$, it's a fixed point on the diagram.

The critical point is the $(T_c, P_c)$ at which the liquid–gas distinction vanishes. Beyond the critical point, the substance is a supercritical fluid with properties intermediate between liquid and gas. The vaporization curve terminates here because there is no longer a phase boundary to cross.

Binary Phase Diagrams

Binary phase diagrams represent two-component systems. They typically plot temperature vs. composition (mole fraction of one component) at a fixed pressure. Since pressure is held constant, the phase rule becomes $F = C - P + 1$ (one fewer external variable).

Key features include:

• Liquidus line: the boundary above which the system is entirely liquid.
• Solidus line: the boundary below which the system is entirely solid.
• Solvus line: the boundary between a single-phase solid solution and a two-phase solid region.

Between the liquidus and solidus, two phases coexist (e.g., liquid + solid). To determine the composition of each phase at a given temperature, you draw a horizontal tie line across the two-phase region. The endpoints of the tie line, where it intersects the phase boundaries, give the compositions of the coexisting phases.

The Lever Rule

To find the relative amounts of each phase in a two-phase region, use the lever rule. If the overall composition is $x_0$, and the tie line endpoints are at $x_\alpha$ and $x_\beta$:

$\frac{n_\alpha}{n_\beta} = \frac{x_\beta - x_0}{x_0 - x_\alpha}$

The ratio of the amount of one phase to the other is inversely proportional to the "distance" from the overall composition to that phase's endpoint. Think of it like a seesaw: the phase whose composition is farther from the overall composition is present in a smaller amount.

Phase Transitions and Their Characteristics

Types of Phase Transitions

Phase transitions occur when a substance moves from one phase to another in response to changes in temperature, pressure, or composition. The common ones for a pure substance are:

• Melting (solid → liquid): Requires the enthalpy of fusion ($\Delta_{\text{fus}}H$). Occurs at constant temperature for a pure substance at fixed pressure.
• Vaporization (liquid → gas): Requires the enthalpy of vaporization ($\Delta_{\text{vap}}H$), which is typically much larger than $\Delta_{\text{fus}}H$ because intermolecular interactions must be almost entirely overcome.
• Sublimation (solid → gas): Bypasses the liquid phase entirely. $\Delta_{\text{sub}}H = \Delta_{\text{fus}}H + \Delta_{\text{vap}}H$ by Hess's law.
• Condensation and freezing are the reverse processes of vaporization and melting, with enthalpy changes of equal magnitude but opposite sign.

These are all first-order phase transitions: they involve a discontinuous change in the first derivative of the Gibbs energy (entropy and volume), and they have a nonzero latent heat.

Solid-Solid Phase Transitions and Binary Systems

Solid-solid transitions involve rearrangement of atoms or molecules within the solid, often changing the crystal structure. Examples include the transition between different allotropes of carbon (graphite ↔ diamond, though this requires extreme conditions) and the ferromagnetic-to-paramagnetic transition in iron at its Curie temperature.

In binary systems, several special invariant points arise where three phases coexist at a fixed temperature and composition:

• Eutectic point: A liquid solidifies into two distinct solid phases simultaneously. The eutectic temperature is the lowest melting point for any composition in the system. This is why salt lowers the melting point of ice on roads: you're moving toward a eutectic composition.
• Peritectic point: A solid phase reacts with a liquid phase to produce a different solid phase upon cooling. This is common in metallic alloy systems.
• Monotectic point: A liquid phase decomposes into a second liquid phase and a solid phase. This occurs in systems where the two liquids have limited miscibility.

At each of these invariant points in a binary system, $F = 2 - 3 + 1 = 0$ (at constant pressure), so the temperature and phase compositions are all fixed.