In thermodynamics, the phase rule is a general principle governing multi-component, multi-phase systems in thermodynamic equilibrium. For a system without chemical reactions, it relates the number of freely varying intensive properties () to the number of components (), the number of phases (), and number of ways of performing work on the system ():
:<math>F = N + C - P + 1</math>
Examples of intensive properties that count toward are the temperature and pressure. For simple liquids and gases, pressure-volume work is the only type of work, in which case .
The rule was derived by American physicist Josiah Willard Gibbs in his landmark paper titled On the Equilibrium of Heterogeneous Substances, published in parts between 1875 and 1878.
The number of degrees of freedom (also called the variance) is the number of independent intensive properties, i.e., the largest number of thermodynamic parameters such as temperature or pressure that can be varied simultaneously and independently of each other.
An example of a one-component system () is a pure chemical. A two-component system () has two chemically independent components, like a mixture of water and ethanol. Examples of phases that count toward are solids, liquids and gases.
Foundations
- A phase is a form of matter that is homogeneous in chemical composition and physical state. Typical phases are solid, liquid and gas. Two immiscible liquids (or liquid mixtures with different compositions) separated by a distinct boundary are counted as two different phases, as are two immiscible solids.
- The number of components (C) is the number of chemically independent constituents of the system, i.e. the minimum number of independent species necessary to define the composition of all phases of the system. When the system enters the two-phase region, it is no longer possible to independently control temperature and pressure.
right|thumb|280px|Carbon dioxide pressure-temperature phase diagram showing the [[triple point and critical point of carbon dioxide]] In the phase diagram to the right, the boundary curve between the liquid and gas regions maps the constraint between temperature and pressure when the single-component system has separated into liquid and gas phases at equilibrium. The only way to increase the pressure on the two phase line is by increasing the temperature. If the temperature is decreased by cooling, some of the gas condenses, decreasing the pressure. Throughout both processes, the temperature and pressure stay in the relationship shown by this boundary curve unless one phase is entirely consumed by evaporation or condensation, or unless the critical point is reached. As long as there are two phases, there is only one degree of freedom, which corresponds to the position along the phase boundary curve.
The critical point is the black dot at the end of the liquid–gas boundary. As this point is approached, the liquid and gas phases become progressively more similar until, at the critical point, there is no longer a separation into two phases. Above the critical point and away from the phase boundary curve, and the temperature and pressure can be controlled independently. Hence there is only one phase, and it has the physical properties of a dense gas, but is also referred to as a supercritical fluid.
Of the other two-boundary curves, one is the solid–liquid boundary or melting point curve which indicates the conditions for equilibrium between these two phases, and the other at lower temperature and pressure is the solid–gas boundary.
Even for a pure substance, it is possible that three phases, such as solid, liquid and vapour, can exist together in equilibrium (). If there is only one component, there are no degrees of freedom () when there are three phases. Therefore, in a single-component system, this three-phase mixture can only exist at a single temperature and pressure, which is known as a triple point. Here there are two equations , which are sufficient to determine the two variables T and p. In the diagram for CO<sub>2</sub> the triple point is the point at which the solid, liquid and gas phases come together, at 5.2 bar and 217 K. It is also possible for other sets of phases to form a triple point, for example in the water system there is a triple point where ice I, ice III and liquid can coexist.
If four phases of a pure substance were in equilibrium (), the phase rule would give , which is meaningless, since there cannot be −1 independent variables. This explains the fact that four phases of a pure substance (such as ice I, ice III, liquid water and water vapour) are not found in equilibrium at any temperature and pressure. In terms of chemical potentials there are now three equations, which cannot in general be satisfied by any values of the two variables T and p, although in principle they might be solved in a special case where one equation is mathematically dependent on the other two. In practice, however, the coexistence of more phases than allowed by the phase rule normally means that the phases are not all in true equilibrium.
Two-component systems
For binary mixtures of two chemically independent components, so that . In addition to temperature and pressure, the other degree of freedom is the composition of each phase, often expressed as mole fraction or mass fraction of one component.
:<math>F = C - P + 1</math>
This is sometimes incorrectly called the "condensed phase rule", but it is not applicable to condensed systems subject to high pressures (for example, in geology), since the effects of these pressures are important.
Phase rule in colloidal mixtures
In colloidal mixtures quintuple and sixtuple points have been described in violation of Gibbs phase rule but it is argued that in these systems the rule can be generalized to <math>F =M+ C - P + 1</math> where <math>M</math> accounts for additional parameters of interaction among the components like the diameter of one type of particle in relation to the diameter of the other particles in the solution.
References
Further reading
- Chapter 9. Thermodynamics Aspects of Stability