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thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of ther ...
, the phase rule is a general principle governing "pVT" systems, whose
thermodynamic state In thermodynamics, a thermodynamic state of a system is its condition at a specific time; that is, fully identified by values of a suitable set of parameters known as state variables, state parameters or thermodynamic variables. Once such a set ...
s are completely described by the variables
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and ...
(),
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The def ...
() and
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
(), in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In ther ...
. If is the number of
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, is the number of components and is the number of phases, then :F = C - P + 2 It was derived by American physicist
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
in his landmark paper titled ''
On the Equilibrium of Heterogeneous Substances In the history of thermodynamics, ''On the Equilibrium of Heterogeneous Substances'' is a 300-page paper written by American chemical physicist Willard Gibbs. It is one of the founding papers in thermodynamics, along with German physicist Hermann ...
'', published in parts between 1875 and 1878. The rule assumes the components do not react with each other. The number of degrees of freedom is the number of independent intensive variables, i.e. the largest number of thermodynamic parameters such as temperature or pressure that can be varied simultaneously and arbitrarily without determining one another. An example of one-component system is a system involving one pure chemical, while two-component systems, such as mixtures of water and ethanol, have two chemically independent components, and so on. Typical phases are
solid Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural r ...
s,
liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, an ...
s and
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
es.


Foundations

*A phase is a form of matter that is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, ...
in
chemical composition A chemical composition specifies the identity, arrangement, and ratio of the elements making up a compound. Chemical formulas can be used to describe the relative amounts of elements present in a compound. For example, the chemical formula for ...
and
physical state In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. Many intermediate states are known to exist, such as liquid crystal, ...
. Typical phases are solid, liquid and gas. Two
immiscible Miscibility () is the property of two substances to mix in all proportions (that is, to fully dissolve in each other at any concentration), forming a homogeneous mixture (a solution). The term is most often applied to liquids but also applies ...
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. *The number of degrees of freedom (''F'') in this context is the number of intensive variables which are independent of each other. The basis for the rule is that equilibrium between phases places a constraint on the intensive variables. More rigorously, since the phases are in thermodynamic equilibrium with each other, the
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
s of the phases must be equal. The number of equality relationships determines the number of degrees of freedom. For example, if the chemical potentials of a liquid and of its vapour depend on temperature (''T'') and pressure (''p''), the equality of chemical potentials will mean that each of those variables will be dependent on the other. Mathematically, the equation , where ''μ'' = chemical potential, defines temperature as a function of pressure or vice versa. (Caution: do not confuse ''p'' = pressure with ''P'' = number of phases.) To be more specific, the composition of each phase is determined by intensive variables (such as mole fractions) in each phase. The total number of variables is , where the extra two are temperature ''T'' and pressure ''p''. The number of constraints is , since the chemical potential of each component must be equal in all phases. Subtract the number of constraints from the number of variables to obtain the number of degrees of freedom as . The rule is valid provided the equilibrium between phases is not influenced by gravitational, electrical or magnetic forces, or by surface area, and only by temperature, pressure, and concentration.


Consequences and examples


Pure substances (one component)

For pure substances so that . In a single phase () condition of a pure component system, two variables (), such as temperature and pressure, can be chosen independently to be any pair of values consistent with the phase. However, if the temperature and pressure combination ranges to a point where the pure component undergoes a separation into two phases (), decreases from 2 to 1. When the system enters the two-phase region, it becomes no longer possible to independently control temperature and pressure. In the
phase diagram A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseou ...
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 A supercritical fluid (SCF) is any substance at a temperature and pressure above its critical point, where distinct liquid and gas phases do not exist, but below the pressure required to compress it into a solid. It can effuse through porous so ...
. Of the other two-boundary curves, one is the solid–liquid boundary or
melting point The melting point (or, rarely, liquefaction point) of a substance is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends ...
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 In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at which the subli ...
. Here there are two equations , which are sufficient to determine the two variables T and p. In the diagram for CO2 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 Photograph showing details of an ice cube under magnification. Ice Ih is the form of ice commonly seen on Earth. Phase space of ice Ih with respect to other ice phases. Ice Ih (hexagonal ice crystal) (pronounced: ice one h, also known as ice-p ...
,
ice III Ice III is a form of solid matter which consists of tetragonal crystalline ice, formed by cooling water down to at . It is the least dense of the high-pressure water phases, with a density of (at 350 MPa). It has a very high relative permittiv ...
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 In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ...
or mass fraction of one component. As an example, consider the system of two completely miscible liquids such as
toluene Toluene (), also known as toluol (), is a substituted aromatic hydrocarbon. It is a colorless, water-insoluble liquid with the smell associated with paint thinners. It is a mono-substituted benzene derivative, consisting of a methyl group (CH3) at ...
and
benzene Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms ...
, in equilibrium with their vapours. This system may be described by a boiling-point diagram which shows the composition (mole fraction) of the two phases in equilibrium as functions of temperature (at a fixed pressure). Four thermodynamic variables which may describe the system include temperature (''T''), pressure (''p''), mole fraction of component 1 (toluene) in the liquid phase (''x''1L), and mole fraction of component 1 in the vapour phase (''x''1V). However, since two phases are present () in equilibrium, only two of these variables can be independent (). This is because the four variables are constrained by two relations: the equality of the chemical potentials of liquid toluene and toluene vapour, and the corresponding equality for benzene. For given ''T'' and ''p'', there will be two phases at equilibrium when the overall composition of the system (system point) lies in between the two curves. A horizontal line ( isotherm or tie line) can be drawn through any such system point, and intersects the curve for each phase at its equilibrium composition. The quantity of each phase is given by the
lever rule In chemistry, the lever rule is a formula used to determine the mole fraction (''xi'') or the mass fraction (''wi'') of each phase of a binary equilibrium phase diagram. It can be used to determine the fraction of liquid and solid phases for a ...
(expressed in the variable corresponding to the ''x''-axis, here mole fraction). For the analysis of
fractional distillation Fractional distillation is the separation of a mixture into its component parts, or fractions. Chemical compounds are separated by heating them to a temperature at which one or more fractions of the mixture will vaporize. It uses distillation to ...
, the two independent variables are instead considered to be liquid-phase composition (x1L) and pressure. In that case the phase rule implies that the equilibrium temperature (
boiling point The boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. The boiling point of a liquid varies depending upon the surrounding envi ...
) and vapour-phase composition are determined. Liquid–vapour
phase diagram A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseou ...
s for other systems may have
azeotrope An azeotrope () or a constant heating point mixture is a mixture of two or more liquids whose proportions cannot be altered or changed by simple distillation.Moore, Walter J. ''Physical Chemistry'', 3rd e Prentice-Hall 1962, pp. 140–142 This ...
s (maxima or minima) in the composition curves, but the application of the phase rule is unchanged. The only difference is that the compositions of the two phases are equal exactly at the azeotropic composition.


Phase rule at constant pressure

For applications in materials science dealing with phase changes between different solid structures, pressure is often imagined to be constant (for example at one atmosphere), and is ignored as a degree of freedom, so the formula becomes: :F = C - P + 1 This is sometimes incorrectly called the "condensed phase rule", but it is not applicable to condensed systems which are subject to high pressures (for example, in geology), since the effects of these pressures are important.


References


Further reading

* * Chapter 9. Thermodynamics Aspects of Stability {{Authority control Equilibrium chemistry Laws of thermodynamics