Talk:Phase rule

Basic Issues

 * Question asked in 'Comments about the Ratings'. (Click the 'Show' on the blue bar above, which reveals:-

'There is something not explained in the gibbs' equation ... the equation is F=C-P+2 when the pressure OR temperature is constant but this formula is not valid when there is a change in both temperature and pressure the formula will change to F=C-P+1 please revise and reply...thank you for your effort'.


 * Gibbs's phase rule relates natural objects: the 'number of phases, P' and the 'number of intensive variables, C+2'. I'm not clear the phase rule ever changes, unless one adds new forms of work. When p=0, some rewrite it F=C-P+1; but a clearer statement is F=C+2-P, with dp=0 (which reduces the number of intensive variables by one). If one substitutes a natural, univariant curve for the pressure axis (which one always can): it's not so clear that the phase rule 'becomes' F=C-P+1 at a state of constant pressure. This confusion may also arise because we don't normally change scientific formulas: we change the values of its variables.


 * The phase rule F=C+2-P is valid at all equilibrium states (described by, at most, C+2 intensities) in open or closed systems. The rule F=C+2-P is valid when all intensities are kept constant (at an equilibrium, invariant state, illustrated by the triple point of water), when all but one are kept constant (along, for example, an equilibrium, univariant path), ..., or none is kept constant (along a C+2-dimensional family of equilibrium paths within a graphical region), where change is so slow that the system continuously equilibrates with its environment). Enrico Fermi offered a paragraph opining the existence of the last, which geologists see frequently.


 * By modifying the environment at one's will, equilibrium states can be moved or stopped at one's will. These equilibrium states will always be either stable or indifferent equilibrium states (discussed in treatises by Pierre Duhem and a thesis by Paul Saurel), not reversible ones (which are not realized in nature). At all these states, the phase rule F=C+2-P is satisfied.


 * I'm unclear about the confusion being asked about, but I'm hoping these remarks might help. Geologist (talk) 00:43, 28 October 2008 (UTC)


 * Is not Celsius the correct name for what used to be called Centigrade?


 * Should "Gibbs' phase rule" be "Gibbs's Phase Rule" ? Geologist 18:07, 27 March 2007 (UTC)


 * And, what happened to the nice exposition on the homology to Euler's polyhedron, or the question about Poincarre? I hope this stuff is coming back, this is what is interesting to (*) non-chemists.  Since Gibbs based his rule on Euler it's hardly controversial.  On the question of thermodynamic implications of polyhedral holes, I think there is comment by Cliff Joslyn on this.  Just something I vaguely remember.


 * I beg to disagree here - adding these quasi-mathematical detours makes the article much _less_ useful. There should be a clear division between material that a competent natural scientist needs to know, and 'explanations' however beautifal that appeal to pure mathematicians. This article desperately needs a rewrite to apply properly to multi-component systems, which is where its non-trivial content lies - the rule physically is about multiple applications of the equation $$\Delta G = 0 $$ - see any physical chemistry textbook.(eg Moore's Physical Chemistry p101)


 * I must agree with the spirit of the above remark. Gibbs's phase rule was of very great importance in physics & chemistry, and it still is very important in the natural & more applied sciences. Though its similarity to Euler's theorem on polyhedra was, at one time, intriguing (though unrelated to Gibbs's application of Euler's theorem on homogeneous functions); and philosophers took an interest in Poincaré's remark, the theorem's great value deserves a strong presentation of its effect on phase diagrams, and the theorem and diagrams' profound effect on most every science. Igneous and metamorphic petrology, for example, were both founded by this theorem. - The phase rule deserves a much longer encyclopedic entry; and it would first be nice to list those fields of sciences it founded. We might then solicit information from experts in these fields. Is this straying too far?


 * I've inserted references to two proofs below. However, neither uses Euler's theorem on homogenous functions, which I always felt was over-kill. As Gibbs does, they integrate the characteristic potential at constant p, & T, the Gibbs equation, and show that the class of equations it represents also yield a second differential form, the Gibbs-Duhem equation. Because the only similarity to Euler's characteristic is that both have topological interpretations, I suggest that section be deleted. Geologist (talk) 14:07, 24 March 2008 (UTC)


 * Bowen's reaction series founded modern igneous petrology, and the phase rule founds it. Eskola's p,T-classification of metamorphic rocks founded modern metamorphic petrology, and the phase rule founds it. (That the Earth's crustal rocks are now visible because they are in metastable equilibrium was observed by Goldschmidt, and preceded these theorems.) Geologist 18:42, 27 March 2007 (UTC)


 * The in apropos, superficial comparison with the Euler Characteristic of a convex polyhedron appears back. Someone above wrote: 'Since Gibbs based his rule on Euler it's hardly controversial.' Because this statement not really true, the relation of the phase rule to Euler's formula may very well be controversial. If you must have this, I suggest you apply it to the Tammann-Saurel theorem: a theorem in A 19th Century paper by Gustav Tammann and an early 20th Century paper by Paul Saurel. These relate a copunctal bundle of rays in the space of intensities to an object in the dual space of extensities: a polytope. (The dual of the plane dp,dT has great importance, but this, to my knowledge, isn't published.) Now Euler's formula makes some statement about the region around an invariant point, whose variables are f, c, & p. Can you find a reference to a paper on this? Geologist (talk) 00:35, 21 March 2008 (UTC)


 * The above I worked out (though in my head only), and obtained only some trivial combinatorial formulae. Perhaps I'm missing something. Geologist (talk) 14:07, 24 March 2008 (UTC)


 * This page should be merged with Gibbs phase rule -- till we *) 01:42, Aug 26, 2003 (UTC)


 * The relation of the universal gas law to the Gibbs' phase rule looks to be a bit tenous to me - can the gas laws be related to the thermodynamics of the Iron-carbon phase diagram??? Of course, if you want to talk fugacity in perfect systems that might be difference, but _this_ is about degrees of freedom, I think. Linuxlad 19:25, 20 Feb 2005 (UTC)


 * The ideal gas law is an artificial model of a gas. One wouldn't design it so it violated Gibbs's phase rule; so, the fact that it 'obeys' the phase rule has, I believe, no scientific meaning. Geologist (talk) 00:35, 21 March 2008 (UTC)

Degrees of freedom..
I would like some clarification on the term 'degrees of freedom'. Following the links doesn't really give a satisfactory explaination as to what this means in relation to the phase rule. As simple as possible would be good.. my concept of physics is limited...

Degrees of Freedom
Let me offer a geologist's interpretation or description of 'degrees of freedom'. This can also illustrate something of its breadth of application in the natural sciences.

Consider a rock. :-) One usually names it using all its 'essential' minerals. Let's assume these are all its minerals. The rock ameliorates perturbations it encounters during its path in the Earth as best it can. The number of tools at its disposal to do this are f. The value of f is called its 'degrees of freedom' (or thermodynamic flexibility by me). There are only f independent variations of thermodynamic variables drawn from among a pool of c+2 potentially independent thermodynamic variables: c variations in compositional escaping tendency are possible (accomplished my moving materials), one variation in temperature is possible (by moving heat), and one variation in pressure is possible (by performing work).

Gibbs's expression of his phase rule: f = (c+2) - p
The rock can experience f independent perturbations of any c+2 combination of these variables. (These are perturbations in natural variables, unlike dμi, dT, & d(-p), which are perturbation in artificial laboratory variables.) The rock would have all c+2 independant variables at its disposal: f tools; but one relation (described by the Gibbs-Duhem equation) is imposed by each phase within the rock, needed to keep that phase thermodynamically stable. In fact, there are p of these. So, f = (c+2) - p. As the number of independent perturbations by the environment increase, phases are dropped to increase f.

Ignoring reactions
There is a problem when attempting to use the phase rule. C is not constant. In fact it often requires a complicated calculation (such as that by S.R. Brinkley, in 1946) be made continually along its path. To drop a phase, species react. However, Gibb's huge project (creating physical chemistry) was greatly simplified by ignoring reactions. This little abstraction in no way changes those vast number of theorems he derived. (Adding reactions, in fact, creates more. :-)

De Donder's expression of Gibbs's phase rule: f = (s-r) + 2 - p
It is not at all obvious from the way Gibbs constructed c, but it was later shown equal to s-r, the number of stoichiometric species in the rock (which is a fixed number of species) minus the number of reactions among them (imposed by the conservation of matter).

Counting the number of species, s
However, we count s in a special way when examining each mineral and fluid: s is really the number of dμi, which is equal to the total number of species capable of independent variation in that phase alone, at constant T and p. It is not nearly as difficult to count s as to calculate c. Examine each mineral & fluid (each phase), and note it lies in the convex span of several compositional entities, or formula units (like H2O). Because the mass fraction of these formula units must sum to 1, their variations sum to 0; so, we subtract one from the number of these formula units to find the s independent variations of species contributed by that mineral alone at constant T & p. We commonly refer to these formula units as stoichiometric species when speaking of the system & environment, rather than an individual phase..

Calculating the number of independent reactions among the species, r
After we count the value s contributed by each mineral, we sum them to create s for the system. Now, we count r by creating an independent set of reactions among s. This is done by solving SR=0, where S is a matrix composed of s columns. The solution set is the nullspace of this matrix, best found (IMO) by row reduction to the Hermite matrix. See Talk:row echelon form.

Basic reactions (not published, but helps explain why reactions are written as they are)
If the algorithm to row-reduce matrix S is chosen carefully, the resulting reactions (the non-zero columns of R, which will be I-H), will each contain no more than c + 1 non-zero coefficients, termed a basic solution. If one phase was a double salt, it was convenient to select 4 rather than 3 formula units, contributing one reaction among the formula units of that one phase alone. For one phase to not violate the phase rule, it can contribute at most c independently variable species. Each 'basic reaction' contributes at most c + 1 - 1 = c independent chemical variations.

Chemical reactions in classical thermodynamics
De Donder's expression of the phase rule also works for systems without reactions, for s-r = c. It was 'developed' by Th. de Donder in early 20th Century Belgium, and popularized by I. Prigogine & R. Defay in their 1954 treatise. Many more references are needed by someone with access to the literature. Different sciences write chemical reactions differently, for good reasons. The column vectors of matrix S contain the amounts of each component in one unit of that species.

Use gram-atoms as species units
Common choices for the components are elements, oxides, or cations; common choices for the unit of species are the gram-formula unit (mole), gram-atom unit, or gram-cation unit. The best choice for reactions in chemical thermodynamics is the relative amounts of gram-formula units; for these satisfy many chemical rules or models. The best choice for reactions in classical thermodynamics is the relative amounts of gram-atom units; for these satisfy the lever rule and other obscure, but very important exact thermodynamic theorems. (One calculates the relative amount of gram-atoms of H2O from the relative amount of gram-formula unit of H2O by dividing the coefficient by the sum of the subscripts of the elements in the formula, then multiplying to create integers.) The coefficients of the latter kind of chemical reaction sum to zero and illustrate clearly the conservation of matter: they are sometimes called conservative chemical reactions.

From intensities to intensive variables
'Intensity' is a handy term, little used today, that is one class of thermodynamic variable. Specific equations, such as Clapeyron's, can be easily generalized by substituting any intensity & conjugate density. Other generalizations are (generalized) densities, extensities, and energies (characteristic potentials). One can find some use of these in the late 19th Century thermodynamic literature, and in Bryan's admirable little attempt to generalize thermodynamics using geometry and (unfortunately) Energetics.

Note, however, the phase rule applies to intensive variables (intensities & densities), variables that don't change their values when the system is replicated. (Using the Gibbs-Duhem equation & conservative chemical reactions to calculate the values of accessible directions on an intensity diagram is a wonderful application for students of elementary linear algebra, and appears in an early paper by Gibbs. Geologist (talk) 10:51, 21 March 2008 (UTC)

Electric and magnetic effects
How is the statement of rule changed when electric and magnetic effects are considered--84.232.141.36 (talk) 19:27, 17 January 2011 (UTC)?
 * The equation becomes F = C-P+n, where n is the number of variables other than composition variables which are necessary to specify the state of the system. See M.A. White "Properties of Materials" (Oxford Univ. Press 1999, p.172), which says that of electric and/or magnetic field are important in some circumstances, for example cholesteric liquid crystals (for electric field).
 * Also Laidler and Meiser ("Physical Chemistry", Benjamin/Cummings 1982, p.213 specify that F = C-P+2 is valid when phase equilibrium is unaffected by gravity, electric forces, magnetic forces or surface forces. One could find examples where each of these is important. Dirac66 (talk) 01:05, 18 January 2011 (UTC)

Vandalism?
There is a substantial drop in quality in the Examples section, between the 10 March 2006 and 17 March 2006 revisions. I don't know whether to add a cleanup tag or revert to the 10 March 2006 version (implying deliberate vandalism). Comments and help please? Sentinel75 06:14, 11 May 2006 (UTC)


 * The examples presented are all closed systems. (When investigating these, it makes more sense to combine the Gibbs's Phase Rule with Duhem's Theorem.) The Phase Rule applies to open systems as well. Geologist (talk) 19:31, 22 April 2008 (UTC)

Alternative derivation
In our thermodynamics class, we saw a different, more elaborate derivation of the Gibbs phase rule. It is this:

A system with C components in P phases, can be specified using the following intensive variables:
 * Temperature and pressure for each phase
 * Mole fraction of each component, for all phases.
 * In total: 2*P + C*P

The relations you can come up with, are the following (letters standing for components, numbers for phases):
 * in equilibrium:
 * T1 = T2 = ... (P-1) relations
 * p1 = p2 = ...
 * x1_a = x2_a = ...
 * x1_b = x2_b = ...
 * (P-1) * (C+2) relations
 * (P-1) * (C+2) relations
 * (P-1) * (C+2) relations


 * always:
 * x1_a + x1_b + ... = 1
 * P relations
 * P relations
 * P relations

This gives us ( 2*P + C*P ) - ( (P-1)(C+2) + P ) = C - P + 2 degrees of freedom. I don't know which derivation is most logical; the one depicted here or the one currently in the article. Please comment

Alternative derivation
The proof depicted here is the more logical, if one uses the variables used in texts today, xi. Gibbs, I believe, used the Gibbs-Duhem equation to derive, but not to prove, the phase rule. He chose not to prove it, though his argument is always cited as proof. (Similarly, his description of a phase (planar sides, &c) is not the definition he used: his definition was a region homogeneous in densities. Gibbs was a mathematician, and his is the only treatment I've read that clearly states both the necessary & sufficient conditions, not just sufficient, for a statement to be true. He argues for a phase rule, using intensities only, such dp, dT, & dμi. However, he states that the equation f = (c+2) - p applies to generalized densities as well (all intensive variables). His equation is local, appying within the neighborhood of a point on a surface.

More advanced texts, such as Denbigh's, use proofs such as yours - using global variables. Each intensive variable T, (-p), & xi is presumably a curve over a domain. (Some people prefer to use scalar stresses, such as dT, because 'relative values have absolute significance' -P. Bridgman.) There are at least two proofs in the primary literature; yours comes, I believe, from an early German paper by Wind. There was also a claim by Helm that the 1st law was necessary & sufficient to prove the phase rule. Other names associated with early papers on the phase rule are Natanson, Riecke, Duhem, de Donder, Planck, Saurel, Wind, Meyerhoffer, Nerst, Perrin, Raveau, and Trevor. H.W. Bakhuis-Roozeboom wrote a nice, qualitative thirty page article on the phase rule as a preface to his famous treatises on phase diagrams.

It would be nice to finally clarify all this, for I've never seen a review of proofs. Geologist 17:58, 27 March 2007 (UTC)

Two possible 1901 proofs found
Google's Books has a review of the 1901 physics literature, Die Fortschritte der Physik im Jahre 1901, that reviews two significant papers. One reviewer claims Paul Saurel (in 1901, 'On the Phase Rule'.J. Phys. Chemistry, v.5, p. 401-3) has extended Gibbs's phase rule from intensities to intensive variables: 'Temperatur, Druck, und Concentration der Phasen'. Saurel's works are flawless, so let's hope 'Concentration der Phasen' means independently variable concentrations within the phases.

The same abstracting journal reviews a paper by C.H. Wind in 1901, 'Sur la règle des phases de Gibbs'.'' Arch. Néed. v. 4, p.323-31''. The review of this paper contains an equation that very closely resembles Gibbs's phase rule as developed by de Donder, but for a wrong sign. I have Saurel's paper, I know, but I don't believe I have Wind's original paper at hand. If I have misread the German reviewer's definitions of Wind's variables, these two 1901 papers may contain the first proofs of the two expressions of Gibbs's phase rule described under 'Degrees of Freedom'. Geologist (talk) 13:48, 24 March 2008 (UTC)


 * There is an earlier reference, whose part I have (missing the last few pages) is leading to a correct derivation of the phase rule. It even employs Euler's theorem on homogeneous functions. Pierre Duhem, 'On the General Problem of Chemical Statics'. J. Phys. Chemistry: an English translation of the French manuscript was published around 1900. The part I have does offer an excellent positivist definition of equilibrium, one that could be used with profit today. Because phase rule applies only at equilibrium, a good proof should probably examine the independence of differentials d(Mi/MP) near states of equilibrium. Duhem's paper takes this approach, studying δMi. Geologist (talk) 21:27, 12 April 2008 (UTC)

Suggestions
1. A definition of 'degrees of freedom'.

In thermodynamics 'degrees of freedom' points to the number of intensive properties that may be freely set.

On simple monophasic hydrostatic systems (C=1, P=1) this number is two. Usually temperature and pressure, for the sake of simplicity.

When the system exhibits two phases in equilibrium (for instance water boiling at 100 celsius and standard pressure) the number of degrees of freedom reduces to one by Gibbs phase rule (C=1, P=1). This means you may freely change the temperature (for instance) of this system while preserving phase equilibrium. But, pressure will change accordingly in a way which is not due to the observer but to the thermophysical properties of water, Ie: through the coexistence line of vapour and liquid.


 * You meant to write P=2 here, I think. Geologist (talk) 11:13, 21 March 2008 (UTC)

When the system exhibits three phases in equilibrium (triple point) you get no degrees of freedom by Gibbs phase rule (C=1, P=2).


 * Again, P=3. (It must have been late.) Geologist (talk) 11:13, 21 March 2008 (UTC)

Meaning: the temperature and pressure of this triple point is determined by the thermophysical properties of the system (see triple point of water, for instance) and, in no manner, by the will of the observer. Yet, you may well change extensive and specific properties of the system at the triple point. For instance you may change the volume of the system, or energy, or enthalpy... just by changing the amount of liquid, solid and vapour present at the triple point thus leading to a line of triple point if volume (or energy, or entropy ...) is pictured. But, notice all these lines, states, collapses on a single value of the intensive parameters ---pressure and temperature---

2. The example pV=nRT is poorly presented since V is not an intensive property and can not be accounted for the number of degrees of freedom. Three intensive variable set would be pressure, temperature and chemical potential. Just two are freely choosen, the third being determined by the Gibbs-Duhem relation.


 * Yes, those intensities were the intensive variables originally used by Gibbs. There are probably much better examples to draw from than this equation of state, though it does satisfy the phase rule: p(V/n)=RT, (V/n) being intensive. The phase rule does apply to closed systems (where f is usually 2); but although this equation of a closed state is closed, it's not a system. Every model of a substance (equation of state) must satisfy the phase rule to be useful, but the phase rule's power is in its generality: all equilibrium states of all systems, real or hypothetical, satisfy it. Geologist 18:02, 20 April 2007 (UTC) ]

3. Nothing gets complex at the critical point. That paragraph should be erased.

Etaoin Shdrlu 13:11, 28 March 2007 (UTC)


 * I agree this should be removed, and nothing gets complex at the critical point. I'm not sure, however, things don't get difficult. I've long worked with a projective geometry of classical thermodynamics, which completely treats triple points. My initial finding is that projective geometry can not treat critical points: a concept of perpendicularity is needed. This suggests a difference in something. Geologist 22:59, 1 November 2007 (UTC)]

Possible improvement
Does anyone have an early reference to Gibbs work or writings on the phase rule. I cannot find any. I am looking at Max Plank "Treatise on Thermodynamics," 1945 unabridged republication of 6th/7th edition ca. 1926 (original preface dated 1897). Planck does not go into degrees of freedom or variability and doesn't invoke Gibbs Duhem however he uses Eurler relation to collapse the equilibrium expressed by (T,P) and [S - (U + pv)/T]. Perhaps this is actually Gibbs Duhem.


 * That would be interesting, for the derivation of the Gibbs-Duhem equation using Euler's relation is not in Gibbs, and the equation itself is not in either of Duhem's celebrated treatises. (So many mathematical relations in chemical thermodynamics were derived in Gibbs's treatise below that one needs to distinguish them: attaching the name of someone who later derived it independently or developed its implications is almost a necessity. In the preface to Duhem's Thermodynamique et chemie, one can read of his great enthusiasm for Gibbs's work. Though Duhem's work differs (he says) in containing more mathematics than typical works of the time on the phase rule, I can't find the Gibbs-Duhem equation in it.) The equation is very clearly derived in Gibbs, however. What is not derived in Gibbs is the phase rule. Gibbs uses his equation 97 (the Gibbs-Duhem equation) to quickly argue its likelihood, but never changes his variables from intensities (the same in each phase) to concentrations (different in each phase). This was left to someone else - Wind, I think, though Natanson comes to mind as well - in the early German literature. Here, I think, is a Google reference to Gibbs's phase rule (a term he never uses):

Gibbs's Original Derivation of his Phase Rule


 * There is also a forgotten work published in 1900 by H.W. Bakhuis-Roozeboom that serves as a very long preface to his many volumed work on the applications of the phase rule. I don't have a reference, but I'm sure it's also in Google Books (somewhere).


 * Planck's unappreciated work was never too attractive to me (it was originally a doctoral dissertation), because I never knew when approximate gas equations were mixed into his relations. He doesn't refer to Euler's derivation of anything in his early versions (in German), but his equation 151 is reminiscent to the Gibbs-Duhem equation. If so, he passes it over without a second thought. Oh, yes; unless I'm mistaken, Max Planck refers in his earlier copies of his treatise to 'phases', but he never mentions the name 'Gibbs'. :-) Geologist (talk) 14:01, 18 January 2008 (UTC)

Etaoin Shdrlu comment is nice. Maybe this is trivial. What happens to the theory above the critical point. for the gas, an infinitesimal amount below the PVT critical point there are 2 phases so df = 3 - 2 = 1. An infinitesimal amount above the critical point there is 1 phase so there are 2 degrees of freedom.


 * Yes. :-) Even more interesting, in a single-component substance, like H2O, the molecular amounts (I believe) of water & vapor become the same at the critical point, then ...murkiness and poof. This is very unlike moving a p,T-point off a univariant curve or invariant point, when a reaction slowly progresses until a 'species' is exhausted. Geologist (talk) 14:01, 18 January 2008 (UTC)

I would like to see more complete treatment of the mathematics and examples for both vapor liquid/gas and alloys, etc.Danleywolfe (talk) 23:26, 11 January 2008 (UTC)


 * Indeed. The phase rule is of profound importance not only in physical chemistry, but in many other sciences. We have a key application of it in geology by V. Goldschmidt. How to get scientists in many fields contribute a paragraph or two would appear a challenge. Also, before someone posts a (correct) mathematical derivation of it, it would be nice to have a reference. :-) Geologist (talk) 14:01, 18 January 2008 (UTC)

Perhaps I hadn't examined the actual article, and assumed the improvements in the talk section has been added, but the article is riddled with inaccuracies and irrelevancies:-


 * 1) The theorem is Gibbs's phase rule, not Gibbs' phase rule. (In English, one leaves off the final s only when it's difficult to pronounce.)
 * 2) Gibb's phase rule first appeared in the Transactions of the Connecticut Academy of Sciences in the paper 'On the Equilibrium of . Heterogeneous Substances', published in parts between 1875 & 1878. It was brought to the attention of European scientists by the lectures of J. Clerk Maxwell, and subsequently translated into French by Le Chatelier and part into German by Ostwald (or vice-versa).
 * 3) Water is a liquid.
 * 4) H2O is a formula unit, used to express chemical composition; it is not a molecule.
 * 5) The 'thermal analysis technique' (if you must have it) takes place at constant pressure.
 * 6) 'volume of a gas' is not an intensive variable, but V/n is.
 * 7) 'universal gas law' should read 'ideal gas law' (if you really must have it)
 * 8) Above the critical point, there is one phase:a fluid.
 * 9) There is (IMO) no 'condensed phase rule'. That is, I have objections to it. (1) First, it is poor diction. The word 'rule' has two uses in science: an exact relation and an approximate relation. Switching usage from exact to approximate is confusing. (This is not the case with 'Goldschmidt's Phase Rule', from geology.) (2) There are many applications of the phase rule at constant pressure, f = c + 1 - p; but this is just that. One application is the iron,titanium-oxide geo-thermometer (in petrology): this is valid for two iron-titanium oxides & O2 (or H2O) in solution, where it is universally found that the two oxides' rims had equilibrated near atmospheric pressure. Here, f = 3 + 1 - 2 = 2. The thermometer fails in plutonic rocks, where total pressures vary, despite extremely low vapor pressures. (3) What would again appear to satisfy the requirements of a 'condensed phase rule' is the equilibrium assemblage kyanite-sillimanite-andalusite: these aluminosilicate minerals have very low vapor pressures at high temperatures; but their vapor pressures change differently with pressure. It's this variation that causes the p,T-diagram to have curves that change greatly with pressure, in apparent 'violation' of any 'condensed phase rule'. Geologist (talk) 19:54, 22 April 2008 (UTC)
 * 10) No one, to my knowledge, has found a substantive relation to Euler's formula (see, however, a reference to the Tamman-Saurel theorem above).
 * 11) Throughout most of the 20th Century, the rule was written F = C + 2 - P. Because pressure is represented by a lower-case, italic p, I'm not aware of the need for Greek symbols, though they are used in recent editions of a popular text in Chemical Engineering. Are these symbols used in the literature of various sciences?

Might I strongly urge the phase rule be written: F = C + 2 - P, which is much easier to remember, making F = ( S-R ) + 2 - P easier to remember.

Geologist (talk) 23:00, 20 March 2008 (UTC)

November 2008 Rewrite
I found the page in a sad state and did an extensive rewrite. I tried to not only correct the errors, but to put the rule in context. When you look at the basis, you can see why there is a different rule for condensed phases.

It should be clear that the rule is of no help in predicting when mulitple phases will form and does not give equations of state. That you get no phase transition above the critical point relates to phase diagrams, but is not relevant on this page. There are all kinds of different phase behavior and the rule only tells you what to do after you have figured out the number of phases by some other means. The reference to Euler's formula was apochryphal, but somehow persisted from the first draft until I nixed it. The talk about degrees of freedom was confusing and I suspect came out of someone editting the page without quite understanding the rule or being able to articulate their understanding.

I should have mentioned the assumption of no chemical reactions. My source assumes no chemical reactions, but there is a comment above about this and I suspect the only chemical reactions that would matter would be ones that change the number of species.

A derivation of the rule would be nice, but I am going to leave that to someone else. I think I see a good one in the talk above: assumptions that T, P, and xi specify each phase corresponds to my Duhem's Rule applied separately to each phase. Add in phase equilibrium and the derivation is just a few steps.Paul V. Keller (talk) 06:30, 15 November 2008 (UTC)

I (or someone) needs to explain more about intensive variables and relate the thermodynamic state and Duhem's Rule to a description of a system state that is entirely in terms of intensive variables. Also needed is a definition of "independently variable" and an explanation of why temperature and pressure are not independently variable in a multiphase system under that definition.Paul V. Keller (talk) 16:12, 15 November 2008 (UTC)

Another re-write
I'm sorry to say that the last re-write falls way below an acceptable level of quality. It appears that the editors involved have little expertise in the subject matter. I have done a complete re-write based on two chapters of a standard text-book in physical chemistry. These chapters include all the diagrams in phase diagrams and many additional applications of the phase rule. Indeed a case could be made for merging phase rule and phase diagram, but I am not going to propose that right now.

I hope that previous editors will not be offended and that they will see that the present text does much more justice to the topic than previous texts. Petergans (talk) 11:36, 22 November 2008 (UTC)


 * Not offended. If it was not clear to you, it failed in its intended purpose.
 * I still like the accuracy of my verison. Instead of pushing back, I made a few corrections to yours.  Hopefully, we can agree.
 * Boiling is a misleading term when talking about a two phase system of steam and watter. As you change the temperature or pressure, your get some condensation or evaporation (maybe even boiling) during the transient.  The point is that the ability of water to redistribute between the phases keeps you on the phase line until you have exhausted one phase or the other.
 * An explanation of where the rule comes from would not only put the rule in context but lead to a better appreciation for its limitations.Paul V. Keller (talk) 05:53, 24 November 2008 (UTC)
 * Your corrections are perfectly acceptable. I have added a couple of examples of eutectic alloys. Regarding your last point, I am not aware of any limitations. What have you in mind? Petergans (talk) 09:33, 24 November 2008 (UTC)
 * The main rule is limited to closed systems in which the thermodynamic state can be completely described by specifying temperature, pressure, and the amounts of each species charged. That covers a lot of systems, but assumes things like electric fields and inertial fields are not important. You would want to be careful applying the rule to a liquid crystal system or a system in a centrifuge.
 * The other side (fewer independent variables) is already covered to a large extent. The condensed phase rule reflects that pressure makes little or no difference in many systems involving liquid or solid phases, although there are condensed phase systems where the pressure does matter and the main rule would still be the one to use.  Diamond formation comes to mind, although I am not sure that is a good example.  Fewer variables are needed when some of the compenents interchange through a reversible chemical reaction, but that is covered by specifying "independent" components.Paul V. Keller (talk) 14:37, 24 November 2008 (UTC)

Examples should include calculation of F = C-P+2
I am glad to see that Petergans and Paul Keller are converging on a useful set of examples (or "consequences"). I would like to suggest that the examples would be clearer if each (or most) specified the explicit calculation of F using the phase rule. Sample format: for the liquid-vapour equilibrium of a pure substance, C=1, P=2, F = 1-2+2 = 1 so that T and p cannot vary independently. Dirac66 (talk) 16:19, 24 November 2008 (UTC)

Pure systems much improved, binary systems to redo.
After the edits by PVKeller yesterday, the material on pure systems is much improved. The material on binary systems has been removed for the moment, since it implied incorrectly that the phase rule is responsible for the existence of features such as azeotropes and eutectics. I think the next step is to rewrite more correct material for binary systems, possibly based on a physical chemistry text or texts. Dirac66 (talk) 02:36, 9 December 2008 (UTC)

In this rewrite, it would still be useful to apply the phase rule to a simple phase diagram for a binary system. I suggest the boiling point diagram at right from the article on Phase diagram. The essential point is that a given T, the compositions of liquid and vapor are not independent since the chemical potentials of these two phases must be equal. Note that I have now chosen a diagram with no azeotrope, since (as PVKeller has pointed out) the application of the phase rule is the same at azeotrope points (or eutectic points) as at other points, so that no special mention of azeotropes is needed in this article. Dirac66 (talk) 14:47, 9 December 2008 (UTC)


 * One should tread gently when applying the phase rule to systems of more than one component; and possibly pause and consider fundamentals. (I quite enjoyed Dr Keller's very brief rewrite.) On the two-component phase diagram, many important objects appear graphed, but the system is not.


 * The graph of the binary system is a point somewhere within the diagram illustrated. Your discussion suggests the system being discussed is somewhere on the tie-line connecting the liquid and vapor compositions. Its exact position is unimportant here, because every univariant system (two-component vapor over liquid at a specific T & p) is 'indifferent' to the relative quantities of the two phases. This argues for the separation of "Gibbs's Phase Rule" from "Thermodynamic Phase Diagram". Confusion arises because of the use of 'system' for a point and 'System' (capitalized by Ricci) for a region. It might be good to sit back and review the meaning of 'independence', as Dr Keller suggested, and use a limited vocabulary. 'Functional independence', 'implies' & 'existence' could be borrowed from mathematics.


 * G.N. Lewis & W. Bancroft had an unnecessarily acrimonious split over the value of phase rule, which is, geometrically speaking, a topological theorem: it tells one little about low-component systems (and the nature of bonds), but it tells one much about systems of many components. For this reason, it is still of great importance in the natural and applied sciences while it is on the brink of being dropped from physical chemistry texts. It might be better if this article attracted more attention by astronomers and viticulturists rather than only chemists and physicists. The two fundamental theorems of petrology, Bowen's Reaction Series and "Eskola's Mineral Facies", are mere applications of Gibbs's phase rule to rocks.

Geologist (talk) 01:06, 22 July 2009 (UTC)
 * BTW, Why are four curves emanating from the one-component invariant point? The invariant point and critical point are natural objects, but the horizontal and vertical lines passing through the critical point are not (.. at least within the theory of classical thermodynamics): they have no substantial meaning.

Geologist (talk) 01:33, 22 July 2009 (UTC)
 * Sorry I didn't notice the lever rule earlier: this is very important to geologists and is a relation among compositions only, like indifferent theorems (the Gibbs-Konowalow Theorem, Saurel's Theorem). Perhaps it belongs in an article devoted to phase diagrams. Also, it is valid only on diagrams whose abscissa is in fractions of total mass or total atoms (see 'Use gram-atoms as species units, above), not mole fractions.

Thanks for your comments. It is true that the article at present is written from the point of view of physical chemistry, and it would help to add more applications to geology etc. I would add those at the end, however, and maintain the logical structure as Foundations, Pure Systems, Binary Systems, followed by more complex applications.

Re Foundations – I think the present version due to Dr Keller is simple and very good, and the discussion of independent variables is quite clear. The previous versions posted last year became so involved with the relation to more abstract mathematics and topology that the meaning of the rule was lost and they were hard to read. I vote for leaving this section alone.


 * Sorry, I was referring to a version by him that appear for only a day or so. It had no mistakes, though any derivation from Duhem's Theorem would be for closed systems only. Geologist (talk) 12:40, 26 July 2009 (UTC)

Re One-component diagram – yes, the four curves emanating from the triple point are confusing. I have just added a short paragraph to explain why there are two green curves, but it would probably be better to redraw the diagram without the dotted curve for water.

The horizontal and vertical dotted lines could also be suppressed in a redrawn diagram. The problem is that this diagram has been taken from another article, and these lines were there to show the position of supercritical fluid, which is not really helpful to this article as it is not a separate thermodynamic phase.

Re: Binary diagram – yes, a system point can be anywhere on the diagram, but the article considers points between the two curves as the two-phase region is more interesting. I have just modified the sentence about the isotherm (or tie line) to specify that it is drawn through the arbitrary system point; that is why the point is on the tie line.


 * The question you might want to ask is: Is the phase rule about a point, or about a diagram? Is it just the lever rule that extends the point to the rest of the diagram? [Comment added by Geologist 12:40, 26 July 2009 (UTC)]

This section only mentions one type of binary phase diagram, as an illustration of the working of the phase rule for C=2. For others, the last paragraphs mentions some possibilities and provides a link to the article on (thermodynamic) phase diagram.

Re: Lever rule – it works for mole fractions as well. Atkins and de Paula (8th edn, p.182) give a proof in terms of number of moles of each component. To obtain mole fractions, just divide both sides of the equation by the total number of moles in the system. Dirac66 (talk) 03:31, 22 July 2009 (UTC)

Lever Rules
There are several 'lever rules': one in which only the mass of the phase is known, one in which the composition of the phase is known, and one in which ... well, nothing fixed is known. The first is nice in that one need not know compositions, and can specify a phase as 'halite'. The second is used in petrology; and it is necessary when crystals precipitate (or bits or rock are added). OK, I won't complain about the third; though association & dissociation vary with temperature, and the 'lever' seems a bit unnecessary. This mimics the Lewis-Bancroft split mentioned above; and is likely illustrates the split between chemists and natural scientists:-) Geologist (talk) 12:40, 26 July 2009 (UTC)

I have now specified that the rule to be used corresponds to the variable on the x-axis, which is mole fraction in the diagram shown. Dirac66 (talk) 21:21, 26 July 2009 (UTC)

Proofs
I do have a comment on this line. 'If four phases of a pure substance were in equilibrium, the phase rule would give F = -1 which is impossible. This means that four phases of a pure substance (such as ice I, ice III, liquid water and water vapour) can never be in equilibrium at any temperature and pressure.'

This is what Gibbs wrote: 'It does not seem probable that P can ever exceed C+2.' -Gibbs, p.97 The article might benefit from a proof; but, because the phase rule can't be proved, that is asking too much. We cannot prove, to my knowledge, that there should not someday be found a relation among the temperature, pressure, & chemical potentials of components in a phase other than the Gibbs-Duhem equation. Gibbs was certainly aware of this, and never proved the phase rule. Wind & Duhem, I believe, both draw upon the equality of chemical potentials (which I'm glad was removed).

The derivation given by me at the top of the page is Wind's, I believe; and Duhem was first to use Euler's theorem to derive the Gibbs-Duhem equation. Here is Gibbs's derivation:

'If a homogeneous body has C independently variable components, the phase of the body is evidently capable of C+1 independent variations. A system of P coexistent phases, each of which has the same C independently variable components, is capable of C+2-P variations of phase.' -Gibbs, p.96.

Notice that the above derivation just counts phases, employing 'intensities', not all intensive variables (which include 'generalized densities'). Geologist (talk) 12:40, 26 July 2009 (UTC)

Because Gibbs can be cryptic, I offer this 'clarification' of his above derivation. (Upon proof-reading, I'm unsure this clarifies anything.-bb) The phase rule is a local 'rule', relating d(mu)1, ..., d(mu)c, d(-p) & dT at every point of stable or indifferent equilibrium in an open or closed system. Only a lever rule extends the point to a diagram.

Gibbs: If a homogenous body [a single phase system] has C independently variable components [in addition to the thermodynamic variables p and T], the phase of the body [a rim of crystal or bubble] is evident capable [by itself] of [only] C+1 independent variations, [because the Gibbs-Duhem relation imposes one restriction upon these variations]. A system of p coexistent phases, each of which has the same C independently variable components [otherwise see Wind's derivation], is capable of C+2-p variations of phase [as imposed by p independent Gibbs-Duhem equations: (delta m)1 ... (delta m)c + (delta V)d(-p) + (delta S)d(T) = 0 ].

One can write the above as an array of p, homogeneous, Gibbs-Duhem equations and subtract the number of rows (p) from the number of columns (C + 2) to obtain the number of independently variable intensities the phase assemblage can span. (This assumes the number of phases & (row) rank are the same, an assumption which appeared to trouble Gibbs.) The use of molar or mass fractions, as used in introductory texts, requires an extension to this derivation which I think is found in Duhem (some of my pages are missing). Cleverly skirting this makes the above derivation seem rather magical to me, at least. Certainly elegant.

I don't think one should rule out F = -1 as a possibility, since Gibbs didn't. One could even build such a system from a 1-component p,T-diagram, if there existed a region surrounded by five invariant points. A substance that dissolves in all phases but the region's phase would destabilize the region, possibly resulting in a point on a 2-component p,T-diagram from with five curves emanate (within our ability to measure or observe otherwise). The variance at the point would then be -1. This would require the rows of the above array being linearly dependent, which has not been proved impossible. Geologist (talk) 06:11, 30 July 2009 (UTC)

When I said F = -1 is impossible, I meant that the concept of "-1 degrees of freedom" is meaningless. When you say not to rule it out, I think you mean that we should not rule out C - P + 2 = -1, i.e. P = 4 for pure systems. I have now reworded the paragraph to separate the two meanings. For P = 4, I mentioned mathematical dependence as a possibility but added that a non-equilibrium system is more likely in practice.

Also I said mathematical dependence rather than linear dependence, because I am not clear in my understanding that the dependence must be linear. I will change it to linear if you are confident that that is correct. Dirac66 (talk) 15:46, 15 August 2009 (UTC)

Justification for use in article
I think that Geologist's points about Proofs are generally valid, but that it is necessary to bring the argument down to a suitable level for the article. An encyclopedia article is intended as an introduction for those unfamiliar with a subject who have seen the term and want some explanations. For the phase rule perhaps the typical reader would be a student with one year of physical chemistry and one year of calculus.

I agree that there is no rigorous proof of the phase rule, since the usual justification/derivation/argument does not exclude the possibility of linear dependence. But to introduce the subject I think it would be useful first to explain why the rule generally (= “almost always”) works for equilibrium systems, and then mention the possibility of exceptions due to linear dependence or non-equilibrium.

In order to make the learning curve as simple as possible for typical 2009 students, I think it should be based on contemporary textbooks and use chemical potentials, even if Gibbs etc. used other functions which are now less familiar. To simplify the discussion even further, we can start as now with the special cases C=1, P=2 and C=1, P=3 but I propose to add a mention of the chemical potential equations, at the level of high school algebra: two equations can be solved to find two unknowns T and p. This would be analogous to what is already in the discussion of binary systems. The argument for the general case (any C) could be added too, but we have given a reference to a leading textbook.

Finally the case of C=1, P=4 (i.e. “F=-1”) does need revision, but I will leave that for next week. Dirac66 (talk) 21:44, 4 August 2009 (UTC)

C=1, P = 2 and 3 done. P = 4 (or F = -1) to come, but first I want to discuss the aluminosilicate system (andalusite etc) further. Dirac66 (talk) 01:13, 12 August 2009 (UTC) Sorry, on rereading your comment, I see that aluminosilicate is not relevant here. Dirac66 (talk) 01:40, 12 August 2009 (UTC)

Perhaps two rules?
Gibbs's original statement, motivated by an example, seems a simple & intuitive derivation. Note that he wrote his rule F = C+2 - p. (I'm not sure the new casting clarifies it any.) Linear algebra wasn't invented in Gibbs's time: Gibbs's treatise used determinant theory. In the 19th Century, this was taught to American secondary school students. (American education has changed.) If you wish to also use a more general derivation containing the equality of chemical potential functions, mole fractions, and reactions, may I suggest the derivation in Prigogine & Defay?

One can safely pull 'escaping tendency' functions out of the air, for this is what Gibbs did. Calling them 'escaping tendencies' makes their equalities intuitive, avoiding the deadly MIT derivation. Chemical engineering texts from MIT inherited Gibbs's derivation of equalities of escaping tendencies for closed systems and consequently derive a phase rule limited to closed systems.

Picture of Gibbs's Rule

Gibbs always illustrated his analytical equations geometrically. Each phase can be represented by a surface in the space p, T, mu(H2O). Three surfaces intersect at a point. The univariant curve liquid-vapor can even terminate when the corresponding surfaces rotate and become a single, fluid sheet. Projected along the mu(H2O) axis, the objects trace the p,T-diagram for H2O already shown. There must be a simple diagram like this somewhere. Geologist (talk) 18:14, 6 August 2009 (UTC)

I may try to find Prigogine and Defay someday, but for now I favor using the simple arguments found in modern textbooks such as Atkins and de Paula. Dirac66 (talk) 01:13, 12 August 2009 (UTC)

Failure of Phase Rule
Sorry, thought I should clarify why I thought Gibbs believed the phase rule isn't a theorem (for he covers most linear dependencies in the paragraph that derives the Gibbs-Konovalow Theorem): he chose to believe the critical point of water to be a phase, so the phase rule consequently fails at the critical point. Denbigh has other ideas; but note this:

'For as every stable phase which has a coexistent phase lies upon the limit which separates stable from unstable phases, the same must be true of any stable critical phase.' Gibbs, p. 131.

In other words (this is like interpreting Scripture), the two phases liquid water and water vapor occupy a curvilinear region on a P,T-diagram that stops suddenly, where there is a tie-line of zero length :-) joining that region with the critical point, occupied by the critical phase: one component, one phase, zero degrees of freedom. The phase rule fails. Two more relations are needed, to stabilize the critical phase.

Gibbs states that the curvature of the characteristic function (the spinodal) is zero there - providing the needed relation (which, by itself, stops any metastable extensions into the fluid region). This choice of a critical phase led to the study of critical points of various orders (the Curie point, superconductive transitions, &c).

A third relation is needed to limit the critical region to a point. Having examined the characteristic function of the critical phase, Gibbs found the tangent zero and the curvature zero. The third derivative, the flatness, must be less than zero (convex - Gibbs chose the entropy function, which was concave). Some greater derivative must be non-zero or the critical phase could not be stable. Now one has three phases imparting one relation each to create a point, and one critical phase imparting three relations to create a point. Gibbs is really simple. :-)

One could claim that the critical phase H2O is not within the domain of classical thermodynamics; but this murky phase we can clearly see in a test tube. This choice of phase creates a clear failure of the phase rule, which Gibbs had to have seen very clearly. Geologist (talk) 00:18, 6 August 2009 (UTC)

It is worth noting that classical thermodynamics, sans molecules, is a mathematical model based upon smooth, differentiable curves. When plopped upon reality, disagreements appear where its domain has been exceeded. One wouldn't expect this on a mundane p,T-diagram of significant amounts of substances. However, when a phase requires higher derivatives of the characteristic function to distinguish it from its neighbor, the substances are likely so similar that classical thermodynamics will fail and a discrete physical theory need be called upon. This is unexpected.

No Rescue in Sight

When failures like this appear, it is common practice to limit the domain of the theorem by qualifying it in some manner. However, the phase rule fails at the critical point of H2O; and the critical phase cannot be ignored, for Gibbs uses it to trace a critical curve in 2-component systems and critical surface in 3-component systems. This counterexample to the phase rule would appear to me a difficult impediment to skirt; unlike many other so-called failures.

My apology about the length given to this.

The simple suggestion that follows from all this is that 'proof' not be found in the article: it is everywhere in the primary literature, and likely some current texts. 67.91.218.205 (talk) 06:25, 6 August 2009 (UTC)

Certainly the word "proof" does not belong in the article since it is not a rigorous proof. But I think the article should explain the origin of the rule, at least for simple cases (perhaps better than now) with a reference for the general (though not quite universal) case. As for the "failures", I suggest adding a section at the end on Limitations (nicer word than failures) of the Phase Rule to point out briefly when it does not work. I think we now have 3 types of failure - non-equilibrium systems, linear dependence and critical points. And perhaps very small systems where thermodynamics is known to fail. P.S. This is my last comment for several days. Will try to edit article next week. Dirac66 (talk) 13:13, 6 August 2009 (UTC)

Many authors have done this, though every theorem and certainly rule has a limited domain of application that it inherits from its derivation and testing. The handedness of molecules that twist light was highly contested for years in the literature; but I found this discussion somewhat hollow, since this was just akin to reactions being inhibited (a more general class of consideration to take into account before applying the rule). The phase rule assumes equilibrium in its derivation (so that's not a limitation), and every linearly dependent assemblage forms an indifferent system (such as an azeotrope). The phase rule applies to all these except the gently merging of one phase into another at critical & tricritical points. Most other problems encountered are based upon choosing components wrongly (alleviated by using the phase rule that incorporates reactions, such as Prigogine & Defay's). Others are based upon reactions that don't proceed to equilibrium; and these can be easily fixed using the same rule. Books with the title 'Phase Rule', from Duhem to Ricci, have long lists from which people can choose. Google Books offers free the complete books by Gibbs (his collected works), Duhem, Bancroft, Meyerhoffer, Trevor, Tammann, & even the more recent MacDougall's. (Could some European universities consider adding the dissertations of Saurel and of Defay?)

I vote only for critical points; and this limitation could be eliminated by adding its application there as a corollary of some sort. (The phase rule even works for osmotic systems, which have two equilibrium pressures.) Geologist (talk) 17:25, 6 August 2009 (UTC)

The Phase rule is Valid for Open Systems (though few agree)
Equalities of chemical potentials, used in most popular dervations, appear earlier in Gibbs, who didn't use them when deriving the phase rule; for a good reason, I think: they follow from the assumption of a closed system. Gibbs's derivation is valid for open systems; it is a local proof (making independence the linear independence of tangent vectors at a point), and his proposition becomes one describing a phenomenon (a change). Geologists have applied the phase rule (and equilibrium calculations of temperatures at which metamorphism stopped) for many decades to rocks undergoing changes in bulk composition without any problems. Geologist (talk) 12:40, 26 July 2009 (UTC)

Only in Guggenheim's text does the characteristic function decrease during a natural change. Fermi, in his lecure notes, claimed that the phases within a natural system can obviously change thermodynamic state slowly enough for the phases and reactions to keep in equilibrium. The system would then follow a classic 'equilibrium path'. That is what has been observed in many rocks, not only metamorphic but igneous, and even volcanic. (One must sometimes choose one's reactions carefully, especially in the latter case.)

Both metamorphic and igneous rocks were, in general, open systems: systems during which the bulk composition changed while the thermodynamic change being studied took place. Early work by Goldschmidt and by Eskola can be re-interpreted to show the phase rule was valid along such paths; but more recent work by Korzhinskii and by Thompson tends to obscure the meaning of 'open' in geological systems. What the current opinion is, one must ask an active petrologist. No one should have doubt however, that if the phase rule is valid at every point on a phase diagram, these include points along equilibrium paths whose bulk composition changes continuously (curves that aren't vertical).

Perhaps someone has a reference that explicitly states the phase rule to be valid for open systems? Geologist (talk) 06:49, 30 July 2009 (UTC)

Provided that an open system changes slowly enough to be considered effectively at equilibrium, then I don't see a problem with using chemical potentials. I suggest that we simply say nothing about the system being closed or open. It seems sufficient to specify that the system must be at equilibrium. Dirac66 (talk) 01:24, 12 August 2009 (UTC)

The Condensed Phase Rule Again
Geologists have been confused about the appearance of three 1-component, densely crystalline polymorphs (andalusite, sillimanite, and kyanite) coexisting in certain geological regions. Sorry, but you can't see the triple point if you discard pressure from the P,T-diagram. (Cough.) (A study of extensities rather than intensities explains this phenomenon, I believe.)

Two possibilities of dealing with the unmeasurable effect of pressure might be to discard the d(-p) term, or set the changes in V to zero if it cannot be measured using today's instruments, or if its effects cannot be observed. Geologists never have use of such a 'rule', and either of the last choices would be an operationally correct way of dealing with the situation.

It would be bad if a geology student recognized the phases as condensed and applied F = C + 1 - P.

Geologist (talk) 16:10, 26 July 2009 (UTC)

OK. I note that you have mentioned this point previously. I have adopted one of your suggestions above and renamed the section Phase rule at constant pressure. I did not want to completely eliminate the phrase Condensed Phase Rule from the article since it does have 20,500 Google hits so readers may search for its meaning, but I have called it "misleading" and specified that it is only applicable when pressure effects are small and should not be used at high pressures as in geology. Dirac66 (talk) 23:38, 27 July 2009 (UTC)

Dirac66, Your point about Google hits is very important. Chemists, who work with beakers over Fisher burners would reasonably use C + 2 - 1, so that seems reasonable; and your use of 'misleading' for the 'Condensed Phase Rule' is the best I description can think of. Geologists, too, have 'Goldschmidt's Mineralogical Phase Rule': an unnecessary modification of Gibbs's Phase Rule to specific systems. Geologist (talk) 00:04, 2 August 2009 (UTC)

My other suggestions, though I believe them true (can, I believe, prove them true), may not be used or even believed by most geologists. These would not have a place in an encyclopedia, only in an ignored monograph. (This is one reason I don't modify Wikipedia articles: the current state of knowledge in Geology and my opinions have rarely agreed:-) Placing provable statements in the discussion as personal opinion enhances the Wikipedia, I feel. Geologist (talk) 00:04, 2 August 2009 (UTC)

OK. I plan eventually to get back to some of your other points, but they require thought (which is a compliment) so it will take some time. Dirac66 (talk) 01:03, 2 August 2009 (UTC)

Corollaries to the phase rule
I was interested to learn from your comment about Goldschmidt's rule, which I found explained at http://serc.carleton.edu/research_education/equilibria/phaserule.html, a geologically-oriented site by Mogk on the phase rule which I have now linked from the article. This also has several thousand Google hits (depending on how one defines the search terms) so could be explained in the article. I am now thinking of including both the constant-pressure rule (alias condensed rule) and Goldschmidt's rule in a section titled Corollaries to the phase rule - a nicer way of expressing your phrase "an unnecessary modification of Gibbs's Phase Rule to specific systems". This would make clear that both these rules are just special cases.

A third corollary for the same section would be the rule for magnetic systems when magnetic field (or intensity) is a significant intensive variable, so that F = C-P+3.

Azeotropes & Eutectics
Truly, I hate to nag. 'Liquid-vapour phase diagrams for other systems may have azeotropes (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. The same is true for liquid-solid phase diagrams which have minima known as eutectics.'

It must have been late. :-) Eutectic points are points of minimal-temperature liquids, but only when the phases differ in composition. Azeotropes are points where curves intersect, and phases become the same composition. The Gibbs-Konovalow theorem assures they are extreme points (minima or maxima).

The phase rule is applicable to an open or closed assemblage of phases in stable or indifferent equilibrium, unaffected by energy fields (external energy).

Geologist (talk) 16:10, 26 July 2009 (UTC)

Yes, the eutectic sentence is wrong. What I wrote on azeotropes is correct for the simple case of two completely miscible liquids considered as an example. It would also be correct for eutectics in a system where the two components are completely miscible in both solid and liquid states, but this case is quite exceptional and unimportant. For the far more typical case of immiscible solids, the liquid composition is equal to the overall (weighted average) composition of the two solid phases. Explaining this here seems too complicated though, so I will just remove the sentence about eutectics, which are discussed in their own article. The azeotrope discussion is sufficient as an example of the application of the phase rule to binary systems.

P.S. Note that the eutectic article incorrectly claims (paragraph 3) that there is a single solid phase which is a "homogeneous mixture". I think this may have helped to confuse me at the time, and it will have to be fixed eventually. Dirac66 (talk) 20:15, 26 July 2009 (UTC)

Newlink
Why does a search for newlink redirect here? I was looking for a company called newlink and instead get redirected to a page with no mention of newlink or new link. Clearly this is not helpful. --Shadebug (talk) 16:08, 29 December 2008 (UTC)
 * That was just a glitch; it was created a few years ago by mistake. I've deleted it. DS (talk) 03:39, 31 December 2008 (UTC)

Vapour vs gas
Today 221.232.151.68 added the words "There exists an error in the phase diagram.The superheated vapour region and gaseous phase should be replaced mutually", which Ddcampayo reverted with the edit summary "I think the diagram is fine the way it is."

I believe that 221.232.151.68 is mostly correct here. A vapour is defined as a gas which is below its critical temperature. So the region T > Tc cannot be described as a vapour and should simply be labelled "gas". The low-T region can be described as "vapour", though "vapour (gas)" would probably be better, since the vapour phase is physically a gas. For the purpose of this article on the phase rule, vapour and gas are equivalent.

As for "superheated", this term is used mostly for steam (water), for which superheated steam is steam above the normal boiling point. Not really a useful term to explain phase diagrams.

So this diagram which appears in several articles should be redrawn. For now I will restore a mention that vapour should be in the low-T region, in the paragraph which refers to the Figure. Dirac66 (talk) 15:44, 27 January 2010 (UTC)


 * The diagrams being discussed seem to have little to do with the phase rule, whose value lies in its description of a heterogeneous phase assemblage at one intensive state. The p,T-diagram of H2O benefits very little from the phase rule; and what is being discussed here has nothing at all to do with the phase rule, but with arbitrary nomenclature. The distinction of gas and vapor, superheated steam and vapor, &c is irrelevant to the article. It might be better to use only the words 'crystal', 'liquid', 'gas', and 'fluid' (near but beyond the critical state). The fluid resembles more a liquid on one side of the slightly extrapolated melting curve, a gas on the other. There is no clear distinction by a meniscus, however. Geologist (talk) 20:57, 2 February 2010 (UTC)


 * It might be clearer to erase all the dotted and dashed lines, and use a substance other than H2O if you don't like its curve. Only the 2-phase curves, 3-phase triple point, and 1-phase critical point (with critical opalescence) are natural. (I prefer 'vapor, but 'gas' is spelled the same in Britain and the United States.) Other objects, and your distinctions among them, are as artificial as the axes 'pressure' and 'temperature'. (These are artificial axes of convenience.) The term superheated steam is a layman's term for hot, 1-phase water vapor in stable equilibrium (ie. points in the gas region above 100 degrees Celsius): 'saturated steam' is hot, 2-phase steam in stable equilibrium (ie. points on the boiling curve above 100 degrees Celsius). Geologist (talk) 20:57, 2 February 2010 (UTC)


 * Because the phase rule is a theoretical relation among natural objects, the relation F = C + 2 - P was originally presented by Gibbs as an existence theorem for natural objects (the boiling curve, triple point), objects that are independent of the coordinate axes. As coordinate directions, one may choose the directions of any two phase reactions at a triple point. The phase rule is still valid. Temperature isn't necessary to derive the phase rule, but it is necessary to delineate, here, 'gas' from 'vapor'. Geologist (talk) 20:57, 2 February 2010 (UTC)


 * I agree that the present diagram is far too cluttered with terms which are irrelevant to the article, as well as extra lines and curves. On reading your comments I remembered that in many chemistry texts the first minimal phase diagram is for CO2, so I checked the carbon dioxide article. The first diagram there seems a suitable replacement, as it is simpler with less extraneous terms (as well as more colorful!). I will tweak the text a little to define very briefly the triple and critical points, but I don't want to tweak the actual diagram myself as I am not up to redrawing it. Dirac66 (talk) 22:51, 3 February 2010 (UTC)


 * OK, I have inserted the CO2 diagram which seems the most appropriate existing diagram I could find, and also modified the text according to the new diagram. Thanks to Geologist for his useful suggestions. Dirac66 (talk) 02:23, 4 February 2010 (UTC)


 * 'Vapour vs gas' contains an excellent lexicographical distinction by 221.232.151.68 and Dirac66 that I wasn't aware of. In geology, I distinguish classes that are 'natural' from 'artificial' by the nature of the separation: if discontinuous, the class of objects is natural; if continuous, the classification is artificial. Geologist (talk) 18:27, 26 March 2010 (UTC)


 * The CO2 diagram should thus have two natural objects labeled 'liquid' & 'vapour', and an artificial one labeled 'fluid'. (The straight boundaries should be erased.) However, draw a horizontal isobar in the 'vapour' region. Because a liquid's volume decreases little with pressure, one can move a liquid on the melting curve to the right until it touches the boiling curve. At this pressure and temperature, the fugacity of CO2 can be estimated: one can, in similar ways, estimate thermodynamic properties of liquids, and calculate those of vapours. This illustrates a nice use of 'artificial' classification. Geologist (talk) 18:27, 26 March 2010 (UTC)


 * Below the critical point, a meniscus separates liquid from vapor; just beyond it, there is a continuous change from liquid-like fluid to vapor-like fluid. Farther from the critical point, these properties converge to 'just fluid'. If one used a very light blue for the vapor, one could darken this continuously during a rotation about the critical point from the horizontal to vertical artificial lines. Solid, liquid, & vapor could even be colored yellow, green, & blue: the fluid would be a gradient of hues. (Next, one might visualize densities with hues: a scientific visualization of an artificial class of thermodynamic states. (Sadly, I'm bad with the GIMP.) Geologist (talk) 18:27, 26 March 2010 (UTC)

Metastability
'In practice, however, the coexistence of more phases than the phase rule allows normally means that the phases are not all in equilibrium, i.e. that one or more is metastable.'

Perhaps 'not stable' would do? ('Unstable' won't.) Metastable is stable: after all, we can't afford to wait around forever. Most curves on petrologic phase diagrams are probably metastable, though one tries for stability. Metastable states can sometimes exist near one or both sides of, in this case, a univariant curve. They can displace a curve or point, but they can't increase their numbers. Geologist (talk) 01:55, 26 March 2010 (UTC)

Gibbs defined stabilities locally, according to the curvature of the characteristic function; so there is no distinction between stable and metastable: both are states separated by unstable regions.

If a system is not to become stable as, say, a liquid reaches its boiling temperature, one must relax conditions and include surface energy (or, as Gibbs wrote, 'capillarity'). As the energetic states of small regions of liquid increase continuously, nuclei of small bubbles form and redissolve. Another way of stating that a perturbation reaches the surrounding unstable state is to write the surface tension becomes low enough to allow the state to 'roll' over the unstable region to a gaseous, equilibrium stable state: the bubbles grow, and the liquid boils. An analogous argument can be made for the growth of crystallites. Simplifying this theory, as in the beginning of Gibbs's treatise, perturbations reach the surrounding unstable or non-stable state exactly at the boiling point or curve. Geologist (talk) 10:38, 26 March 2010 (UTC)


 * I have now removed "metastable" which perhaps doesn't mean what I thought. The article on metastable seems unhelpful also. Rather than get into "not stable" vs "unstable", I have simply said that the phases are not all in true (i.e. not apparent) equilibrium. Dirac66 (talk) 20:13, 26 March 2010 (UTC)


 * Even better! Geologist (talk) 01:56, 27 March 2010 (UTC)

Gibbs's Phase Rule
Gibbs wrote the variance F as C + 2 - P (with a change in notation). Only in the last paragraph of 'On Coexistent Phases of Matter' in his treatise 'On the Equilibrium of Heterogeneous Substances' is Gibbs's phase rule offered: 'Hence, if P = C + 2, no variation in the phases (remaining coexistant) is possible. It does not seem probable that P can ever exceed C + 2.'

This statement follows from the homogeneity of densities defining a phase, and the homogeneity of intensities defining its intrinsic stable equilibrium. If all phases are homogeneous in intensities, the P phases display a heterogeneous stable equilibrium. The general, trivial solution to this square array of linear, Gibbs-Duhem equations is d(-p), dT, dmu(1), ..., dmu(i) = 0.

The solution equates differentials to zeros of intensities, not intensive variables. This more generalized phase rule is owed to people other than Gibbs, such as Wind and Duhem.

One should not suppose newer literature is better than older. The only two phase diagrams in Modell & Reid's 1983 postgraduate text for classical thermodynamics at M.I.T. are incorrectly drawn; whereas beautiful diagrams, flawlessly drawn according to theory, can be found in Bakhuis-Roozeboom's 1904 Book 2, Part 1 of 'Die Heterogenen Gleichgewichte vom Standpunkte der Phasenlehre'. It would be nice to see the simplicity, clarity, and precision of the earlier literature (which I have little access to).

One needs to ask who the audience is. One can first sketch the subject using everyday language for everyone, then clarify it mathematically for more advanced readers. Geologist (talk) 10:38, 26 March 2010 (UTC)

Suggestions for new sections
I think your last paragraph suggests a way forward. The audience is surprisingly large - on the revision history page one can click on Page view statistics - this article had 3832 viewers in February 2010 and 3975 from March 1-26! I would guess that all these viewers would have many different levels of prior knowledge about the subject. So I agree that we should have a simple presentation first, followed by more advanced material.

By initial simple presentation I mean essentially what we have now, which corresponds to undergraduate physical chemistry texts. For this topic I do believe that new texts are better than older, since I understand Gibbs was not a very clear writer whereas modern textbook writers have spent several decades simplifying the subject to be clearer to students. Also new texts tend to be more accessible in (university) libraries for readers who wish to go further. I don't worry about the fact that the presentation is not that of Gibbs - the article is a modern presentation of the phase rule originally due to Gibbs. Just as the article on Newton's laws of motion uses vector notation introduced long after Isaac Newton.

As more advanced material to follow, I suggest two new sections. The first could perhaps be called "Historical development of the phase rule", and explain what Gibbs did and what others did, with mathematical clarification as needed. The second might be "The phase rule in geology", where the systems (or minerals) are more complex than the chemists' homogeneous phases of known composition.

However I do not feel competent to start writing either of these two sections. Perhaps a certain geologist who appears to know a lot about these two subjects could now come off the talk page and edit some first versions. Dirac66 (talk) 16:51, 27 March 2010 (UTC)

Proof, foundations and meaning
I have been reading the Article and Discussion (long time after my first reading) and found it really improved.

Yet, I want to add some points for your consideration concerning proof, foundations and signficance of the rule.

Gibb's rule is proved (see any thermodynamic textbook, for instance those by HB Callen) under a very simple, clear assumption: that the description of the thermodynamic potential of a substance is an homogeneous function of the extensive variables. The thermodynamic potential expresses a full knowledge of the system and it is usually taken as S(U,V,N). The knowledge of this function provides you specific heat, compresibility, thermal expansion coeficients and, in turn, any thermodynamic property of the system, including, theoretically, phase transitions.

That S(U,V,N) be an homogeneous function degree one of extensive propirties means S(aU,aV,aN)=aS(U,V,N). That is, to my knowledge, the Gibbs hypothesis for homogeneity.

After that hypothesis, and including standard considerations on equilibrium conditions, Gibbs' phase rule can be found for a multicomponent system with C components and P phases.

You may recognize that, speaking of pure substance, there are 3 independent extensive properties in the description S(U,V,N). While the degrees of freedom (number of independent intensive properties) ---and here, when I say intensive property I mean properties providing equilibrium conditions and thus excluding specific properties--- is, at most, two.

Extensive properties are aditive properties and, in turn, sizeable properties. They account for the size of the system.

Intensive properties on the contrary are non-aditive, non-sizeble properties. Gibbs's phase rule is saying that the number of independent intensive properties is less than the number of independent extensive properties. Meaning, you should always keep at least one extensive property on your description. That extensive property will ultimately account for the size of the system.

For instance, from U(S,V,N) (a valid representation and also an homogeneous function under Gibbs' hypotheiss) you may perform Legendre transformation and arrive to G(T,P,N). Where N will still account for the size of the system. But never to R(T,p,mu) because that 'potential' will be zero due to Gibbs' hypothesis or Gibbs-Duhem equation (the former being a consequence of the latter).

Incidentally, the Gibbs hypothesis of homogeneity provides a grounding for specific properties. Since the homogenous relation is valid for any a, it is also valid for a=1/N and so U(S/N,V/N,N/N)=U(S,V,N)/N so that U(S,V,N)= N U(s,v,1) where s=S/N, v=V/N are specific entropy and volume. Being U(s,v,1)=u(s,v) the specific energy (energy per one N) we find U=Nu. Less than trivial, the hypothesis provides a scientific background for the intuitive utility of specific properties: we find the behaviour for N=1, we know the behaviour for any N.

Last and not least. The very experimental fact that you never ever observe P=4 on a pure substance shouldn't be related to whether F=-1 is meaningless or not. That very experimental fact is of huge significance since, in turn, it validate the Gibbs hypothesis. As long as we have never ever observed a P=4 point we are pretty sure that S(U,V,N) is an homogeneous function degree 1 of U,V,N which is a fundamental condition for the whole building so-called thermodynamics (dealing with entropy, chemical potentials, specifice properties and so). Meaning, we are pretty pretty sure, none will ever found a P=4 point on a pure substance. You might also want to say that the inexistence fo P=4-point is an 'experimental proof' of the Gibbs phase rule (meaning, the existence of the 4-point would have removed the rule).

An article for the Gibb's hypothesis of homogeneity should be considered. 217.216.118.109 (talk) 10:45, 1 November 2010 (UTC)

Theorem
Since it's proved, the article should be called Gibb's phase theorem like any other proved statements.--79.116.89.242 (talk) 21:00, 3 January 2011 (UTC)-

From the Wikipedia style manual concerning article titles, see section WP:commonname: "Articles are normally titled using the name which is most commonly used to refer to the subject of the article in English-language reliable sources. This includes usage in the sources used as references for the article."

In this case most (or all) thermodynamics texts that mention the subject refer to Gibbs' phase rule, so Wikipedia policy is not to change that, even if theorem might be more a logical name.

Also it is Gibbs' and not Gibb's, because the man's name was Josiah Willard Gibbs. Dirac66 (talk) 22:40, 3 January 2011 (UTC)
 * This one of the cases when a rule, in this case the convention regarding common names can be ignored, as the statement Gibbs' phase rule is a theorem is obvious/straightforward (like water is a liquid) and doesn't require to be sourced.--79.119.209.111 (talk) 18:50, 10 January 2011 (UTC)

Euler
Wanting to learn more about this theorem, I clicked 'Euler Characteristic'. Why did I do that? -Geologist 209.218.108.22 (talk) 07:35, 20 March 2011 (UTC)
 * Is this Geologist who has contributed here before or another geologist? Anyway, there is an apparent analogy between Euler's formula V - E + F = 2 and the phase rule in the form F - C + P = 2 (where F has a different meaning in the two formulas). I am not certain whether this is actually significant or a mere coincidence. There was some discussion of this question on this talk page, but it seems too complicated to add to the understanding of the article.
 * So as you have suggested, it is not really helpful to include the Euler article in See also since the connection between the two topics. is not explained. I will remove it. Dirac66 (talk) 21:55, 20 March 2011 (UTC)

Reacting Systems
The first paragraph of the article states about C+2-P: 'This version of the Gibbs' phase rule is only valid for non-reacting systems.' Is this true? Geologist (talk) 10:21, 7 January 2012 (UTC)
 * Not if C is the number of chemically independent components as stated. This sentence was inserted by 70.36.134.99 on 6 Jan 2011. I think the intended meaning is that only for non-reacting systems can C be interpreted as the total number of components. But that is not what the article now says, so a fix is needed.
 * Probably we should replace this sentence by a brief explanation of C = S - R with an example, perhaps the dissociation of CaCO3. Would this be an improvement? Dirac66 (talk) 18:20, 7 January 2012 (UTC)

Gibbs called S a component of the system, and wrote n for the number of independent components. He wrote the phase rule n + 2 - r, where r is the number of phases. Because n is now used otherwise, I call C, the components, the number of independent constituents (Gibbs's 'n'). The phase rule is thus C + 2 - P, where C is your 'number of "chemically independent" components'. Despite letters chosen, the Gibbs phase rule as stated here works for reacting systems as well. Using S-R rather than C is just a convenience, for C is otherwise difficult to calculate. Perhaps the last sentence in the paragraph could just be removed. Geologist (talk) 00:38, 8 January 2012 (UTC)
 * I'll remove the incorrect sentence, but I think we should also attempt to explain the mysterious phrase "chemically independent" which appears twice but is not defined. I have just looked through four recent texts in physical chemistry, materials science, and solid state chemistry. They all limit the symbols to F, C and P so I will forget about S and R. However they also try to define C more clearly than this article does now, for example as the minimum number of chemical species needed to define the composition of the system. And they all add a simple example or two which give the idea of S-R without introducing new symbols, e.g. NaCl(aq) or CaCO3(s) dissociation or PCl5(g) dissociation. I will insert something soon along these lines. Dirac66 (talk) 02:50, 8 January 2012 (UTC)

A common definition of components are chemical constituents necessary and sufficient to construct the system. (Remember, phase compositions or amounts should vary with T&p.) Although the phase rule appears qualitative in that it relates integers rather than quantities, some elementary linear algebra is really needed for it to make sense. Such a treatment is beyond the level of the article.

The number C is the row-rank of a matrix. The rows are S chemical species whose amounts vary with T&p; the columns are any chemical substances that define these, even units of chemical elements (or ions). The co-rank is R, the number of chemical reactions. The number of components C is the row-rank, S-R. This, I believe, is equivalent to Brinkley's definition of 1946 (J. Chem. Physics). At worst it requires the Singular value decomposition. Because S and R are often simple to count, subtracting them can be easier than writing, compiling, and executing a complex computer program. Perhaps a 'vagued-up' definition of C is better. :-) Geologist (talk) 12:28, 8 January 2012 (UTC)
 * I remembered that the definition is considered in more detail at Component (thermodynamics), so I fixed up that article with additional examples, and added a note here to say where to find them. And I have limited the discussion to simple examples, rather than trying for complete generality with linear algebra. Dirac66 (talk) 17:53, 15 January 2012 (UTC)

Grammar
A noun ending in 's' is made possessive by adding "'s", as in James's Park, Gibbs's Phase Rule, &c. When this becomes unpronounceable, the final 'z' sound is removed, as in Schreinemakers' Rules. 'Gibbs Phase Rule' would be acceptable to everyone; but I suppose it's too late to change the title of the article. Geologist (talk) 03:15, 22 November 2013 (UTC)
 * Hmm. A Wikipedia article can always be renamed, or "moved" in Wikispeak. The question is what is the preferred term in most external sources: Gibbs'? Gibbs's? Gibbs?. I checked the 4 physical chemistry texts I have, and they unanimously avoid the question and just refer to the Phase Rule, and then say in the next sentence that it is due to Gibbs. So perhaps this article should just be named Phase Rule, since the first sentence does say that it was proposed by Gibbs. What do your geology texts call it?
 * The article on Josiah Willard Gibbs uses Gibbs's in most of the article, but again the phase rule is just referred to as the phase rule. I think I will ask for comment from Eb.hoop, who is the chief contributor to the Gibbs article. Dirac66 (talk) 02:02, 23 November 2013 (UTC)
 * I support re-titling this simply "Phase rule", which seems the standard usage in textbooks and in other encyclopedias. And of course in the text we should write Gibbs's, rather than Gibbs', since that is now the standard usage. - Eb.hoop (talk) 21:01, 23 November 2013 (UTC)
 * Thanks. Since renaming is an important change in Wikipedia we are supposed to allow time for input, so I'll leave a notice here for a week. If there is consensus, we can move (rename) the article to Phase rule, and at the same time change the first word of the text from Gibbs' to Gibbs's. Dirac66 (talk) 00:51, 24 November 2013 (UTC)

PROPOSED NEW ARTICLE TITLE
Following the discussion above, it is proposed to move (rename) this article to Phase rule, i.e. to remove Gibbs's name from the article title. The opening line of the article would of course continue to specify that it was Gibbs who proposed the phase rule. Also a search for the old title Gibbs' phase rule would be redirected to the article with the new title.

Further comments are invited here. If there are no major objections, I propose to rename the article in one week. Dirac66 (talk) 00:51, 24 November 2013 (UTC)

Conflicting meanings for N
The intro to this article now has two conflicting meanings for the symbol N. The first paragraph defines N as an alternative symbol for the number of components, usually denoted C. But the second paragraph defines N (since an edit yesterday) as an alternative symbol for the number of non-compositional variables such as temperature and pressure.

This contradiction is confusing and could lead some non-expert readers to think that the number of components is equal or somehow related to the number of non-compositional variables. So at least one usage should be eliminated from the article. My own preference would be eliminate all the alternative symbols and only mention the usual symbol (F, C, P) for each number in the phase rule. What do others think? Dirac66 (talk) 01:14, 29 April 2016 (UTC)
 * No reply after 2 months, so I have gone ahead and deleted the alternative symbols including both meanings for N. Dirac66 (talk) 20:42, 6 July 2016 (UTC)

Assessment comment
Substituted at 18:32, 17 July 2016 (UTC)

Its not independent control of variables
The number of independent parameters should not be described in terms of what a controller of the system can or can not do. This is about defining other parameters by the F number of independent intensive ones, no matter how the system arrived to those values. In fact, pressure ALWAYS changes with the change of tempetature. You can arrive to a different set of T and p by changing T and p in weird patterns. But YOU CAN arrive to any set of T and p within the gas space on the diagram. That's the point. Its not about what you can do with control. But if you observe a system with 2 phases, and T is such, then p is such (one parameter defies this state). Vesta1212 (talk) 02:58, 10 April 2024 (UTC)