Talk:Gibbs' phase rule

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Comments:
  • 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 ΔG = 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)
  • 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)

[edit] 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...

[edit] 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).

[edit] 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.

[edit] 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. :-)

[edit] 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).

[edit] 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..

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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)

[edit] 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)

[edit] 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
  • always:
    • x1_a + x1_b + ... = 1
    • ...
    • + ---------------------
    • 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

[edit] 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)

[edit] 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)

[edit] 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)]

[edit] 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)