The True Meaning of Component in the Gibbs Phase Rule H. F. Franzen Ames Laboratory-DOE, and Department of Chemlstty, Iowa State Unlverslty, Ames, IA 5001 1
The Gibbs phase rule is one of the fundamental bases for considering heterogeneous equilibria. In a previous article' the fact that this relationship is an assumption of, rather than a result of, thermodynamics was discussed. This follows because the assumption that c 2 macroscopic properties fix the state of a macroscopic system in a state of rest in the absence of long-range fields and surface effects (i.e., P - V work only) is fundamental to the development of thermodynamics, rather than a consequence of the laws of thermodynamics. For example, it is necessary to recognize that c 2 variahles fix the state of asystem in order that the Gihhs free energy be taken to be a function of T, P, ni, from which the total differential, dG = -SdT VdP &dni follows. The purpose of this article is to call careful attention to the nature of c. The recognition that c has its origin in the counting of numbers of properties (from the physical point of view), i.e., variahles (from the mathematical point of view), calls attention to its fundamental character as the numher of indenendentlg vnriahle quantities speciiying the rhemicnl runtent of a system. This character is illustrated hv the followinr mobl e i : A metal hydroxide, M(OH)., is d&omposed at"; high temperature such that the system obtained contains a solid nonstoichiometric oxide, MO, (with variable x ) , and gaseous HZand H20. What is c? This problem can he solved by first considering c in a system without the restraint that results from the method of preparation, followed hy a consideration of the restraint implied by the given recipe. In the absence of a stated preparation technique most students would immediately see that the system MO,(s), H2(g), HpO(g)would have three chemical composition variables, for example, quantity of M, quantity of Hz, and quantity of Hz0 (although quantity of M, quantity of H, and quantity of 0 would he equally good). The difference between these two examples is worthy of
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' Franzen, H. F.: Myers, C. E. J. Chem. Educ. 1978, 55, 372.
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comment.'l'he description of the chemical content variahles :n terms of uuantities of M.. H.,. in terms . H,O . is a descri~tion of chemical~suhstancesfrom which one couldreadily make the system, i.e., a recipe such as "mix l o g of M, 2 g of Hp, and 1.5g of Hz)" is one that could he followed by a chemist witha modicum of laboratory facilities. On the other hand, arecipe specifying quantities of M, H, and 0 would present many difficulties in view of the problems associated with the production of H and 0 with analvtical accuracv. The M. Hv. Hz0 description is well adapted to the concept of c as the number of chemical comoonents. whereas the M.. H.. 0 description is more readily grasped by students as a generalization of this concept, the numher of chemical content variables. Ineither case, the unrestrained system has c = 3, and f = 3 - 2 2 = 3. Now a basic question is "What is f i n the restrained system synthesized from M(OH),"? Since we know p = 2.ihis qu&tion is equivalent t o " \ ~ h a tis c in the restrained system"" The answrr to this question is obtained by counting the restraints placed on the-intensive variahles by the method of preparation and subtracting this numher from the numher of chemical composition variables in the unrestrained system. Since all the hydrogen in the H20(g) and the H2(g) arises from the starting material M(OH),,
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and since all of the oxygen in the MO, and in the H 2 0 arises from the startipg material
n w =~
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where WM= moles M in MO,, N&= moles 0 in MO,, h'f,, = moles Hz in vapor, and = moles Hz0 in the vapor. Combining these two yields
m,o
Dividing the numerator and denominator on the left by NBM % and on the right by + Nf,, yields
mZo
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where the X?s are mole fractions in the ao~rooriate ohase .. . (solid or vapor). This relationship among intensive variables is implied bv the method of meparation. and it follows from its existence that the number oiindependent intensive variables in the restrained svstem is one less than the numher in the unrestrained system. The conclusion is that f = 2 for the problem originally stated. I t follows that c = 2 for the system synthesized from M(OH),. This conclusion presentsa problem for the conventional approach of calling c the numher of components. Clearly the system as synthesized is made from a single chemical suhstance: however. it has lone been realized that in determining c any restraints must he adequately considered. The classical examole is the svstem: CaOM and COz(g), synthesized from C ~ C Owhich, ~, in spite of its formation from a single substance, is a c = 2 system. This case differs from the M(OH), case, however, in that i t is clear what choices one has with regard to "components", e.g., CaO and COz would he satisfactory from any point of view. In the M(0H). case, however, i t is not a t all clear what the two components might he; that is, it is not clear how t o descrihe the chemical contents of this system in terms of two chemical substances. Thus, this example provides us with the impetus to use the terminology "chemical content variahle numher" as opposed to "numher of components" for c since the former carries with it no implication that we are able to provide a recipe for synthesizing the system involving c chemical content quantities. According to the ahove it is appropriate t o define c bye = .,,c,, - p where c is the number of chemical content variables in the unrestricted case (in which case c can be called the numher of components without confusion), minus the number (p) of restraints placed upon the intensiue variables hv the method of oreoaration. . . In detail, the procedure for finding c is 1) count the number of chemicallv independent species 1e.e.. 4. uiz. M linsolid ( gthe ) case solution), 0 (in solid shution), ~ ~ (&gd )- ~ ~ ~ (in discussed above); 2) write down a comolete set of indeoendent, uniquely halanceable net reactions (e.g., this s e t con tains only one reaction in the ahove example, and this reaction can he taken to be Hz(g) 0 (in s.s) = HzO(g)); 3) subtract the numher of reactions in 2) from the numher of species in 1); this gives c,,,~,; 4) determine the numher of restraints, p, placed upon the intensive composition vari- p. ables by the method of preparation; and, 5 ) c = c., I t is instructive to consider these steps in greater detail. With regard to I), there might be some concern about miss-
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ing some species. For example, if, in the sample problem, the pressure was sufficiently low and the temperature sufficiently high, the radicals H(g), O(g), and OH(g) would he present. However, with the addition of each species comes an additional reaction in 2), and thus no change in c,.,.. Furthermore, it should be noted that two species are required to descrihe the solid because of the variahle r .. i.e... from a thermodynamic point of view TiO, is just as much asolution, albeit a solid solution, as is ethanol in water. If x were fixed (strictly x can never be fixed, hut stoichiometric compounds are commonly considered and are acceptable fictions), then, taking x = 2.00, MOz should he taken as a single chemical species. In this case, there would he three species (MOz, Hz, H20) with no reactions among them and again c,,,,, = 3. The restraint would arise from the material balance reaction
Thus, p = 1 and c = 2, and M(OH), and MOz would be satisfactory "components", i.e., the system could he swthesized without arbjtrary restraints on-extensiue prope;ties if it were open with respect to M(OH), and M02. With regard to 2) it is important that the reactions he independent and uniquely balanceable, i.e., that no member of the set be the sum of two (or more) other memhers of the set.'l'hiscondition assures that thereexist an many independently variable extents of step (1) as there are reactions in the set, and thus independent relations Xu+, = O for each reacrion. Kach such relation reduces bv one of the number of variables of chemical content, leadin& step 3). Step 4) has been discussed previously. In conclusion, the chemical content variable numher appropriate to the phase rule is e = s - - r - p, where s = the numher of species present in the system a t equilibrium, r = the numher of independent extents of reaction, and p = the numher of restraints placed upon the intensiue variables by the prescribed method of preparation. Acknowledgment The ideas expressed in this article were in large measure stimulated by the consideration of problem numher 3c of Chapter 5 of K. Denbigh's excellent hook, "The Principles of Chemical Equilihrium", Cambridge University Press (1981). The Ames Lahoratory-DOE is operated for the U.S. Department of Energy by Iowa State University under contract No. W-7405-Eng-82. This research was supported by the Office of Basic Energy Sciences, Materials Sciences Division.
Volume 63 Number 11 November 1986
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