Hugo F. Franzen' and Clifford E. Myers2 Ames Laboratory-USDOE and Department of Chemistry. Iowa State University, Ames, Iowa 5001 I
Chemical Thermodynamics and the Phase Rule
The Gihbs Phase Rule is an imnortant orincinle underlvinr much of mineralogy, metallurgy, ceramics, and materials science. Its "usual" derivation is from the perspective of chemical thermodynamics applied to solutions, and it follows from the fact that the chemical potential for each component has the same value in all the phases present in the system a t equilibrium. We wish to show that the Phase Rule can, in a sense, he obtained without using chemical thermodynamics p e r se but by using an assumption which is implicit in applications of chemical thermodynamics. We shall follow this development with some comments on some important thermodynamic relationships with particular emphasis on the distinction between chemical components and chemical species. Thermodynamics is concerned with 1) establishing the existence of properties (internal energy, entropy, temperature, and then enthalpy, Gihbs and Helmholtz free energy, chemical potential, etc.), 2) deriving relationships among the properties, and 3) establishing the relationships of the properties to nhvsical nrocesses. With regard to 21 it is necessarv to know how ma& properties are independently variable,-and thus thermodynamics must deal with the number of independent variahles necessary to specify the state of the system. This point is generally not given the attention it deserves. Let us first consider a system of one component. If we take the effects of lonp-range fields to be uniform throughout the system, we mayspecify states of the system by fixing the values of any three of the variahles P, the pressure, V, the volume, T, the absolute temperature, or n , the number of moles, since it is a fundamental assumption that these four variahles are related through the appropriate equation of state. For a system of c components, one frequently used eauation of state relates P.. V.. T.. and n;. where t h e n ; are the nimhers of moles of the components.'~hetotal number of variahles which must he fixed to soecifv . .a state of the svstem is one less than the total number of variables in the equation of state.. i.e... c 2. I t is common practice to make a distinction between kinds of variahles among the c 2, namely to distinguish those which serve to fix the intensive properties of the system (the independent intensive variables, of which there are f ) and those which serve to fix the quantity of each phase (i.e., the masses or volumes of each phase). The number of the latter. p, is equal to the number of phases. Thus, of the c 2 properties which fix the state of the system if the system is described in a state of rest, there are f independent intensive properties and p extensive properties, and
+
+
+
Work performed for the US. Department of Energy, Division of Basic Energy Sciences. Author to whom correspondence should be addressed. On leave from Department of Chemistry,State University of New York at Binghamton, N.Y., 13901. Wenbigh, K., "The Principles of Chemical Equilibrium," 3rd. Ed., Cambridge Univ. Press, London, 1971, pp. 45,76-82. 'Lewis, G. N., and Randall, M., "Thermodynamics," 2nd Ed., (revised by Pitzer, K. S., and Brewer, L.), McGraw-Hill, New York, 1961, p. 203. 372 1 Journal of Chemical Education
f=c-p+2 which is the Gihhs Phase Rule. Since thermodynamic functions are state functions, they can he among the c 2 variahles used to specify a state of the system, i.e.. in an eauation of state. Hence while a "usual" equation of state may be given as V = V(P,T,n,), the equation U = U(S,V,n, ) is also an equation of state. The latter is particularly useful in its differential form
+
dU = TdS - PdV
+ 2 u,dn, ,=I
where
Denbighhnd others have pointed out that this is a fundamental equation for systems in the absence of surface effects and long-range fields. It contains knowledge ohtained from the first and second laws since it includes changes in IJ (heat and work effects), T (temperature), changes in S (heat and the permissible direction of change), and fii (chemical potentials). Indeed, eqn. (1) was used by Gibbs to define the chemical p ~ t e n t i a l . ~ Let us now examine the concents of chemical comnonent and chemical species in terms of eqn. (1). A component is understood to be one of the minimum number of substances for which a chemical formula can be written and from which arhitrarv of each nhase in a svstem in anv of its " auantities . states of interest can he synthesized in principle by processes which occur naturallv in the svstem, these processes involving exchange with the surroundings only of matter composed of these suhstances. There mav he a varietv of choices of components, and the chemical ~"hstancesc h k e n as components may he negligibly present in the system. For example, asystem in which Hz0 is the only component will contain uery little of the species H20(g) a t very high temperatures and very low pressures (i.e., such that H20 is highly dissociated). We wish to distinguish between species (chemical substances demonstrably present in a system and with variable mole numbers in a closed system) and components (for which the mole numbers are invariant in a closed system). In this discussion we shall use the superscripts c or s to refer, respectively, to a substance as a comnonent or snecies. As an instructive example, let us consider two cases involving Hz, 02, and Hz0 under conditions such that thereaction
+
HP lI2O2= H 2 0
(21
does not proceed in the absence of a catalyst. In Case a) a catalyst is absent; in Case b) a catalyst is present. Clearly, in Case a) the reaction written above does not proceed to equilibrium and c = 3. This requires that the species Hz, On, and HzO must all be considered as components. Equation (1) for Case a ) must be written d U = TdS - PdV + uSH,dnk,
+ &dnb, + &0dnrH1o
(3)
Here the superscripts is used to indicate that all three are demonstrably present as species. But because all three are also independently variable, fiidni for each must he included. The dnj thus can he either changes in mole numbers due to reaction (addition of a catalyst) if the system is closed or due to addition of new material if the system is open. The reaction is maintained in equilibrium in Case b), for which c = 2, and eqn. (1) becomes d U = TdS
-PdV+
ph,dnf,,
+ &dnb8
(4)
When the states of the system are limited to those a t which the reaction is a t equilihrium, dU, TdS, and P d V are the s a n e in the two cases, given the same change in the state of the system. Hence, from eqns. (3) and (4) &,dm&,
+ &dn& + r k a d n k a = ~ k ~ d n+f flb2dnbZ ,~
(5)
Now, by the stoichiometry of the reaction dnb, = dnb, - %dnka
and dnhi = dnhz- dnkSo
Substituting in eqn. (5) and rearranging gives
Now for any given overall composition of a system in Case a ) we close the svstem to transfer of mass with the surroundings and introduce the catalyst. Then equilibrium will be reached. Then if the svstem is perturbed in anv wav, the changes in then. c m bt>eicritr.
