P. G. Wright
The University Dundee, Scotlond
I
Introduction of Set Theorv info Chemical ~ h e r m o d ~ n k i o
I t was argued by Landsberg' that the conventional general thermodynamic theory of Clausius, Kelvin, and Carath6odory could with great advantage be re-expressed in the terms of set theory. Gain of clarity was thereby to be attained in respect of the need for, and formulation of, particular axioms, and also in respect of the details of derivations. The field of chemical thermodynamics clearly invites a similar tran~cription.~
Sets in Which Certain Kinds of State are Represented Equilibrium Equilibrium states only, but Non-equilibriom states striotiy metastable atates [at leaat and equilibrium states certain kinds of] are allowed no others are allowed states of a cioad ayatem
7
B
7
[having sub-set* [ h + v i n ~aub-sets m any one of p 14 Bny one of whiob themass of whmh the mass of each substance is each substance is eonstantl eonstantl
L
'I
6
Sets of Points Which Can Usefully be Defined in the Thermodynamic Theory of Chemical Equilibrium
Landsberg's sets y and 6 are open sets (sets from which boundary points are excluded) in which there are represented respectively In 7, equilibrium states of a closed system, and no other states, the term "equilibrium states" being construed strictly and metastable equilibrium states disallowed. In 8, equilibrium states, and 841.50 (subject to certain resttictions) non-equilibrium states, of an open system.
More than one distinct issue is raised by bringing in a set 6. The set on which some function is defined is thereby extended to cover metastable equilibrium states of closed systems non-equilibrium states of closed systems equilibrium states of open systems other st,ates of open systems
and each extension (or, a t least, the first three) is worthy of separate consideration. The drawing of a clear distinction between metastable equilibrium states and genuine non-equilibrium states (which necessarily involve non-vanishing rates of change or rates of flow) is of particular significance to chemical thermodynamics. The following sets (see table) are suitable for a more detailed pre~entation.~ In a set e, equilibrium states, strictly construed, alone are represented, but states attained by addition or removal of matter are allowed. In a set 8, equilibrium states of a closed system alone are represented, but metastsble equilibrium states are allowed. In a set r, equilibrium states of a cloned system alone are represented, metastable equilibrium states being allowed, hut there are represented only states in which there is a certain constant quantity of each chemicdsubstancepresent. In a set n, equilibrium states alone are represented, metastable equilibrium states being allowed, and also equilibrium states attained by addition or removal of matter. In a set 7,at least certain non-equilibrium states of a closed systemarerepresented, in addition to equilibriumstates. In a set p, at least certain non-equilibrium states of a closed system are represented, and possibly also equilibrium states,
.
~~~
[All are sets from vhich boundary points have been excluded (open sets).]
but there are represented only states in which there is a certain constant quantity of each chemical substance present. In a set r, there are represented equilibrium states, strictly construed, of s. closed system and magnifications thereof; where by "magnification" in meant that T,p (and any other such intensive variable~)are the same, while the quantity of every chemicslsubstancepresentis altered by the same factor. Sets 8, U are related to a, 7 as r is to 7.
The definition of these sets facilitates the separate consideration of separate issues concerned specifically with metastable equilibrium states; with non-equilibrium states, and with open systems. Sets I?, 8,I1 have relevance especially to the existence of extensive and intensive properties, a matter requiring some such explicit axiom as Landsberg's tentative "fourth law." The Business of the Thermodynamic Theory of Chemical Equilibrium
Any thermodynamic theory of chemical equilibrium is concerned with the deduction of general relations holding when there is equilibrium w.r.t. one or more chemical reactions. Central to such a theory is a deduction amounting to a systematic answer to the questlon Which are the states for which there is equilibrium w.r.t. chemical reaction?
Whichare the states which constitute aset r ?
Given a set 8 (or a set T), which are the states constituting the set 7 contained in it?
The primary business of this part of thermodynamic theory is not the discussion of spontaneous processes, but by considering jictitious processes to draw conclusions about real equilibria. Such is the purpose of the virtual Volume 48, Number 5, May 1971
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variations invoked in Gibbs' treatment of chemical equilibrium. After having gone through the central deduction, it is then ~ossibleto eo on to exulicitlv derived (but logically 'subsidiary) relations concerned with the effect on chemical equilibrium of change of temperature, change of pressure, change of solvent, or electric or magnetic fields. Transcription of Gibbs' Thermodynamic Theory of Chemical Equilibrium into the Terms of Set Theory
Gibbs' treatment of chemical equilibrium, transcribed into the terms of set theory, takes the following form. For simplicity, attention is restricted to the case of a single reaction, in a homogeneous system, having the volume V as sole deformation coordinate. It is assumed, as having been established by Clausius, that 1) For virtual variations which are possible processes linking any two neighhoringpoints of any set 0, the inequality q
5
T6S
holds (to the firsborder of small quantities), 2) In any set y, and in any set r dU
+ pdV = TdS
It is further assumed, some such postulate as Landsberg's "fourth law" here being needed, that 3) Functions U, S for all sets 7, and all sets 1, contained in a given set 7, can be subsumed into common functions U,S, defined on the whole of that set 9, and having the property that the U is a differentiable function of variables (8, V,[mil) which for states represented in the set n constitute a completeset.
It follows directly from assumptions (2) and (3) that: throughout, 7, to the first order of small quantities
--
where vi (bU/bm,)s.v,mj+i. (The symbolm, denotes the mass of the ith species present, and the symbol q, is used to avoidconfusionwith the conventional chemical uotential referred to the mole as unit of quantity of a substance.) Holding throughout 7,this equation necessarily holds throughout any set 0 that is a subset of 7. Bv a~nlicationof anv of the extremum ~rinciules der&edLfromthe inequality (I), it would then foliow, subject to a. reservation made below, that in
x
y
Stater around the critical point of a simple fluid. Cms-hatched area represents points of the set y for the homogeneous fluid. Lined oreo represents points, other than y, which ore contained in the set 0 for the homogeneour fluid.
pidmi = 0
i
The conclusions, drawn at this stage, follow only if for the reaction in question there exist metastable equilibrium states (bonafide, or created by the addition of an inhibiting anti-catalyst). For reactions for which even in principle there are no metastable equilibrium states, and there exists no substance having the properties of an anti-catalyst, the argument fails a t this stage. Gibbs' deduction would then fall back on equations and inequalities assumed (but without the p~ssibilit~y of citing in justification Clausius' deduction of the existence of S) to hold for noints of a set s. Attention to these issues has on various occasions been drawn in other termss, but a presentation in the language of set theory makes the matter especially clear-cut. Chemical Equilibrium in Dilute Systems
If Gibbs' relation in
y
x
at*< = 0
has been established, this does not of itself sufficeq to establish the complete chemical principle of the exisfence of equilibrium constants. To claim, in a purely macroscopic analysis, to have deduced the approximate "constancy" of an appropriate quotient of concenfmtions, it is necessary to proceed by an argument which invokes another assumption 4) The axiom of the ideal behavior of dilute systems.
where the oi are quantities such that when the chemical reaction occurs the masses mr of the various substances present change by amounts which stand in the ratio of the ol
a KIRKWOOD, J. G., AND OPPENREIM, I., "Chemical Thermodynamics,"New York, 1961, p. 104. ' MCGLASHAN, M. L., J. C H E M . E D U C226 . , ~ ~(1966). , 'LEVIS, G. N., AND RANDALL,M., "Thermodynamics," New York, 1923, p. 232.
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Journal of Chemical Education
Such an axiom, implicit in many discussions5,ought to receive explicit formal enunciation in a systematic macroscopic theory of chemical equilibrium. It is of importance that it can be formulated as a purely macroscopic assertion, with an empirical basis which can be constructed from purely macroscopic experimental observations. Heterogeneous Equilibrium
I n the above, attention was restricted to homogeneous systems, in order to concentrate exclusively on features
associated with chemical reaction as such. Chemical equilibrium in heterogeneous systems, and many aspects of phase equilibrium, are largely amenable to treatment in similar terms. Two other specific points may, however, he mentioned. The first is that competent accounts of the significance of phase diagrams can be regarded as patently being exercises in L'do-itryourselfset theory." The second relates to behavior of a simple fluid near the critical point. Expansions in power series about the critical point have now lallen into disrepute. Such expansions ought
always to have been considered suspect, and their failure expected to be more likely than tbcir succrss, for the following reason. The crit,ical point is a boundary point, not only of the set -y for states of a homogeneous fluid, hut t h o of the corresponding 8et 0 [or even the corresponding cct, r, should such he invoked] (see figure). The rxp:~nsions, then, are expansious of a fuuction in power series about a point which is a boundary point of the set on which the function is defined. From this viewpoint, the failure of the expansions is not a failure of classical thermodynamics, hut :I f :LI'1ure to be classical enough.
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