Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 PROCEEDINGS THE ROYAL A MATHEMATICAL, PHYSICAL SOCIETY ib . & ENGINEERING ----------------O F ----------------- SCIENCES Chemical Thermodynamics in Landsberg's Formulation P. G. Wright Proc. R. Soc. Lond. A 1970 317, doi: 10.1098/rspa.1970.0128, published 7 July 1970 References Article cited in: http://rspa.royalsocietypublishing.org/content/317/1531/477#related-u rls Email alerting service Receive free email alerts when new articles cite this article - sign up ® in the box at the top right-hand corner of the article or click here To subscribe to Proc. R. Soc. Lond. A go to: http://rspa.royalsocietypublishing.org/subscriptions Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 Proc. Roy. Soc. Lond. A. 317, 477-510 (1970) Printed in Great Britain Chemical thermodynamics in Landsberg’s formulation B y P. G. W eig h t Department of Chemistry, The University, Dundee {Communicated by A. D. Walsh, F.R.S.—Received 28 1969— Revised 30 January 1970) A presentation is given of the form taken by the thermodynamic theory of homogeneous chemical equilibrium when transcribed into Landsberg’s (1961) formulation of thermo­ dynamics in terms of set theory. Certain sets of points can usefully be defined, in order to deal separately and explicitly with metastable equilibrium states, and with open systems as such, without having simultaneously to consider non-equilibrium states. ‘Narrower’ and ‘wider’ thermodynamic principles can be distinguished in terms of what is the set for which some proposition is asserted to apply, and Landsberg’s formulation makes possible greater clarity in the enunciation of the special axioms of chemical thermodynamics. In particular, the axiom of the ideal behaviour of dilute systems takes a very clear-cut significance. When arguments of the type introduced by Landsberg are used, the assumptions involved in a formal development of the thermodynamic theory of homogeneous chemical equilibrium are clearly apparent. Metastable equilibrium states, and non-equilibrium states,, each have their own distinct place in the theory. Some comments are made in relation to the usual presentation. 1. In tro d u c to ry rem arks Landsberg (1961) has given a formulation of thermodynamics in terms of set theory. States of systems being represented by points in suitable phase spaces, certain sets of points are important in general thermodynamic theory. (An earlier version was first given by Landsberg (1956), and some of the more readily explained aspects have more recently been emphasized by Landsberg (1964).) Ideas which have some resemblance to these appear in Tisza’s (1961) axiomatic thermodynamics of the stability of phases. Landsberg’s and Tisza’s formulations are similar in the respect of each building up the theory from an orderly systematic set of listed postulates, but in other respects there are distinct differences. Landsberg follows Clausius, Kelvin and Caratheodory in basing the theoretical deductions on statements that certain things are physical necessities or physical impossibilities. Tisza, claiming to axiomatize the work of Gibbs, bases the theory on the idea of transfer of quantities satisfying principles of conservation and additivity. The existence of U is then assumed (referring to a first law), and the existence of S introduced by a postulate (Tisza 1961, p. 12) that equilibrium corresponds to the solution of a certain extremum problem. Landsberg (1961) gave a brief account, in his formulation, of chemical thermo­ dynamics, but neither he nor Tisza gave a detailed treatment of chemical thermo­ dynamics in terms of a formulation in set theory. It is here argued that such a detailed account, with special reference to the thermodynamic theory of chemical [ 477 ] The Royal Society is collaborating with JSTOR to digitize, preserve, and extend access to Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. www.jstor.org Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 478 P. G. Wright equilibrium, could with advantage be given. The merit of a formulation following Landsberg appears in respect both of deductions and of the statement of axioms. In such a presentation, the necessity for certain assumptions (sometimes kept hidden) stands out with especial clarity. The phase rule and the Gibbs-Duhem equation are not examined here. These matters do not occupy the central position in chemical thermodynamics, and have already been treated by Landsberg (1961). 2. The g e n e ra l scheme of th e o rd in a ry therm odynam ic THEORY OF CHEMICAL EQUILIBRIUM 2.1. Consecutive steps in the ordinary thermodynamic theory of chemical equilibrium The familiar ordinary thermodynamic theory of chemical equilibrium, basically that of Gibbs, with a few formal modifications following the introduction of activities by G. N. Lewis, involves a small number of essential steps. These will be listed here, in readiness for certain comments in terms of set theory. For simplicity, consider only the case of a single chemical reaction in a homo­ geneous system subject to a uniform hydrostatic pressure, with electrical and magnetic phenomena having no effect. A single reaction in a homogeneous system may be written formally as v u , % where X^ is the chemical symbol for the ith substance, and vi is a coefficient such that when the reaction occurs the number of mole of X^(^) changes by an amount proportional to The deduction then proceeds as follows: (i) To the first order of small quantities bU = TbS—pbV + S 14 i where /I,- = (dU 1дщ)8'УМ}+{). (ii) An extremum principle, based on the inequality of Clausius, is invoked. (iii) It is inferred that ^ ч л (2i> ^ )e q = 0 (the subscript ‘eq ’ denoting that the quantity in question is to take a value applying when there is equilibrium with respect to the reaction). (iv) Each is expressed formally as the sum of (a) a quantity which is in a certain sense ‘constant’, and (b) a logarithmic term ВТ In a^ (v) It follows that a certain quotient (П ail)eo. i is (in a certain sense) ‘ constant5. Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 Chemical thermodynamics in Landsberg's formulation 479 (vi) For sufficiently dilute s3^stems, the relation deduced approximates to (IIc?)eq = 'constant’ i (where ci is some sort of 'concentration5 of the species X^). 2.2 The ordinary thermodynamic theory of chemical equilibrium, for a single chemical reaction in a homogeneous system (i) The situation ordinarily concerned is that of a system in which the pressure is uniform and hydrostatic, and any effects of macroscopic electrical and magnetic phenomena are negligible. The variables {,S, V, and the щ} form a complete set. It is assumed that U is a differentiable function of these variables. Then, trivially, to the first order of small quantities bU = TbS—pbV + i (ii) The inequality of Clausius, in the form q ^ TbS, or bU -w TbS, appropriate to virtual variations, implies such extremum principles as: 'For equilibrium in a closed system of prescribed entropy and volume, it is necessary that the energy be a minimum \ (iii) Any such extremum principle leads to a condition in terms of chemical potentials. Taking, for example, the extremum principle just cited, the argument proceeds as follows. For fixed prescribed values of S and V, a stationary value of U, by (i), corresponds t0 2><<Ч = о. i Since each dщ is proportional to the corresponding vi9 and non-zero Ащ had to be considered in locating the stationary value of U, this stationary value is given by 2 = 0. i It can be verified that in all ordinary cases the stationary value is indeed a minimum (conditional minimum), and not a maximum or a point of inflexion. Thus (i) and (ii), taken together, require that ( S ^ i) e q = 0. i (iv) Each will depend on the temperature and on concentration. Each can then be expressed as the sum of (a) a quantity independent of all concentration variables, and (b) a term which incorporates all dependence on concentration variables. For term (b) the logarithmic form ETlnai is chosen, by analogy with the logarithmic form li.t — a quantity independent of concentration variables + RT In G{, Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 480 P. G. Wright which holds to a good approximation for dilute systems (gases at low pressures, and dilute solutions). The definition of ai is completed by imposing the requirement that in the limit of infinite dilution aijci —> 1. W B y (iU ) < 2 > Л ) .. = 0, i and by (iv) each is given by an expression of the form fii = +ВТ1па€, where /if is a quantity independent of concentration variables. (Each /if will depend on the temperature, and for a condensed phase will depend slightly on the pressure.) These relations together require that (П a?)eq = exp ( - 2 Viflf /ВТ), i i The r.h.s., while dependent on the temperature (and for a condensed phase slightly dependent on the pressure), is necessarily independent of all concentration variables. Therefore, so also must the l.h.s. That is, the quotient (П ail)eq i takes a value which is ‘ constant5 in the sense of being, while dependent on the temperature (and, for a condensed phase, slightly dependent on the pressure), independent of all concentration variables* (vi) It is implicit in the physical background to (iv) that for sufficiently dilute systems J ai я с*. Therefore, for sufficiently dilute systems fflc?)eq i takes a value which is, approximately, 'constant5 in the sense of (v). Extensions and applications The foregoing analysis, and its extension to heterogeneous systems, constitute the central argument of the thermodynamics of chemical equilibrium. With the exception of electrochemical applications, most chemical applications of thermo­ dynamics follow by direct appeal either to the result itself or to its readily deduced consequences van5t Hoff's equation ('isochore5) for the variation of the 'constant5 (Па?)еа with temperature, Planck's equation for the variation of the 'constant5 i (П ^ :)eci (for a reaction in solution) with pressure, and van’t Hoff-Dimroth relation i for the dependence of the £ constant5 (П <^)eq on the solvent. The estimation of the Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 Chemical thermodynamics in Landsberg’’s formulation 481 ‘constant’ (n ai*)eq from purely calorimetric data combines appeal to the result Ъ with appeal to the third law. Indeed, of significant non-electrochemical applications of ‘post-first-law’ thermodynamics to chemical reactions, the only one not dependent on the above deductions is the direct application of a thermodynamic inequality to determine the direction of possible change (an application especially useful for heterogeneous systems). 2.3. Comments on the usual presentations of the thermodynamic theory of chemical equilibrium Matters relating to the existence of 8 In stages (i) and (ii) of the argument, many of the customary presentations do not make it altogether clear how the relations written down are connected with more fundamental general thermodynamic principles. It is evidently assumed, at least implicitly, that 8 exists for all states which have to be considered. The under­ lying more fundamental reasoning is one or the other of two arguments concerning S. (a) In some accounts the existence of 8 is taken (as it clearly may be) as a theorem deduced, either following Clausius or following Caratheodory, from a physical axiom which is taken as fundamental. The deductions cannot demonstrate the existence of 8 for states other than equilibrium states, and it thus becomes necessary to inquire whether or not the existence of 8 has been demonstrated for the states which have to be considered in the argument which is to follow in (iii). The mathematical procedure natural to such an inquiry is a classification expressed in terms of set theory. (b) In other accounts (see, for example, Guggenheim 1957) the existence of 8 is- itself taken as a fundamental axiom. The logical propriety of so doing has repeatedly been defended by Guggenheim in particular. For such applications of the axiom as the thermodynamic treatment of chemical equilibrium, it is physically necessary to have a specification of what are the states for which for a system there exists a function 8—for example, a specification of whether 8 exists for certain non­ equilibrium states as well as for equilibrium states. Unless such is given, it is not clear whether or not 8 is to be taken to exist for the states to be considered in the argument of (iii). The natural mathematical procedure is to define certain sets of states, and to examine the logical consequences of the postulate that there exists an 8 defined on such-and-such a set. Indeed, a postulate that ‘there exists a function S’ is not a mathematically well formed postulate unless completed by ‘defined on any set у ’ or ‘defined on any set etc. Clarification of such matters can be, and sometimes has been, given without using the terminology of set theory. In so doing, however, it is the notations rather than the notions of set theory that have been avoided. The thermodynamic theory of chemical equilibrium clearly invites a transcription into the terms of set theory which Landsberg so advantageously employed with general thermodynamic theory. Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 482 P. G. Wright Matters relating to ideality In stages (iv) and (vi) of the argument, there is either implicit reference or direct appeal to general properties of dilute systems (gases and solutions), and this appeal is in fact to beliefs (concerning 'ideality’) which are taken as axiomatic. Properly, then, there should be explicit appeal to a formally enunciated axiom. This, however, is rarely done. Two recent clarifying discussions are worthy of note here. McGlashan (1966) has criticized text-books which purport to carry the argument to the conclusion of stage (vi), but which fail to point out that only by appeal to statistical mechanics or to experiment is it possible to justify the assumed logarithmic dependence of the chemical potential on concentration. Ben-Naim (1962), in a context not involving chemical reactions, has given a particularly clear analysis of how meaningfully to define activities ai9 replacing the customary e dual definition5 (concentration-dependent part of /^) = RT In ai ailci ~~y 1 f°r infinite dilution by a ‘single5 definition based on empirical knowledge that for sufficiently dilute systems chemical potentials exhibit a very nearly logarithmic dependence on concentration. The use of set theory When, following Landsberg, we present thermodynamic analysis in terms of set theory, a number of ambiguities are automatically avoided. If a function is men­ tioned, a specification has to be made of a set on which it is defined; and, if an equation or inequality is written down, a specification has to be made of a set for which it holds. In chemical thermodynamics it is necessary to consider, beyond the conventional 'laws of thermodynamics’, axioms concerned respectively with: the existence of differentiable thermodynamic functions for states of open systems; extensive and intensive properties, the existence of thermodynamic functions for non-equilibrium states, the behaviour of dilute systems. In such terms, and only in such terms, it is possible to give a full systematic treat­ ment of the macroscopic theory of chemical equilibrium. If, first, a treatment is given in which the existence of 8 for non-equilibrium states is not assumed, the six stages of the usual argument take the following form. (i) There is an equation of the usual form. It holds for any set 7} [a set in which there are represented equilibrium states of open systems, metastable equilibrium states being allowed]. The validity of the equation is justified from an 'ordinary5 second law together with the axiom concerning thermodynamic functions for states of open systems. (ii) There is an inequality of the usual form. It holds for virtual variations linking points of any set 6 [a set in which there are represented equilibrium states of closed Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 Chemical thermodynamics in Landsberg'" s formulation 483 systems, metastable equilibrium states being allowed]. The extremum principles following from the inequality allow of the identification of which of the points of a set в constitute the set у contained in it. [In a set у there are represented only Чгие ’ equilibrium states of a closed system.] That is, by appeal to the inequality it can be decided which of the equilibrium states of a closed system are ‘true’ equilibrium states, and which are metastable equilibrium states. (iii) Two cases arise, according to whether or not there are metastable equilibrium states in addition to ‘true’ equilibrium states. If there are not, the argument can proceed no further. If there are, then the algebraic argument follows the lines of the usual presentation. (iv) All that is involved is a formal transcription of the result of (iii), the algebraic form used being guided by the explicitly formulated physical axiom of the ideal behaviour of dilute systems. (v) The algebra proceeds as in the usual presentation. (vi) Explicit appeal is made to the axiom of the ideal behaviour of dilute systems. It is possible to consider an analysis (see Buchdahl 1966) in which the inequality of (ii) is not used, and (iii) is replaced by an argument using equalities alone. By so proceeding, only a weaker result can be established. Certain special points in principle arise if the system is one in which more than one chemical reaction can occur. If reactions are considered for which there are no metastable equilibrium states in addition to ‘true’ equilibrium states, stage (iii) of the argument just outlined cannot be carried through. For such reactions, a thermodynamic theory of chemical equilibrium seems incapable of development unless it is further assumed that a function S exists which is defined not just on sets в and tj but on corresponding sets 7T and 8 in which there are represented non-equilibrium states (in addition to equilibrium states, ‘true’ and metastable). These various arguments will now be developed in detail. 3. C e r t a in s e t s o f p o in t s w h ic h c a n u s e f u l l y b e d e f in e d IN THE THERMODYNAMIC THEORY OF CHEMICAL EQUILIBRIUM Definitions of particular sets of points 3.1. Landsberg (1961) defined two physically very different kinds of set. In sets /? and y, equilibrium states of a closed system are represented, and no others, the term4 equilibrium states5 being construed strictly and metastable equilibrium states disallowed. (For the complete definitions, see Landsberg (1961, pp. The 31, 47-9). distinction between a set /? and a set у is that the latter is an open connected subset of the former, a set or sets у being obtained from a given set /? by excluding any boundary points of /? that may be included in /?.) In a set 8 (Landsberg 1961, pp. non-equilibrium states also are represented (subject to certain restric­ 129-30), tions), and states in which matter has been added or removed. While Landsberg’s 8 undoubtedly makes possible all deductions which may be Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 484 P. G. Wright needed in chemical thermodynamics, the postulation of the existence of such a set raises more than one distinct issue. It extends the definition of a function S to metastable equilibrium states of closed systems, non-equilibrium states of closed systems, equilibrium states of open systems, other states of open systems, and each extension (or, at least, the first three) might be deemed worthy of separate consideration. The following sets (table 1) are proposed as being suitable for a more detailed presentation. Table 1. Sets in which certain kinds of state are represented equilibrium states, equilibrium states [at least certain strictly con­ only, but meta- kinds of] non­ strued, and no stable equilibrium equilibrium states others states are allowed are allowed states of a closed system 7 [having subsets [having subsets i in any one of p in any one of which the mass which the mass of each substance of each substance is constant] is constant] f open without V states of any explicit an open restriction system open only w.r.t. Г & П magnification [All are sets from which boundary points have been excluded (open sets).] Table 1 (Sttppl.). The sets a, /?, and у (Landsberg 1961, pp. 22-3, 31, 47-9) Sets of points representing states of a closed system; there being allowed only equilibrium states, strictly construed, and no others СС p 7 any two states represented adiabatic adiabatic adiabatic by points of this set are in the restricted linked by a physically sense realizable process which can occur with the system enclosed by a wall that is no point that can no point that can consists of all points be in the set is be in the set is that can be in­ excluded excluded cluded in a con­ nected subset of the interior of /? In a set e, equilibrium states, strictly construed, alone are represented, but states attained by addition or removal of matter are allowed. Downloaded from rspa.royalsocietypublishing.org on November 22, 2013 Chemical thermodynamics in Landsberg's formulation 485 In a set 0, equilibrium states of a closed system alone are represented, but metastable equilibrium states are allowed. In a set i, equilibrium states of a closed system alone are represented, metastable equilibrium states being allowed, but there are represented only states in which there is a certain constant quantity of each chemical substance present. In a set 7/, equilibrium states alone are represented, metastable equilibrium states being allowed, and also equilibrium states attained by addition or removal of matter. In a set 7Г, at least certain non-equilibrium states of a closed system are repre­ sented, in addition to equilibrium states. In a set p, at least certain non-equilibrium states of a closed system are repre­ sented, and possibly also equilibrium states, but there are represented only states in which there is a certain constant quantity of each chemical substance present. In a set Г, there are represented equilibrium states, strictly construed, of a closed system and magnifications thereof; where by ‘magnification’ is meant that T, p (and any other such intensive variables) are the same, while the quantity of every chemical substance present is altered by the same factor. Sets 0, П are related to в, n as Г is to y. 3.2. Remarks on metastable equilibrium states In the present context, it is of fundamental importance that metastable equi­ librium states are to be treated neither with equilibrium states strictly construed, nor with non-equilibrium states, but as a distinct separate class. Salient charac­ teristics of the several types of state are summarized in table 2. Metastable equilibrium states and non-equilibrium states The most obvious distinction among states of systems is that between states in which the observable properties of a system are constant in time (quiescent states), and states in which observable properties are changing. For the latter, rates of change may be included among variables employed. Similar4 additional ’ variables (such as rates of diffusive flow) arise for any genuine non-equilibrium state, including non-equilibrium steady states in which all obser­ vable properties of a system are constant in time (and which are evidently to be classed as quiescent states). In contrast, for all equilibrium states, both 'genuine5 and metastable, there are no variables which measure rates of change or rates of flow. In more qualitative operational terms, equilibrium states are in principle distinguishable from non-equilibrium steady states as follows. Consider a system in a quiescent state, and consider what (if anything) happens if the system is then isolated. If some observable property changes, then the system was originally in a non-equilibrium steady state. If the system was originally in an equilibrium state, ‘true’ or metastable, it will remain quiescent after isolation.