CHEMISTRY A N D ITS LAWS.' B Y F . \V.kLD.
( A y / m f i / o iii Boltriiiin, A u s f i i a . ) 1. Lipoti the invitation of hlr. Trevor I communicate in the following a r P s / i r i / ~of my views and more recent results. In order to appreciate properly the peculiar position occupied b y chemistry in the study of the formation of mixtures, it is first iiecessary to recall tlir iiiore geiieral propositions conceriiiiig the equilibrium of miscible substances. propositions which we owe to J. W. Giblis.' T h e different hoiiiogeiieous lioclies whicli can be produced from a group of 11 coiiipoiieiits, aticl whose forill and quantity do iiot iiecessai-ily enter into consideratioil, are desigiiated by Gibbs as jltosc.~. Phases wliich can exist together separated by plane smfaces, and \vliose eq~~ililiriuiiiis iiot depetident upoii passive resistences,:' are teriiiecl tvc,.\.is/c,iif. T h e 71 coiiiponeiits of a phase are ititlepentlentl~~ variable wlieii the pliase o f I / I iiidepeiitlent variations. A system of I' coexistent phases having as R whole 71 indep,-iicleiitly \xrial)le coinpoileiits is capable, according to tlie Gililis phase rule, of )/ -3--1' ititlepencleiit variations.' Respecting the cliiaiitities whose variations can be coiisideretl together, the studies of Gibbs 1,ield in his equations (99) to (103)~'a Ixrietl choice ; the consideration of the qnaiitities i , /s, ;I,> :I., . . ,( I ,. being the best for most purposes. W e have now to coiicerii ourselves with specifically chemical questioiis, so for this reason it will be well to make the selection iii a iiiaiiiier best corresponding to tlie
+
+
' '~raiislated,Frorii'tlie author's Geriiiaii manuscript, by J. E. Trevor. 'J. I\'. Gihhs. 0 1 1 the Equilil)ri~iiiiof Heterogeiieous Sul)stailces. Tralisactioiis of tlie Cotiiiecticut Acadeiii>.. 1.01. 3 . V, (., pages I 1 0 - 1 1 2 of the Eiiglisli Erlitioii. q/. c, pages 152-153. V, l ' , page 143.
F. 1Vald.
22
methods of chemical operations. T h e chemist operates usually with independently variable temperature 1, variable pressure h, and variable proportions' of tlie 11 coniponents. Equation ( 1 0 2 ) exhibits the relation in which these quantities stand to the function Z , a function which, however, shall not be considered here. T h e phase rule presupposes that the iiumlier of indcpeiidently 1-ariable components is known, while frequentljr in chemical practice (and always a t the outset of the stud). of ).et uiiinvestigated bodies) a ktiowleclge of these must first be sought. In order to take account of this circuiiistance, as well as for other reasons, it will be conrrenient to 1iai.e a peculiar name for the A' substances of which the system of phases is made up and which are not iiecessarilj. identical with the 71 components. They shall be ternied silsfriii-coiizpo/zeiils,' to denote that possibly the phases can not be prepared si)/,y(j~froin them because the actual coniponents may occur in the system-coinpoiieiits in such distribution that, for the forination of a given phase froin the systemcomponents, the transfer of certain quantities of these latter to a n adjoining phase woultl be necessarj.. This will lie the case, for example, when a phase contains less of a component than does either of the s!.sterii.compoiients. W e shall consider first, however, a case in which the systenicomponents can be regarded as at tlie saiiie time coniponetits of the phases (and accorclinglj, A - ) I ) ; but we wish to emphasize that systems of phases are also possible in which the sei.eral proportions of the systeiii-compoiieiits can have df,hei/dr)//!i, xrrltrblt values, although one considers usuall\, onlj. sj.steriis of phases in wliich the dependence of certain \variations of the masses, a s deniandetl b ~ rthe phase rule, is compatible ~ i t htlie constancy or even possible arbidependence applying trary variation of these proportions,-this then o i i l ~ ,to the potentials p of the coniponentss. .
~
~~-
~~~
~~
~
~ ~ l f i sU d) / /~ S Z J P Y / L ~ issc. ////
,
?For IVald's Systeiii-coiiipoiieiits t h e translator ~ ~ o i ~suggest ltl the tertii CoIislitciciits,-whicli has beeii employed during the past two years iii his lectpres on iiiatlieinatical cIieiilistr>~, i l l making tliis tiistitiction hetween coiistitueiits aiid coiiiponents. JIf tlie proportioiis, by weight, of depeiiileiit co1iipJn:iits are arbitrarily varial,le, the correspontliiig n ~ u i i l ~ eofr coiiiponetits, in unaltered state, olivionsly iiiust lie a t tile saiiie time ~//u.sP.s of the systelii ;-their quaiitity is of course immaterial.
Chemistvy a d its Lazes.
23
T h e n- I independent proportions of the components receive dejendenfly zlaviable values when the components are miscible in all proportions. W e shall presuppose accordingly that the systeni' components can form a homogeneous phase at any temperature, under any pressure and in any proportions. Suitable examples can be found in hoinologous series of organic substances, as for exaniple the alcohols, or mixtures of alcohols which are employed in t h e formation of new mixtures. I n order to characterize our Ar=n system-components as at the same time components in general, we further assume them to differ in such wise that no one of thein can be produced through combinain all other respects tion of the others into a single phase-while their chemical character can remain entirely unknown-and we assume also that every phase in the system can be produced siiigly from the system-coniponents. I n tlie employment of system-components with so unlimited miscibility, every variation in their relative proportions will obviously result in the disappearance of a phase unless it be conipensated by J'-r variations, as the phase theory indeed iu gmeval demands. If phases are given and (for r s n ) j and f are independent, every independent variation will necessitate the changing of r - f proportions to new values. But this would require very unusual and exceedingljs difficult operations, allied to ((titration)), which would have to be undertaken siniultsneously with I' substances in order nierely to ensure tlie existence of the phases. I t niay be noted in passing that in the interval which would be required for the execution of these operations, the coexistence of Y phases could not subsist ; and that then all Y phases necessarily would have dependently variable composition. 2. Although cases of the presupposed kind can doubtless be realized, there exists probably (because of the enormous experimental difficulties) no single example of an investigation relating to the conditions of coexistence of phases so essentially instable. All actually executed investigations concerii cases in which the proportions of the system-components are either entirely arbitrary, or at least may remain constant' ; the iiiiscibility of natural substances is ~.
~
.~
~
'Independent variations of the several proportions do not necessarily correspond to independent variations of the system of phases ; see last foot-note.
24
F. Wald.
accordingly often far less than we Iia1.e assumed, or than might geiierally be the case so far as the phase theory is coiiceriied. A n attempt to follow up the causes of this more limited miscibility of the majority of accessible substances would probably not be worth while ; possibly these substances behave cliffereiitly upon other heavenly bodies or at higher temperatures. I n any case, we human beings, our bodies being complicated coinplexes of phases, could not exist under such conditions. This limited miscibility niariifests itself i n graduallj, differing ways. Often merely the dependent variations of tli: several proportions disappear, and the circunistance that the corresponding sj’steni-componeiits may be 11 niteci to a homogeneous phase only in certain limited (though still variable) proportions, is very probably related closely therewith. I n other cases the phases no longer can be produced singly froin the s).stetii-compotients, but onlj. siniultaneously in greater number, so that the system-components themselves niust be regarded as made up of yet other substances. T h e miscibility can be even so far restricted that at times i tiipassible limiting values are set to the proportions i n which the compoiients Cali appear. T h e titimber I / of the cotiiponeiits is therefore not always idetitical with the number N of the sj.steni-coniponents ; it must be determined empirically. -?, Not all degrees of limited miscibility arouse equal interest in the investigator ; the frequent compatibility of constant or even arbitrarily chosen relative proportions of tlie s!.stetii-cotiipoiieiits with the coexistence of phases during other variations, is so coninion a plienontetioii that it is regarcled as almost self-evident. T h e circumstance too that not all substances can be united it1 all proportions to a homogeneous phase seetiis even less striking. On the other hand, it must attract attention that frequently phases are obtained which can not be produced siug$f froni the spsteni-componetits, for such phases have often been entirely unkiio~vnbodies. These cases are the ones which chemistry has selected as the f i y s f objects of its study. An exact discussion of the thertiiodynaiiiic conditions under which the phenomena of liiiiitecl miscibility can appear may lead 1w-y probably to the coticlusion that these phenomena are closely
related to the existence of /iuri/iiig ii/i.v/uws, whose coiitetit in the single coniponents can not, a t least in oiie direction, be exceeded. Be that a s it may, it has heen determined by Proust that such mixtures exist, and ever since that t'iiiie cheniistr), has iiiatle their stud\. its srmird task. T h e cheiiiical study of the natural substances falls accordingljr into two distinct stages :-in the first, one seeks the nuiiiber of independent coiiiponeiits of the giizeii systeiti-compoueiits antl of the phases formed from tlietii, or, in other words, the least nuiiiber of siibstances requisite for the separate preparation of each of the SJ'Stem-coiiipotients as well a s of each of the phases which are produced ; in the second stage, those pliases are sought whose material character remains uriinfluenced l q v tlie independent variations of the systenis of phases in which they arise. Clieniistr)., according to this exposition, is concerned solelj. with the iiiore striking effect:< of limited iiiiscibilitj., atid therefore with an arbitrary (althougli very suitable) sclr~limrof phases and of systems of phases ; and it has deteriiiined a series of peculiar laws which apply to them. T o account for these laws, it has heen deemed necessary hitherto to accept certain hypotheses. ill). \view, 011 the contrary, is tliat these laws are siiiipl), conseqiieiices of the conditions uiider which the substances in qiiestiori have ariseii aiitl through which they are distinguished froiii phases in general. T h e stoichiometric laws hold, not for all phases whatever, but otil). for very definitely characterized ones ; there appl), accordingly to the latter not on159 all laws to which phases in general are subjected, h i t also i i i addifioii certain spscial relations which must hold good i f the phases in question are to satisfy the requirements through which phases is obtained. their exceptional position as o'/(~i~/i(d 4 . I n tlie analytical stage of chemical iiivestigations, which must be iiiiagiiied a s beginning iri a time when iio cheiiiical kiioivledge whatever had been attained, iV sul)stances are niixetl in variable proportions by weight, at variable temperature antl under a n y (usually accidental) pressure, and t h e number of phases which can exist under the given circumstances is deterniined. If none of tlie A' substances cnii be obtained as single phases by rnixing the others, i t follows that each iiiust contain a t least one conipoiient peculiar to
26
F. bUald.
itself. Now it is to be expected that the mixture will be one homogeneous phase : if the experiment yields more than one, each differing froin the systein-compoiieiits, me must assuiiie I . - 1 further components,-and froin this it niust be concluded that a t least When single phases tiirii out u1N+ I*- r components are presetit. to be identical with certaiii s).steiii-co~iipotientsthe). are not counted ; if it appear on the other hand that apparently identical sj.stemcomponents, according to their origin or other circumstances, do give differing sj.stenis of phases under otherwise like conditions, then so many inore components must be counted as dependent variations of this kind are observed. These methods of reasoning are still employed in the chemical investigation of unknown substances : they embody a n empirical form of the phase rule,-they yield the number of independent components from the observed nuiiiber of phases and the number of the independent variations consistent with the existence of these phases. If one is coiivinced that these last data have been correctly determined, he feels equall!. certain about the number of independently variable components, and sets, with the above mentioned possible corrections,
)~=N+Y--I T h a t one can be deceived in this, however, appears from the fact that independently Irariable components are often discovered even in very thoroughly investigated substances, be it for the reason that single phases have been overlooked because of their small quantity or of their similarit). with others, or that the observed nuinher of independent variations is too small. I t niay also transpire that entirely unsuspected components of these substances can be discovered upon mixing the latter with other bodies. If the new components thus found render possible iiide#endeiit variations of the system of phases, they of course require an increase in the number I I formerly accepted as correct. Otherwise these components can be ignored so long a s one is concerned with but a single definite system of phases. But if it be desired to consider siniultaneously a number of such systems, it niay become necessary to employ the elaborated phase-formula I L - ~ + Z - Y of Gibbs, where h denotes the number of coniponents whose quaiitit). in the sj.stem
of phases stands in a n invariable ratio by weight to the quaiitit). of others. 5. Chemical syteniatics is concerned with substances, the nuiiiber of whose cotiipoiieiits usually iiiaj. be regarded as known and which suffer no change of composition as a result of an!. dependent variations of the system to which they as phases belong. 111 order that a substance iiiay be termed a (.hri)zird i u d h d u a l : ( 1 . It must arise from a group of phases in which tlie several proportions of all the s).stem-componeiits are independently variable, whereby, however, as has been noted often alreadjr, the potentials :!. of certain coiiiponeiits (or possibly of all ) may !.et be dependent ; tein of phases must admit at least om' independent \.ariation, from which follows that it can contain at most r = ) r + ~ , or T = ) L - ~ + I , phases respectivelj. : (., Atid finall)., during all Irariations, its character (but not its quantit), ) must reniain apprecialil!. the same a s though it. as individual phase, were enclosed in a shell impermeable for every sulxtance. This latter condition is identical with the requirement of constant composition, but the form of statement here adopted has the advantage of making clear that a substaiice iiiaj. be declared a cheniical individual before any idea h a s been foriiied regarding either its components or the relative proportions in which these components ,appear. Accortlinglj., when we speak of a chemical individual, the question reiiiaiiis entirely open whether tlie substance is actually undecotiiposable, or whether we do not )ret know how to effect a decomposition of it. It has not yet been found possible to free from iiitlepetitlentl~~ variable coniponents certain bodies which are regarded as chemical individuals, but on the assumption that, nevertheless, it is possible to do this, the phase in question is treated as an uiiclwiiiical mixture of the yet uiikiiowii chemical iiitli\ridual with these components. To teriii the composition of a phase d / e i u i m / b constant is intended to signify briefly that the above enumerated conditions a r d fulfilled. If substances w1iic:li are completely iiiiscible within certain limits are iiiixed always in the same (although ar1)itrarj.j p o p o r -
28
tions liy weight, a product of invariable composition is of course always obtained ; but this k i d of constancy is, according to t h e above defiiiitioti, not at all a chemical one. If, therefore, it should happen that one component of a body is kiioivti to be producible froiii other components, and it is desired to employ these latter, it iiiust lie perniissible either to employ these components theniseli.es i l l iiitlepenclent relations li!. weight, or else’ to introduce the coiiiplex coniponeiit, a s formed froni them, into the systeiii in t h e form of a chemical compound. 111 the latter case then tlie criteria of chemical indivicluality must lie applicable i i i the first place to the system-coiiipoiieiit in question. Similarly, when we are called upon to applj. the phase rule in the form ~--/1+2*--1., the phases of constant composition which are obtained may he regarded a s cheinical indivitliials onljv when ever). one of the components included uiider /r is eniplo)retl in a chemical coinpound. Sulistances already recognized a s cheniical iiidi~~itluals can be prepared of course in any other iiiaiiiier suited to the purpose in hatid : Imt for ea’ch of these bodies methods of preparation must exist through which it is proved to be a chemical individual,-aucl it is precisel!. these iiiethotls which alone have ititerest for tis. It is to be noted that in the systems produced 1))~ these iiiethotls, the variations which m a y he regarded a s intlepeiitlent car1 lie 110 longer chosen arliitraril)., Of the physical parameters fi arid 1, one at least i-ariations it niust he iridepentlent, and i f tlie sjrsteiii adiiiits niust be possible to i‘ary both ,h and f independently. If yet other variations are permissible, then, together with ,h and f all the conipotieiits \rliicli may be preserit onlj. iri chemical tinion with others teni-coiiipoiieiits are clependeiitl!. .i.arialile, a n d of the sJ.steiii-coiiiponeiits theniselves all those which at tlie same time form phases of the system. 6 . T h e aim of clieiiiistry is to resolve all mixtures into chemical iiitlivitlunls. and this result has beeii reached i n surprisingly 11ia11y cases. W e ma!‘ lie justified in assuriiitig, therefore, that all 0111’ s!.steiii-coiiipoiieiits are c1iemicnl iiidiricI\ials, ant1 tliat the pliases o i met1 froiii tliein are either themselves cl aiiical individuals or CRII t i l 7 0
‘If this procedure should he i ~ ~ c o ~ ~ ~ p a11t iitl) ) lthe e sul)siste~iceor coilstant cutripositiciii of tlie phase.
lie resolved into such. If a systeni contain phases of \.arial)le coiiiposition we ilia!’ iiiiagiiie these separated into clieiiiical iiitlii.itluals liy ( ( seinipertiieable walls ) ) , nrhereli!. oiie phase with depentleiitl~~ variable osmotic pressure is obtained for ever!’ iiiclepeiideiitly Phases variable relative proportion of the s!,stem-conipoiieiits. whose cliiantity can iieither lie increased nor tliiiiiiiished I)), cheiiiical reactions aiiiong tlie other phases ma!. be ignored entirely, since we components. But if further otiiit tlieiii froin the tiiiiiilier of the phases then reniaiii with tlepencleiitl!. varialile coniposition-as, for example, in dissociating n’ater vapor, from which oxj’geii aiitl hydrogen can be separated, but pure water can not-we can tlisregard this system of phases altogether. For i f its cheniical components are all k i i o n ~ ithe!. must have lieen recognized as cheniical intlivicluals froiii other systems i n which no phases with dependent composition appear. In the s)rstenis which need Iw considered all phases are accorclingly cheniical inrlividuals~ ant1 all systeiii-coiiiporients are at the same tiiiie pliases : the chemical reactiotis occurring in these sj,steiiis niust therefore be tlieriiiod!~iiaiiiicall~reversible. In tlie place of a pressure coiiiiiioii to all phases ive have possibl!. a iiuniber of osmotic tensions, yet, regardless of this, we may speak of o / / ( a pressure p which is either coiiiiiioii to se\-eral phases or coticeriis but a single --arbitrarily chosen-one. T h e tiunilier of phases can not lie greater than I ) r ( o r -h respectiirel!. i , and not less than I I ( o r //-I/). No further liiiiitatioii is imposed upon the character of the phases than that they niii.\t lie chemical iiitli\~itlunls; hut in no wise can anything I)e stated a s to whether one or another contains otie coniponent or se\,eral, It must be admitted, therefore, on the other hatitl. that ever!. kno\vii coiiipound of se\.eral cciiiipoiieiits caii belong to a s!.stein of phases of the kind here presupposed. i n ivhich moreover all of tlie phasesfor the reaction whetlier iiiitinl or siibsequeiitl!. producetl-iiecessar!. are n ~ i u f l o ~ c i / t J o, vf the i / . or of the I / - / / , coniponeiits. T h e correctness of this proposition lieconies iiiore clear wlieii its coiitrary is assuiiietl ns correct : that every reaction of cheiiiical coiiipouiids involves a co-operatioii of the free coniponeiits. No\\,. for tis, oiil!. the so-called elements are uticlecoiiiposal~le,and intleetl
+
+
30
'
these ma>' not be so to-morrow ; but, apart from this, 71-1 eleiiientary components would then have to form phases in every system having u-h independent components. W e should be forced to conclude that even when h beconies zero two cheinical compounds a t most of the same su1)stances could be capable of existence under In chemical independent temperature and dependent pressure. analysis, where tlie pressure is independently Irariable also, only oizc cheniical conipouncl could exist, and 110 compound could contain dependent (h-) components. But these propositions are obviously incorrect, both in themsel\~es and because we halve no guarantee that the so-called elements are incapable of further decomposition. But in order that every conipound can be made oiilj. from other compounds, there must exist certain relations among tlie conipositions of all, relations which are readily seen to be identical with the familiar laws of simple and of multiple proportions. If it had ever beeri noticed that chemistry begins its operations with substances about whose components (either as to their number or their properties) nothing whatever is known, the existence of the stoichiometric laws never could have appeared especially striking. But it seemed more suitable to iniagine all chemical compounds a s individual phases arising from supposedly undecomposable components : and in consequence of this all the relations were forgotten which are necessarily hound u p with the far more frequent production of these coinpounds from others. 7. If a system lia~7e / * - ) / + I phases the pressure is independent,-as well a s the osmotic tensions eventually to be considered. But the system-components are transformable iiitp the remaining phases and the reirerse, and, further, the absolute quantity of each phase is imniaterial : the pressure therefore indeed will be constant a t constant temperature. but the volume of the system undergoing reaction will \Tar).. When only ~ = i i phases are necessary for a reaction the pressitre is still intlepetideiit, and if we disregard the necessity of osmotic \ITalls there retilaills no further iiieaiis of bringing about the reaction. ( T h e reaction is of cotirse supposed possible.) I t follows therefore an entirely unililportat1t that the reaction here is tliertiiocl~~ianiicall~ matter, and that i t can entail neither change of volunie nor devel-
opment' of heat. For if, tor example, a change of pressure produces no reaction the reaction cannot, on the other hand, effect a change in the volume. If osmotic nalls are necessary for the separation of the system into chemical individuals, then this conclusion obviously niust be modified ill such wise that the sum of the quantities of work concerned in the reaction niust disappear. Reactions in nrhich the number of the interacting substances is equal to the number I I (or n d / respectively) of the components can be realized accordingly, with ordinary cheniical meaiis, only when either the substances themselves form niixtures of variable composition, or when bodies are present which, although not directly participating in the reactions, yet render possible the formation of such mixtures. It follows, from the thermal indifference of these reactions, that the sum of t h e specific heats of the products of the reaction must lie equal to the sum of the specific heats of the sq'steiii-coiiipotieiits ;but the necessity of osmotic walls for the separation of substances into cheniical individuals ilia). iiitroduce apparent exceptions. I n any case the bodies inust all be considered i t i the state in which they participate in the reaction ; for example, that the specific heat of car1)on will not follow certaiii rules, is related very probably to the circuinstance that there are few reactions if any which supply pure carbon reversibly and a t the same time satisf!. the condition / ' = I / . If tiow it can be proved tliat by combiiiatioiis of reactions when r = / / + r a total reaction with Y = I I niust arise, there will follow for the latter a ~ioluiiie-neutrality,or a work-neutrality respectivel!.. In (~oii~equetice of this, the quantities of work which are iiivolvetl in the single reactions must stand in rational nunierical ratios. But for this to be true it is merely necessary tliat soiiie one o i the pliases to a t least two of the reactions, and that be coninioii. with / . = / I + the two systems shall be niechanically connected in such wise that the quantity of this phase which is produced by one of the reactiotis shall he consunied I?!, the other. T h e stoichiometric laws of cheiriical illasses a n d voluiiies are referred through these corisiderations : a . T o the peculiar properteins of phases which have beeii arbitrarily choseii I ?
~~
.
'Positive or negative.
32
F. IVald.
as objects of chetnical stud!,. and accordingly to tlie fact that phenoiiiena of this kind esist : 6. To tlie fact that ((siniple))components need not necessarily occur in every group of phases ; c. T o general laws or groups of phases. Mj. conceptions, while avoiding all li>.potheses, possess the advantage of permitting conclusions to be drawn concerning a series of relations whicli already have been recognized in part, but have remained without explaiiatioti, and which i n part have not yet been even suspected. 6. W e shall 1 ) ~ s ourselves ~. next for a iiiomeiit with reversible reactions, for which I,= /L : theriiiodynaiiiicall~.these are wholly indiffereiit. If we set out from a definite state of a system, subjecting the latter successivel), to an isothermal and to an adiabatic change of state ant1 then returning it to the initial state bjr an isotliernial and a n ac1ial)atic change, we obtain a Carnot's cj.cle, which rcmains unaffected so far a s all essential data are concerned if yet another clieniical transforiliatioil forwards and backwards should be introduced at tlie two stages of constant temperature. But this condition ceitainl!, can be satisfied only when for the one independent pressure of the system the Marriotte-Gay-Lussac forniula p:l= R T holds ;-onl~. here S J denotes ti0 longer the volume of a single phase. but the volume of a coniplex of phases. Since it remains open to us which of tlie eventual osmotic pressures we desire to regard as iiidependetit, this last equation must hold true for every one of tliein. Noiv such reactions often involve a number of phases whose volumes are relatively ver!. small and nearly independent of the temperatttre and tile pressure : thes- are the lion-volatile substaiices. If their volumes are neglected, the pressure-volume law follows for tlie remaining phases a s approximatel\. true. It is then not necessary that the slight variations which remain slioultl be accounted for through hypotheses they can be referred to entirelj, real circumstances. Even though no numerical evidence for the correctness of this \.ie\v is available as yet, one may, nevvrtheless, iiot cast it aside a s inatlniissible or unfruitful. 9. 111 Gililx's pliase theory t h e fundaiiiental equatioiis o f tlie phases are looked upon a s indepeiiclent of one another ; and froiii these equations follow both the values assiuned i l l the iiit1ivitlu:il
phases by the relative proportjons of tlie system-components, and those assumed by the phase-coinpotleiits themselves. Now in all clieniical phenoniena these quantities take particular values, which can hold good only when the fiiiidamental equations satisfy definite conditions. Their deduction is a purely mathematical matter but we ma!. nevertheless draw attention to the circumstance that for a long time certain relatioris among the properties of cheniical individuals, and accordingly also ainong their funtlaniental equations, have empirically been recognized. Entire classes of clieniical conipounds in fact exhibit analogous chemical properties, a s do acids, bases and salts for example, or alcohols, ketones atid other organic bodies. Even the elements follow rules,-which find expression in the periodic system. These relations could not possibly exist if tlie eqnations of contlitioii of the cheniical individuals were as independent of one another a s are those of phases in general ; so they also seem to me to furnish a ver). trustworth). support for my views. I A7lat2‘?10,.4i(grisf r , 1896. 1 1 1 this article, OII page 2 1 , the words “is capable” are missing after the \vord “phase” 0 1 1 line 14. 0 1 1 page 32 also, line 4,“or” shoald lie “of” ; and on the seventh line from lielo\r a coniiiia should appear after “hypotheses.” I
