Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 36
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In the latter case the mixing process must be accompanied bycooling, if the temperature is to be maintained. The former case requiresheating lest the mixture cool off during mixing; that case is the interestingone, because the Gibbs free energy can become non-convex, if the heat ofmixing is big enough.It makes sense to consider the special case that only the liquid phase isaffected by the heat of mixing, while the vapour – whose molecules are farapart – is ideal.
In such a case we have Gibbs free energies Gƍ and GƎ of thetype shown in Fig. 5.6left. That figure corresponds to a fixed pair of pressure30J.W. Gibbs: loc.cit. pp. 172–187.Phase Diagrams151Fig. 5.6. Left: Gibbs free energies of vapour and liquid. Right: Phase diagram with amiscibility gapand temperature and obviously there is the possibility now for two commontangents. When the temperature drops, the graph Gƍ comes down withrespect to GƎ and there is the limiting case when the two tangents growtogether and a three-phase-equilibrium exists between two liquids and avapour. At still lower temperatures a common tangent can connect theconvex branches of the liquid graph so that two liquids coexist, solutions aand b, which are 2-rich and 1-rich respectively and which do not mix: Thereis a miscibility gap in the liquid phase.The phase diagram is constructed as before by projecting the commontangents into a (T,m1)-diagram onto the appropriate isotherm. When theend-points of the projections are connected we obtain a diagram of the typeshown in Fig.
5.6right. The point denoted by E is called the eutectic point. Foralloys the eutectic composition has the lowest melting point and that is whateutectic means in Greek: eutektos = easy melting.In solutions the existence of the miscibility gap helps to separate theconstituents and since one solution is invariably lighter than the other one, itfloats on top and may be scooped off – like the fat from the milk.Gibbs did not draw phase diagrams, but he did know the theory.
Thus thethree-phase equilibrium at the eutectic point in a binary solution allows F =2 – 3 + 2 = 1 degree of freedom, see above. This means that the eutecticpoints form a line in a three-dimensional (p,T,X1)-diagram.Phase diagrams of the type shown in Figs. 5.5 and 5.6 – and morecomplex ones – for solutions and alloys are being measured and refined tothis day. The first person to make this part of Gibbs’s work explicit, anddraw conclusions from it, was Hendrik Willem Bakhuis Roozeboom (1854–1907). He learned about Gibbs from van der Waals and made manyexperiments to confirm the phase rule.
The modern physics and chemistryof alloys started with his work.1525 Chemical PotentialsLaw of Mass Action for Ideal MixturesWe recall the law of mass action as derived by Gibbs and combine it withthe form of the chemical potentials valid for ideal solutions. Thus we obtainthe law of mass action for an ideal solutionQJXDD1aDª«exp « ««¬QJD¦D1aºPD gD (T , p ) »kT»»»¼(a1,2...n)The right-hand side is independent of composition and it is thereforeoften called the chemical constant K a, although it depends on T and p.Of course, there are n such constants, one each for every independent reaction.The left-hand side – when written out – is the quotient of mol fractions ofresultants and reactants with the appropriate exponents.
It is customary toconsider the stoichiometric coefficients of the resultants as positive, andthose of the reactants as negative. Also the most negative one among thestoichiometric coefficients is often set equal to –1.31 Thus for the reactionC 12 O2 CO or C 12 O2 CO0the law of mass action has the form – with K(T,p) as the chemicalconstant –X COX C X O2K ( T , p ),whereX C X O2 X CO1,so that a decrease of the mol fraction X O2 will increase the output of CO.Early chemists like Torbern Olof Bergman (1735–1784) or Claude LouisComte de Berthollet (1748–1822)32 were somewhat confused about massaction (sic), i.e. the shift of the chemical equilibrium upon the addition ofmass of a constituent to a mixture.
Bergman had conceived of affinitiesbetween substances, such that substance A reacted with B but not with C, if31There is a certain arbitrariness here. The convention mentioned in the text is notuniversally accepted. Some people prefer to set the smallest absolute value among the ȖĮaequal to 1, thus avoiding fractional coefficients. Both conventions – and still others – areperfectly good, but they must not be mixed. And before we use tables – for chemicalconstants, or heats of reaction (say) – we must know which one of the conventions wasemployed in the compilation of the table.32 Berthollet was yet another scientist ennobled and made a senator by Napoléon, whogenerally knew how to create loyal followers. In the case of Berthollet, however, he madea mistake, because the chemist later voted for the deposition of the emperor.
That gainedhim a peerage under the returning Bourbons, cf. I. Asimov: “Biographies….” loc.cit.Law of Mass Action for Ideal Mixtures153the affinity between A and B was great, while the affinity between A and Cwas small. Bergman prepared tables of affinities that were much used at histime. But then Berthollet observed that A and C would react after all, if onlyC was present in sufficiently great quantity. Thus we see how the somewhatstrange name of mass action came about.
Berthollet wrote a book about hisfindings,33 in which he showed deep insight into the nature of chemicalreactions, but did not quite arrive at the proper form of the law of massaction.The correct formulation of the law came in a paper by Cato MaximilianGuldberg (1836–1902) and Peter Waage, both professors of chemistry atthe university of Christiania – now Oslo – in Norway, and brothers in law.The paper was written in Norwegian and remained unnoticed by mostchemists. Even the French translation in 1867 did not help and it was onlythe German translation in 1879 that made the work known. So for once,Gibbs lost his priority for the law of mass action; he whose memoir made somany other people lose their priority in subsequent years.The argument of Guldberg and Waage is extremely simple: They arguedvery sensibly that a reaction can occur only, when the molecules of allreactants meet at one point in the numbers required by the stoichiometricequation.
They considered – again plausibly – the probability for a moleculeof reactant Į to be at a certain point as being proportional to XĮ. Thereforethe probability for the forward reaction reactant ĺ resultant should begiven byQCo X DPoJD,D 1where Ȟ- is the number of reactants and Cĺ is a factor of proportionality.Accordingly the probability for the backward reaction resultant ĺ reactantshould beQPmCmJXDD QD, 1In equilibrium both probabilities ought to be equal and so Guldberg andWaage came to the condition of chemical equilibrium in the formQ:DJD1D%m.%oThe nature of the right-hand side – and its dependence on T and p – couldnot be determined in this simple manner. The law is truly a law of massaction and not, as it were, of pressure action, or temperature action.And so, although Gibbs was anticipated by Guldberg and Waage withrespect to mass action, his discovery went beyond that of the Norwegians,because he knew the structure of the right hand side, viz.
of Ka(T,p):33C.L. Berthollet: “Essay de statique chimique” (1803).1545 Chemical PotentialsÈ νØaÉ Ç γ α µα gα (T , p ) ÙÙ.K a (T , p ) exp É α 1ÉÙkTÉÙÊÚWe have argued before that g (T,p) can be determined by (p,V,T)measurements and by measurements of heat capacities CV(T,Vo) for one V0.Actually we showed that such measurements leave us with unknownadditive constants in U and S. Therefore Ka(T,p) contains a linear functionof T of the type'hRa T 's RaQQD 1D 1¦ J Da µD hD (TR , p R ) T ¦ J Da µD sD (TR , p R )with unknown coefficients, the specific heat of reaction 'hRa and 's Ra , thespecific entropy of reaction.It is worth mentioning, perhaps, that those constants do nowhere play a role inthermodynamics, except when it comes to chemical reactions. Indeed, when aconstituent vanishes or emerges, then energy and entropy of the constituent vanishand emerge along with the mass, and that includes the additive constant terms inenergy and entropy.The heat of reaction a, viz.
'hRa can be measured by measuring how muchheating or cooling the reaction requires, if temperature and pressure are tobe maintained. And after 'hRa has been obtained in that way, the entropy's Ra of reaction results from a quantitative analysis of the reactionproducts. A systematic experimental campaign was needed for that and thatwas not Gibbs’s thing.Anyway, Gibbs had done enough. And we have not even considered hiscontribution to the thermodynamics of solids, where elastic stresses take over therole of the single pressure in fluids in determining the working term in the first law.Nor have we considered Gibbs’s work on thermodynamic stability or the largesecond part of his memoir which is entitled “Theory of Capillarity”, where Gibbsdeals with surface effects and treats droplets, bubbles and inclusions.
These are allimportant contributions to thermodynamics, but they represent collateral tributariesin the history of the field rather than the main stream.Measurements of the heat of reaction had already been made by Lavoisier.And Germain Henri Hess (1802–1850) measured enough of them topronounce Hess’s rule in 1840 which states that the heats of reaction insuccessive reactions must be added. This rule helped to determine valuesfor reactions which are difficult to investigate directly. After Gibbs’s workthe Hess rule became a corollary of that work.Law of Mass Action for Ideal Mixtures155Heats of reaction are usually measured in calorimetric bombs, i.e. strongchambers, capable of enduring high pressures at constant volume.
PierreEugène Marcelin Berthelot (1827–1907) measured hundreds of heats ofreaction, while Hans Peter Jörgen Thomsen (1826–1909) measuredthousands of them. So we may assume that heats of reaction were availableto chemists at large. This is not to say, however, that the significance of thequantity was universally recognized. Berthelot in particular was confusedabout the role of heats of reaction. He considered them the sole drivingforce for a reaction, such that only exothermic 34 reactions – those with anegative ǻu35 – could proceed spontaneously.
The idea is plausible and,indeed, it is very often true. When it is not true, it is because the entropy ofreaction interferes: Its growth ǻs during the reaction may be so big that – atthe prevailing temperature – it may offset a positive value ǻu and still allowthe necessary decrease ǻf of free energy.A well-known example is the reactionH2 + I2 ĺ 2 HIkJand yet – at about 450 °C –which is endothermic with 'J4 25 mol4hydrogen iodide makes up /5 of all molecules in equilibrium. Indeed, theJtables provide the value 'U 4 166 mol, so that the entropy moves upwardsby the formation of HI, while the free energy moves downwards. That is thedesired direction for both of them. So, how can H2 and I2 survive at all,albeit in the small proportion of 20%? The answer lies in the T- and pdependent part of 'J and 'U .
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