Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 34
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Thus oneobtains for the osmotic pressurePp II p Iȡ Q Q 1kT.¦ ȡDPDȡS D 1QThe ratio of ȡȞ and ȡȞS, the density of the solvent in the solution, is verynearly equal to 1 in a dilute solution, so that van’t Hoff’s law emerges fromGibbs’s thermodynamics, at least approximately.Having said this, I must qualify: One can easily become over-enthusiasticascribing discoveries to Gibbs. It is true that Gibbs had the general ruleabout the continuity of the chemical potential.
Also he had the form of thechemical potential in a mixture of ideal gases. But he did not conceive ofideal mixtures other than mixtures of ideal gases so that he could not get asfar as van’t Hoff’s law for dilute solutions.1425 Chemical PotentialsVan’t Hoff’s extrapolation of ideal gas properties to solutions must haveseemed a wild guess to himself and his contemporaries, and it seemed quiteproperly to be a dubious assumption to the chemical establishment. But itwas also a lucky guess and the question is why? The answer, or at least agood motivation, can be found in Boltzmann’s molecular interpretation ofentropy, cf. Insert 5.2.Entropy of mixing in a solutionWe have seen that the specific termkµνT ln X Q comes from the entropy of mixingof ideal gases, namelySMixQN k ¦ N D ln D .D 1NBut then the entropy has a molecular interpretation, see Chap.
4, and we mayconsider SMix in the present case as klnW, where W is the increase in the number ofrealizations during the mixing process. Assuming a homogeneous distribution {Nx}of particles at position x in V after mixing, and homogeneous distributions {NxĮ} inVĮ before mixing we haveS MixÈØνÉN!Nα ! Ù Ç lnÉ k lnÙ , with 0· Nx ! α 1N α !Ù·Éx°Vx°Vα x ÚÊ0:8and N αxNα,XVαwhere X is the factor of proportionality between the number of positions in V and Vitself.21 It follows by use of the Stirling formula that we haveQVαS Mix k ln ¦ N α ln.α 1VV/VĮ is equal to N/NĮ in gases but not necessarily in liquids, unless the particles ofall constituents are equal in size.
With this proviso Boltzmann’s interpretation ofentropy supports the entropy of mixing of ideal mixtures.Insert 5.2Raoult’s LawFrancois Marie Raoult (1830–1901) was one of the founders of physicalchemistry. He observed experimentally that – in liquid-vapour phaseequilibrium of a mixture – the partial pressure of a vapour constituent is21Recall this kind of quantization in Boltzmann’s arguments, see above. Since X drops out atthe end, the argument may be considered as a calculational auxiliary.Raoult’s Law143proportional to the mol fraction of that constituent in the solution.Obviously, if this is true, we must have 22pĮƎ = XĮƍ pĮ(T) ,where pĮ(T) is the saturation vapour pressure of the single constituent Į.Therefore carbonated mineral water – water with CO2 in solution – iskept in the bottle under CO2-pressure; upon opening the bottle we hear thehiss when the gas escapes and we see the CO2-bubbles that are released bythe water under the lowered CO2-pressure.If the vapour is an ideal gas mixture under the pressure p, we havepĮƎ =XĮƎ p and thus we obtain Raoult’s lawXĮƎ p = XĮƍpĮ (T)(Į=1,2…Ȟ) .Raoult found this law in 1886 and he was lucky indeed to find it at all,because there are few solutions which satisfy this law.
The exploitation ofGibbs’s phase rule for two phases, viz.gĮƎ(T,p,mȕƎ) = gĮƍ(T,p,mȕƍ)reveals the conditions under which the law is valid:x the solution must be ideal,23x the liquid constituents must be incompressible,x the vapour must be a mixture of ideal gases,x the vapour densities must be much smaller than liquid densities.However, when Raoult’s law is valid and when it is applied to a binarysystem, the two equations allow the calculation of X1ƍ and X1Ǝ – henceX2ƍ = 1-X1ƍ and X2Ǝ= 1 – X1Ǝ – as functions of p, when T is prescribed. Usuallythese functions are plotted inversely as p(X1ƍ;T) and p(X1Ǝ;T). The analyticform of Raoult’s law then readspp2 (T ) ( p1 (T ) p2 (T )) X 1 ' and pp2 (T )1 1p2 (T )p1 (T )X "1and the graphs are shown in Fig.
5.3left. That figure represents the prototypeof all (p,X1)-phase-pressure-diagrams with separate boiling andcondensation lines and the two-phase-region in-between. The diagram isdrawn for the case that constituent 1 is the high-boiling liquid andconstituent 2 is the low-boiler: As single liquids they boil at high and lowtemperatures respectively.22As on some occasions before we characterize the liquid by a prime and the vapour by adouble-prime.23 CO dissolved in water does not form an ideal solution. Therefore the above discussion of2mineral water must be taken with a grain of salt.1445 Chemical PotentialspTT122(p)p2(T)boilingcondensationboilingIT2(p)condensationp1(T)1 X1X1IIX1I 1 X1Fig.
5.3. Left: Phase-pressure-diagram. Right: Phase-diagramIf the equations are solved for T – at fixed p –, we obtain the curvesT(X1ƍ;p) and T(X1Ǝ;p), which may be plotted in the (T,X1)-phase-diagram,albeit not in analytic form, since the vapour pressure functions pĮ(T) are notknown analytically. Fig. 5.3right shows a (T,X1)-phase-diagram qualitativelyDiagrams of this type are important tools for the chemical engineer andfor the metallurgist, because they provide them with the knowledge neededfor enriching solutions or alloys in one of their constituents, or even toseparate the constituents.24 Let us consider this:We start at point I in Fig. 5.3right with a feed-stock solution of molfraction X1I – as it was found or provided – and at low temperature, wherethe liquid prevails.
Then we increase T until the boiling line is reached. Thevapour that is formed there has the mol fraction X1II, i.e. it is enriched inconstituent 2. Consequently the boiling liquid grows richer in constituent 1.At the new composition the solution needs to be hotter for boiling and at thehigher temperature the new vapour is not quite so rich in constituent 2 asthe old one, but still richer than X2I = 1 – X1I. When the process ofevaporation continues, the state of the remaining solution moves upwardsalong the boiling line and the state of the vapour moves upwards along thecondensation line until X1I is reached in the vapour, and the solution is allused up. Further heating will only make the vapour hotter at constant X1.The clever chemical engineer interrupts the process at an intermediatepoint and comes away with a 2-rich vapour and a 1-rich liquid.
Both mayserve his purpose.If we wish to separate both constituents completely, the feed-stocksolution must be fed into a rectifying column consisting of many levels of24Metallurgists are dealing with alloys, and solid-melt equilibria. The thermodynamics ofsolutions and alloys is nearly identical despite the different appearances of thosesubstances. To be sure, neither melts nor solids are much affected by pressure andtherefore metallurgists prefer the (T,X1)-diagram over the (p,X1)-diagram.Raoult’s Law145Fig. 5.4.
Schematic view of a rectifying columnboiling liquid, cf. Fig. 5.4.25 The vapour rising from the feed level is ledthrough the liquid solution on top and there it condenses partially, primarilyof course the high-boiling constituent. After passing through several – ormany – such levels, the vapour arrives at the top, where it containsessentially only the low-boiling constituent. That vapour is condensed in acooler which it leaves as a virtually pure liquid constituent, the distillate.Similarly the liquid solution, enriched in the high-boiling constituent by thepartial vapour condensation, overflows the rim of its level and drops intothe solution of the next lower level, enriching it in the high-boilingconstituent beyond the degree of enrichment that was the result of theevaporation. After several such steps the liquid at the bottom level becomesnearly pure in the high-boiling constituent and is led out.
In the stationaryprocess the liquid at each level is boiling at the temperature appropriate toits composition.25In the jargon of chemical engineering to rectify means to purify, or to separate intoconstituents as pure as possible. The process in a rectifying column is also calledsuggestively distillation by reverse circulation.1465 Chemical PotentialsRectifying columns are up to 30m high, 5m in diameter and may contain30 levels.
Unfortunately the method does not work well for complex multiconstituent solutions like mineral oil. For such solutions one has to becontent with obtaining certain fractions like benzine, petroleum, or heavybenzine, etc. which are not pure substances, but pure enough for efficientuse in automobiles (say).The rectifying column represented in Fig. 5.4 and similar modern designsare developments of engineers working in the chemical industry and tryinghard to optimise the process for output and energy consumption. The process itself of rectification by distillation, however, is age-old. So old in fact,that no inventor can be identified. To be sure, whoever the inventor was, hewas not concerned with mineral oil.
Rather he worked in order to satisfy thepressing need – of himself and others – for high percentage hard liquor,such as brandy, whiskey, gin, rum, grappa and the likes. This requiresseparation of alcohol from water by boiling fermented fruit juices or grainmash, and then condensing it. The process was – and is – carried on indistilleries, vulgarly known as stills.Alternatives of the Growth of EntropyOne of the sections in Gibbs’s memoir is entitle: “On the quantities ȥ,Ȥand ȗ” 26 and in that section Gibbs explains what happens to a body when itssurface is not adiabatic and at rest. We proceed to discuss that point.We know from Clausius that the entropy of a body with an adiabaticsurface wV grows, and if the body reaches an equilibrium, the entropy ismaximal.
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