Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 33
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In that sense the kinship of chemical potentials andtemperature is put in evidence: Temperature measures how hot a body isand the chemical potential gȖ measures how much of constituent Ȗ is in thebody. Both measurements are made from outside, by contact.On Definition and Measurement of Chemical PotentialsHowever, Gibbs’s definition of chemical potentials has nothing to do withsemi-permeable membranes. He writes15Definition. – Let us suppose that an infinitely small mass of a substance isadded to a homogeneous mass, while entropy and volume are unchanged;then the quotient of the increase of energy and the increase of mass is thepotential of this substance for the mass under consideration.Obviously this definition is read off from the fundamental equationQ6 d5d7 R d8 ¦ I D d O DD 1and Gibbs blithely ignores the fact that the increase of energy is unknownbefore we have calculated it from the knowledge of the chemical potentialsgȖ(T,p,mȕ).This is the same type of logical somersault, which also defines temperature as( wwUS )V , and which ignores the fact that U(S,V) is unknown before we1415All it takes for that is (p,V,T)-measurements and measurements of cv(T,v0) for one v0.J.W.
Gibbs: loc.cit. p. 149.1385 Chemical Potentialshave calculated it from measurements that involve temperature measurements. Ihave done my best to discredit this procedure before, cf. Chap. 3.Having said this and having seen that the implementation of semipermeable membranes – although logically sound – is strongly hypothetical,we are left with the problem of how to determine the chemical potentials.There is no easy answer and no pat solution; rather there is a thorny processof guessing and patching and extrapolating away from ideal gas mixtures.Indeed, for ideal gases we know everything from Dalton’s law, seeabove. In particular we know the Gibbs free energy explicitly asνpˆ k ˘ÈÊkk T kG = Âmα Íuα (TR) + zα (T -TR) - T Ásα (TR, pR) + (zα + 1) ln - ln α˜ + T˙µµµµα ˚˙TpËÍRR¯αααα =1 ÎνÈ˘Êkk T k pˆ kkG = Âmα Íuα (TR) + zα (T -TR) - T Ásα (TR, pR) + (zα + 1) ln - ln ˜ + T + T ln Xα ˙µαµα TR µα pR ¯ µαµαË˙˚α =1 ÍÎThe last term represents the entropy of mixing, see above.
By thefundamental equation we thus obtain the prototype of all chemicalpotentials, viz.g D (T , p, mE )wGwmDg D (T , p ) kµDT ln X D ,where gĮ(T,p) is the specific Gibbs free energy of the single ideal gas Į at Tand p; XĮ= NĮ/N is the mol fraction of constituent Į. So, in this special caseof ideal gases we may indeed use the Gibbs definition, because we do knowthe functional form of G(T,p,mĮ), which generally, we do not know.And yet, this specific form has become the prototypical expression forchemical potentials, considered applicable sometimes even for solutionsand alloys. To be sure, in those cases gĮ(T,p) are the Gibbs free energies ofthe single liquids or solids, respectively, rather than of the single gases.Originally that extrapolation was a wild guess, made by van’t Hoff and bornout of frustration, perhaps.
When the guess turned out to give reasonableresults occasionally, – often for dilute solutions – the expression wasadmitted, and nowadays, if valid, it is said to define an ideal mixture; such amixture may be gaseous, liquid, or solid.But, even when our mixture, or solution, or alloy is not ideal, the idealgas-expression still serves as a reference: The departure from ideality isrepresented by correction factors ȖĮ or ijĮ and we writeI D (6 , R , O E ) I D (6 , R ) M6 ln(J D : D ) ,PDorOsmosis139È pØT ln Éϕα Xα Ù .µαÊ pα (T )ÚThe former is primarily used for liquid solutions, because the activitycoefficient ȖĮ(T,p,mȕ), if it is different from 1, represents the deviation froman ideal solution. The latter expression is mostly used for vapours, becausethe fugacity coefficient ijĮ(T,p,mȕ), if it is different from 1, represents thedeviation of the vapour from a mixture of ideal gases; pĮ(T) is the vapourpressure of the single constituent Į.We shall not go further into this matter.
Suffice it to say that an army ofchemical engineers are busy determining activity coefficients and fugacitycoefficients, and they lay down their results in books of tables. Their toolsare varied. They use semi-permeable membranes whenever they exist,otherwise they use temperature measurements of incipient boiling andcondensation, and occasionally they use the integrability conditions for thechemical potentials, mentioned in Insert 5.1.
Their task is important, buttheir life is hard. It is worlds removed from the lofty positions of thetheoreticians who think that they have understood thermodynamics whenthey have understood the properties of monatomic gases.16gα (T , p, mβ ) gα (T , pα (T )) kOsmosisAlthough good semi-permeable membranes are rare, there are someefficient ones, for water particularly. Wilhelm Pfeffer (1845–1920), abotanist, experimented with them. He invented the Pfeffer tube which issealed with a water-permeable membrane 17 at one end and stuck – with thatend – into a water reservoir, cf.
Fig. 5.2. The water level will then be equalin tube and reservoir. Afterwards some salt is dissolved in the water of thetube; the membrane is impermeable for the sodium ion Na+ and the chlorideCl- into which the salt dissociates upon solution. One observes that thesolution in the tube rises, because water pushes its way into the tube in aprocess called osmosis.18 For reasonable data, viz.2 litre reservoir, 1 cm2 tube diameter, 1 g salt, T = 298 K, p =1 atmthe solution in the tube rises to a height of nearly 10 m (!).1916These practical people have their own pride in their work though, and rightly so: They liketo ridicule the theoreticians as suffering from argonitis.17A ferro cyan copper membrane.18 The Greek word osmos means to push.19The Pfeffer tube is nowadays a popular show piece in high-school laboratories.
Thesolution does usually not reach its full height during the lab session.1405 Chemical PotentialsFig. 5.2. Pfeffer tubeAfter equilibrium is established, the membrane has to support aconsiderable pressure difference, the osmotic pressure P = pII – pI.Pfeffer reported his experiments in 1877, just in the middle of the twoyear-period when Gibbs published the two parts of his great paper. HadPfeffer known Gibbs’s work, he could have written a formula for thecalculation of the pressure pII on top of the membrane, namelygWater(T,pI) = gWater(T,pII,mNa+,mCl-,mWaterII)and, of course, he would have had to know the functions gWater in orderto calculate pII or, in fact, to calculate the osmotic pressure P = pII – pI.As it was, Pfeffer did no calculations at all, nor did he present anyformulae.
However, he knew how to measure the osmotic pressure and henoticed that – given the mass of the solute – the pressure decreased with thesize of the dissolved molecules. Being a botanist he dissolved organicmacro-molecules, like proteins, and he was thus the first person to makesome reasonably reliable measurements on the size of giant molecules.20It is not by accident that it was a botanist who concerned himself withsemi-permeable membranes.
Plants and animals make extensive use ofosmotic phenomena in order to transport substances, often water, throughcell boundaries, and life would be impossible without them.Thus the roots of trees lie in the ground water and their surfacemembranes are permeable for the water. The water can therefore dilute thenutritious sap inside the roots and, at the same time, push it upwardsthrough the ducts that lead from the roots to the tree tops.
It has beenestimated that in a tree this osmotic effect can overcome a height differenceof 100 m.In animals and humans the cell boundaries are also permeable for waterand the osmotic pressure across the membranes of blood cells amounts to7.7 bar (!). Therefore the cells would burst, if we injected a patient withpure water. The fluid in the drips fixed to hospital beds is a salt solution –8.8g per litre water – which balances the osmotic pressure in the cell byexerting itself a counter-pressure of 7.7 bar.
The solution is known as thephysiological salt solution; physicians say that it is isotonic to the contentsof the cell.20I. Asimov: “Biographies....” loc.cit p. 441.Osmosis141Dilute solutions are analogous to ideal gases in some respect. At leastthat was the hypothesis made by Jacobus Henricus van’t Hoff (1852–1911),a chemist of note and physical chemist, who was the first Nobel prizewinner in chemistry in 1901.
Van’t Hoff assumed that the molecules ofȞ – 1 solutes move freely in a solution much in the same way as gasmolecules move through empty space. Thus the osmotic pressure of asolution on a semi-permeable membrane – permeable for the solvent Ȟ –should be given byQ 1PkUD P T ,¦DD1as if it were the pressure of a mixture of ideal gases. That relation is knownas van’t Hoff’s law.Van’t Hoff’s suggestion met with heavy disapproval among moreconservative chemists; but then he produced experimental evidence and itturned out that the law was sometimes true.
Van’t Hoffpublished it in 1886 and, of course, he had been anticipated – at leastpartly – by Gibbs. Indeed the continuity of the chemical potential of thesolvent Ȟ across the semi-permeable membrane, and the assumption of anideal solution reads, according to Gibbs, see aboveM+++6 ln : Q .IQ (6 , R ) IQ (6 , R ) PQIf the single solvent is incompressible, with ȡȞ as density, gȞ(T,p) is alinear function of p with 1/ȡȞ as coefficient, and if the solution is dilute, wehaveln X QQ1 N D| ¦,D 1NSQwhere N QS is the number of solvent molecules in the solution.
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