Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 32
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This is what we have set outto show. The homogeneity of the pressure p follows from the momentumv = 1/ȡ is the specific volume.On the European continent g is also called the specific free enthalpy.10This assumption is known as the principle of local equilibrium since – as we recall – theGibbs equation holds for reversible processes, i.e. a succession of equilibria. Gibbs acceptsthis principle remarking that it requires the changes of type and state of mass elements tobe small.89Gibbs Phase Rule133balance because, when the motion has stopped, the condition of mechanicalequilibrium reads xpi = 0.One might be tempted to think that, since u, s, and v – and hence g – areall functions of T and p, the homogeneity of g should be a corollary of thehomogeneity of T and p, – and therefore not very exciting.
But this is notnecessarily so, since g(T,p) may be a different function in different parts ofthe body. Thus one part may be a liquid, with gƍ(T,p), and another part maybe a vapour with gƍƍ (T,p). Both phases have the same temperature, pressureand specific Gibbs free energy in equilibrium, but very different values of u,s, and v, i.e., in particular, very different densities.
And since the values ofgƍ(T,p) and gƍƍ (T,p) are equal, there is a relation between p and T in phaseequilibrium: That relation determines the vapour pressure in phase equilibrium as a function of temperature; it may be called the thermal equation ofstate of the saturated vapour or the boiling liquid.Gibbs Phase RuleA very similar argument provides the equilibrium conditions for a mixture.To be sure, in a mixture the local Gibbs equation cannot readTd(ȡs) = d(ȡu) – gdȡ ,as it does in a single body, because s and u may generally depend on thedensities UD of all constituents rather than only on ȡ. Accordingly, one maywriteQTd(ȡs) = d(ȡu) – ¦ gD dUD;D 1³Xthe gĮ’s may be thought of as partial Gibbs free energies, but Gibbs calledthem potentials and nowadays they are called chemical potentials.11 Obviously they are functions of T and ȡȕ (ȕ = 1,2…Ȟ).
Let us consider theirequilibrium properties.Thermodynamic equilibrium means – as in the previous section – a maximum of S, now under the constraintsνρÈØODUD d8 (Į = 1,2…Ȟ), and U Ekin Ô É ρu Ç α α 2 Ù dVÊÚ2α 1V8in a volume with an adiabatic impermeable surface at rest.11The canonical symbol for the chemical potential of constituent Į, introduced by Gibbs, isµĮ. I choose gĮ instead, since µĮ already denotes the molecular mass. Moreover, thesymbol gĮ emphasizes the fact that the chemical potential gĮ is the specific Gibbs freeenergy of constituent Į in a mixture.1345 Chemical PotentialsAs before we take care of the constraints by Lagrange multipliers OOand ȜE and obtain as necessary conditions for thermodynamic equilibriumĮX e = 0,1TandOE , andgDDDTOm .Thus in thermodynamic equilibrium all constituents are at rest, and T ,and all gĮ (Į = 1,2,…Ȟ) are homogeneous throughout V.
The pressure p isalso homogeneous; as before, this is a condition of mechanical equilibrium.And once again – just like in the previous section – if the body in V is allin one phase, liquid (say), the homogeneity of T and gĮ means that alldensities ȡĮ are homogeneous. However, if there are f spatially separatedphases, indexed by h = 1,2…f, the homogeneity of gĮ implieshhgD (T , UD )ffgD (T , UD )(D1,2,...v ), ( h1,2,...
f 1)so that the chemical potentials of all constituents have equal values in allphases. This condition is known as the Gibbs phase rule.Since the pressure p is also equal in all phases, so that p = p(T, ȡĮh) holdsfor all h, the Gibbs phase rule provides Ȟ(f-1) conditions on f (Ȟ – 1) + 2variables. That leaves us with F = Ȟ – f + 2 independent variables, ordegrees of freedom in equilibrium.12 In particular, in a single body thecoexistence of three phases determines T and p uniquely, so that there canonly be a triple point in a (p,T)-diagram. Or, two phases in a single bodycan coexist along a line in the (p,T)-diagram, e.g. the vapour pressure curve,see above, Inserts 3.1 and 3.7. Further examples will follow below.Law of Mass ActionIf a single-phase body within the impermeable adiabatic surface at rest isalready at rest itself and homogeneous in all fields T and ȡĮ, the Gibbsequation may be written – upon multiplication by V – asνT dSdU Ç gα dmα .α 1While such a body is in mechanical and thermodynamic equilibrium, itmay not be in equilibrium chemically.
In chemical reactions, with thestoichiometric coefficients ȖĮa, the masses mĮ can change in time accordingto the mass balance equations131213Sometimes this corollary of the Gibbs phase rule is itself known by that name.Often, or usually, there are several reactions proceeding at the same time; they are labelledhere by the index a, (a = 1,2…n). n is the number of independent reactions.
There is somearbitrariness in the choice of independent reactions, be we shall not go into that.Law of Mass ActionnmD (t )135amD (0) ¦ J D PD R a (t ) ,a 1aso that the extents R of the reactions determine the masses of allconstituents during the process. And in equilibrium the masses mĮ assumethe values that maximize S under the constraint of constant U. We use aLagrange multiplier and maximize S-ȜEU, which is a function of T and Ra.Thus we obtain necessary conditions of chemical equilibrium, viz.wSwU OEwTwTνÈ SÇ ÉÊ mα 1α λE0henceU Ø aγ α µαmα ÙÚ1TQ0 hencegD J D¦DOEaPD0 , (a = 1,2…n).1The framed relation is called the law of mass action. It provides as manyrelations on the equilibrium values of mĮ as there are independent reactions.Gibbs’s fundamental equationIn a body with homogeneous fields of T and ȡȕ the local Gibbs equationQTd ( UU )d ( UW ) ¦ I D dUD holds in all points and, if we consider slowD 1changes of volume V – reversible ones, so that the homogeneity is not disturbed –,we obtain by multiplication by VQQUd7 U (W 6U ¦ I D D )d8 ¦ I D dO D .D 1D 1UIn a closed body, where dmĮ = 0, (Į = 1,2...Ȟ) holds, we should haveTdS = dU + pdV and this requirement identifies p so that we may write6d5u Ts pUUQ¦ g D D and hence 6d5UD 1Qd7 R d8 ¦ I D dO D .D 1Alternatively for the whole homogeneous body we haveQQd) 5d6 8dR ¦ I D dO D .Ghence¦ gD mDD 1D 1The first one of these relations is called the Gibbs-Duhem relation and theunderlined differential forms are two versions of the Gibbs fundamental equation;they accommodate all changes in a homogeneous body, including those of volumeand of all masses mĮ.
However, the last two equations implyQ¦ O D dI DD 15 d6 8 dR ,1365 Chemical Potentialsso that gĮ(T,p,mȕ) can only depend on such combinations of mȕ that are invariantunder multiplication of the body by any factor; they may depend on theconcentrations c ȕ ȡ /ȡ for instance, or on the mol fractions X ȕ N / N .ȕȕIf we know all chemical potentials gĮ(T,p,mȕ) as functions of all variables, wemay use the Gibbs-Duhem relation to determine the Gibbs free energyG(T,p,mȕ) of the mixture and hence, by differentiation, S(T,p,mȕ), V(T,p,mȕ),and finally U(T,p,mȕ).The integrability conditions implied by Gibbs’s fundamental equation viz.wg DwgDwgDwg HwSwV,,wm HwmDwTwmDwpwmDhelp in the determination of the chemical potentials gĮ(T,p,mȕ).Insert 5.1Semi-Permeable MembranesThe above framed relations, – the Gibbs phase rule, and the law of massaction – are given in a somewhat synthetic form, because they are expressedin terms of the chemical potentials gĮ.
What we may want, however, arepredictions about the masses mĮ in chemical equilibrium, or the massdensities ȡĮh of the constituents in phase equilibrium. For that purpose it isobviously necessary to know the functional form of gĮ(T,p,mȕ). In generalthere is no other way to determine these functions than to measure them.So, how can chemical potentials be measured?An important, though often impractical, conceptual tool of thermodynamics of mixtures is the semi-permeable membrane. This is a wallthat lets particles of some constituents pass, while it is impermeable forothers. One may ask what is continuous at the wall, and one may betempted to answer, perhaps, that it is the partial densities ȡĮ of thoseconstituents that can pass, or their partial pressures pĮ.
However, we knowalready that the answer is different: In general it is neither of the two; ratherit is the chemical potentials gĮ(T,p,mȕ).This knowledge gives us the possibility – in principle – to measure thechemical potentials: Let a wall be permeable for only one constituent Ȗ(say). Then we can imagine a situation in which we have that constituent inpure form on side I of the wall at a pressure pI, while there is an arbitrarymixture – including Ȗ – on side II under the pressure pII.
We thus have inthermodynamic equilibriumgȖ (T,pI)= gȖ(T,pII,mȕII) .On Definition and Measurement of Chemical Potentials137The Gibbs free energy gȖ (T,pI) = uȖ(T,pI) – TsȖ(T,pI) + p vȖ (T,pI) of thesingle, or pure constituent Ȗ can be calculated – to within a linear functionof T – because uȖ(T,p), and sȖ(T,p), and vȖ (T,p) can be measured andcalculated, the former two to within an additive constant each, see Chap.3.14 Thus a value of gȖ(T,p,mȕ) can be determined for one given (Ȟ+2)-tupel(T,pII,mȕII). Changing these variable we may – in a laborious process indeed– experimentally determine the whole function gȖ(T,p,mȕ).In real life this is impossible for two reasons: First of all, measurementslike these would be extremely time-consuming, and expensive to the degreeof total impracticality.
Secondly, in reality we do not have semi-permeablewalls for all substances and all types of mixtures or solutions. Indeed, wehave them for precious few only.But still, imagining that we had semi-permeable membranes for everysubstance and every mixture, we can conceive of a hypothetical definitionof the chemical potential gȖ as the quantity that is continuous at a Ȗpermeable membrane.
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