Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 35
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That is the case, for instance, when the adiabatic surface is at rest,so that the energy U + Ekin is constant. The question arises, however, whathappens when the surface is not adiabatic, or when it is not at rest, or both.The easy answer is, that in such cases generally equilibrium will not beapproached.However, that is too pat for an answer. There are special boundaryconditions – other than adiabaticity and rest – for which equilibrium can beapproached and some of them may be characterized as follows:x Homogeneous and constant temperature To on V and body at rest there,x adiabatic boundary V and homogeneous and constant pressure there,x homogeneous and constant temperatures To and pressure po on V.We refer to Chap. 4 and recall the equations of balance of energy andentropy26J.W. Gibbs: loc.cit.
p. 144.Alternatives of the Growth of EntropyEnergy:27Entropy:d(U E kin )dt ³ qi ni dA wV147³ pX n dAiiwVqndSt ³ i i dA .dt wV TIt is then fairly obvious that under the three stipulated sets of conditionswe obtainxxxd (U E kin To S )d 0,dtd (U E kin poV )dSt 0,0 anddtdtd (U Ekin To S poV )d 0.dtThis means thatx U + Ekin – ToS ĺ minimum for To constant and w8 at rest,x S ĺ maximum for an adiabatic surface,x U + Ekin – ToS + poV ĺ minimum for To constant and po constant.The first and last conditions are alternatives of the growth of entropy,appropriate for the stipulated conditions. The validity of these trends towardequilibrium is independent of how far the body is away from equilibrium;indeed, initially the process in wV may be characterized by turbulent flowfields and strong gradients of temperature and pressure.
At the end,however, when equilibrium is near, we know that Ekin is negligible and thefields of temperature and pressure are very nearly homogeneous, apart frombeing constant. That is the situation considered by Gibbs.Indeed, Gibbs uses a method akin to the method of virtual displacementknown in mechanics. The kinetic energy never occurs and temperature andpressure are always equal to their boundary values.
Therefore he concludes:x Free energy F = U – TS is minimal in equilibrium compared to itsvalues in other states with the same T and V.x Entropy S is maximal in equilibrium compared to its values in otherstates with the same p and enthalpy H = U + pV.x Gibbs free energy G = U – TS + pV is minimal in equilibriumcompared to its values in other states with the same T and p.Free energy, enthalpy and Gibbs free energy are the quantities ȥ, Ȥ and ȗin Gibbs’s work.
He does not name these quantities apart from calling ȥand ȗ force functions under the appropriate conditions of constant (T,V) and(T,p) respectively. I have introduced the now common names and chosen27The working term is simplified here, because we do not account for viscous stresses.1485 Chemical Potentialsthe symbols F, H, and G which are most often used in the modernliterature.28The question is, of course, what it is that can change when T and p arealready equal to the constant boundary values. One possibility is that themasses mĮ of a chemically reacting mixture can change and at constant Tand p they will change so as to minimize G; see above, where we havederived the law of mass action.
Another possibility is that different phasesin a body can readjust themselves – at constant T and V – so as to minimizeF and to make the chemical potentials homogeneous.Entropy and Energy in CompetitionThe knowledge, that the free energyF = energy – T · entropytends to a minimum as equilibrium is approached, is more than the result ofsome formal rearrangement of equations and inequalities. Indeed, the knowledge provides a deep insight into the driving forces of nature. Obviously, adecrease of energy and an increase of entropy are both conducive to makingthe free energy small.
If T is small, such that the entropic part of F isnegligible, the free energy tends to a minimum because the energy does.And, if T is large, so that the entropic part of F dominates, the free energybecomes minimal, because entropy tends to a maximum. Those are theextremes; at intermediate temperatures it is neither energy that reaches aminimum, nor entropy that reaches a maximum.
Both quantities have tocompromise and the result of the compromise is the minimum of the freeenergy.The Pfeffer tube provides an instructive example for that situation, cf.Fig. 5.2. The energy – gravitational potential energy in this case – tends toadjust the levels of liquid in tube and reservoir to be equal; that is thesituation where the energy is minimal. The entropy, on the other hand, tendsto pull all the water from the reservoir into the tube, because that meansmaximal entropy of mixing of water and salt. Neither energy nor entropysucceed; they compromise and as a result some water remains in thereservoir, – less for a higher temperature.The phenomenon is also interesting for another aspect: Obviously it isessentially the water that pays the cost, as it were, because its potential28It is not uncommon though to see the free energy be denoted by ȥ, as in Gibbs´ work;others prefer the letter A for available free energy. The letter H for enthalpy stands forheat content which is the literal translation of the Greek word enthalpos: en inside +thalos heat.
This is a good name, since the enthalpy comes closest among allthermodynamic quantities to what the layman calls heat. The G for the Gibbs free energyis, of course, in honour of Gibbs himself.Phase Diagrams149energy rises considerably; and it is the salt that profits because its entropyincreases with the larger volume of the solution in the tube. We concludethat nature does not allow the constituents of a mixture to be selfish: Thesystem as a whole profits by decreasing its free energy.Even closer to home is the case of our atmosphere: The potential energyof the air–molecules would be best served, if all of them lay at rest on thesurface; but the entropy would be best off, if all molecules were spreadevenly throughout infinite space. The compromise of minimal free energyin this case provides earth with a thin layer of thin air. If the earth werehotter, like the planet mercury, that atmosphere would have left us, and if itwere smaller, like mars, the atmosphere would be even thinner.29Considerations like these help to create an intuitive feeling for the significance of Gibbs’s force functions.Phase DiagramsLet the Gibbs free energy G of a binary mixture with a fixed mass m =m1 + m2 at some fixed values of T and p be represented – as a function ofm1 – by the convex graph of Fig.
5.5left. It follows from the relations ofInsert 5.1 that the graph begins and ends at g2(T,p) and g1(T,p) respectivelyas indicated in the figure. Moreover, if we draw the tangent at some pointG(T,p,m1*), the intercepts of that tangent with the vertical lines m1 =0 andm1 = m represent the chemical potentials g2(T,p,m2*) and g1(T,p,m1*),respectively, cf. figure.Now, let there be two such graphs, corresponding to two phases ƍ and ƍƍ(say).
These are shown in Fig. 5.5right for a (T,p)-pair for which theyintersect. If the two phases are to be in phase equilibrium, the Gibbs phaserule requires that the chemical potentials gĮƍ and gĮƎ (Į = 1,2) be equal.That requirement provides an easy graphical method for the determinationof m1ƍ and m1Ǝ in phase equilibrium: Indeed, m1ƍ and m1Ǝ are the abscissaeof the point of contact of the common tangent of the graphs Gƍ and GƎ, seeFig. 5.5right.For fixed p and changing T the common tangent shifts, since the endpoints g2(T,p) and g1(T,p) of both phases change in their own ways. At hightemperatures the Gibbs free energy GƎ of the vapour phase is everywherebelow Gƍ so that the body minimizes its Gibbs free energy by being in thevapour phase.
Similarly, at low temperature we have Gƍ < GƎ, irrespectiveof the value of m1 and the liquid phase prevails, since it has the smallerGibbs free energy. More interesting is the case where Gƍ and GƎ intersect sothat two phases can coexist with the masses m1ƍ and m1Ǝ corresponding to29These and other examples have been worked out by Müller and Weiss in a recent book. I.Müller, W. Weiss: “Entropy and energy – a universal competition.” Springer, Heidelberg(2005).1505 Chemical Potentialsthe end point of the common tangent. For m1ƍ<m1< m1Ǝ the Gibbs freeenergy has values on that tangent, because those values are lower than thevalues of either phase.Fig. 5.5. Left: Gibbs free energy and chemical potentials. Right: Common tangent for phaseequilibriumAll this is described – not in an optimal fashion – by Gibbs, who has alarge chapter on “Geometric visualization”.30 After Gibbs it has becomecommon practice to project the common tangents for different temperaturesonto the corresponding isotherms in a (T,m1)-diagram, or a (T,X1)-diagram.The end points of those projections are then connected and form boiling andcondensation curves like those of Fig.
5.3right.The convex graphs of Fig. 5.5 are appropriate for ideal solutions, or idealalloys, where SMix is the only non-zero mixing quantity. When, on the otherhand, UMix and VMix are non-zero, they combine in the Gibbs free energy toHMix = UMix + pVMix, the heat of mixing. The heat of mixing can be bothpositive and negative. It is due to the fact that unequal next neighboursamong molecules are respectively either unfavourable or favourableenergetically.
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