Chemical potential gauge invariance

Chemical behaviour depends on chemical potential and electromagnetic interaction. Both of these factors depend on the local curvature of space-time, commonly identified with the vacuum. Any chemical or phase transformation is caused by an interaction that changes the symmetry of the gauge field. It is convenient to describe such events in terms of a Lagrangian density which is invariant under gauge transformation and reveals the details of the interaction as a function of the symmetry. The chemically important examples of crystal nucleation and the generation of entropy by time flow will be discussed next. The important conclusion is that in all cases, the gauge field arises from a symmetry of space-time and the nature of chemical matter and interaction reduces to a function of space-time structure. 

T.l.cf) is related to the chemical potential of pure i, corrected for by a quantity that is readily evaluated by use of (3.5.2) this is a direct consequence of gauge invariance. Equation (3.5.13) also specifies in detail how to evaluate the standard chemical potential referred to molarity. 

There finally remains the problem of guaranteeing that the chemical potentials specified so far shall actually remain gauge invariant with respect to the use of reference or standard chemical potentials which differ from those adopted in the present section. This matter is to be handled in Exercise 3.5.5. That such invariance can always be maintained should become clear on reflection of the meaning of Eqs. (3.5.5),... 

Equivalence of the Various Formulations Gauge Invariance Requirements for Reference Chemical Potentials... 

Any formulation that satisfies (3.5.4d) represents an acceptable expression that guarantees the invariance of ixi under a change of composition variables. By analogy to other field theories we shall refer to this process as maintaining the gauge invariance of the chemical potential. There obviously is considerable leeway in how to proceed hence, we note that (3.5.4d) may sensibly be broken up into two portions a part which relates the various reference chemical potentials fXi(T, P, qf) at fixed compositions, and a part involving relations between composition variables / Tln(y, /), in which qi =Xi,Ci(T, F),orm/. 

Equation (3.5.13) bears the important message that iXiiT, l.c ) is related to the chemical potential of pure i, corrected for by a quantity that is specified by Eq. (3.5.2) this is a direct consequence of gauge invariance. Eq. (3.5.13) shows in detail how to determine standard chemical potentials in terms of molarity. Eq. (3.5.13) is also consistent with Eq. (2.5.12b) as applied to pure materials. Next, equate (3.5.12a) with (3.5.12b), taking account of (3.5.13) we obtain ... 

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