Chemical potential discontinuous

The second term of the r.h.s of Eq. (27) permits to avoid the chemical potential discontinuity [4, 24], and thus the equalization principle can be fulfilled [3]. To understand it, let us consider two fragments 2 and within a molecular framework which at equilibrium must obey the condition i.e., the chemical potential of the donor... 

It is generally assumed that isosteric thermodynamic heats obtained for a heterogeneous surface retain their simple relationship to calorimetric heats (Eq. XVII-124), although it may be necessary in a thermodynamic proof of this to assume that the chemical potential of the adsorbate does not show discontinu-... 

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

The interface region in a composite is important in determining the ultimate properties of the composite. At the interface a discontinuity occurs in one or more material parameters such as elastic moduli, thermodynamic parameters such as chemical potential, and the coefficient of thermal expansion. The importance of the interface region in composites stems from two main reasons the interface occupies a large area in composites, and in general, the reinforcement and the matrix form a system that is not in thermodynamic equiUbhum. 

Second-order phase transitions are those for which the second derivatives of the chemical potential and of Gibbs free energy exhibit discontinuous changes at the transition temperature. During second-order transitions (at constant pressure), there is no latent heat of the phase change, but there is a discontinuity in heat capacity (i.e., heat capacity is different in the two... 

In a foregoing section, we mentioned that field forces (e,g., of the electric or elastic field) can cause an interface to move. If they are large enough so that inherent counterforces (such as interface tension or friction) do not bring the boundary to a stop, the interface motion would continue and eventually become uniform. In this section, however, we are primarily concerned with boundary motions caused by chemical potential changes. From irreversible thermodynamics, we know that the dissipated Gibbs energy of the discontinuous system is T-ab, where crb here is the entropy production (see Section 4.2). Since dG/dV = dG/dV = crb- T/ A < ), we have with Eqn. (4.8) at the boundary b... 

Discontinuous systems. The membrane is regarded as a surface of discontinuity, hindering the movement of the different ions and molecules. The driving forces are in this case the differences in electrical potential, pressure and chemical potential (765, 166) [see equation (4)]. 

To differentiate between the variety of phase equilibria that occur, Ehrenfest proposed a classification of phase transitions based upon the behavior of the chemical potential of the system as it passed through the phase transition. He introduced the notion of an th order transition as one in which the nth derivative of the chemical potential with respect to T or p showed a discontinuity at the transition temperature. While modern theories of phase transitions have shown that the classification scheme fails at orders higher than one, Ehrenfest s nomenclature is still widely used by many scientists. We will review it here and give a brief account of its limitations. 

In a first-order transition, the first derivative of Gm (chemical potential, p) shows a discontinuity. Since... 

Anywhere a chemical potential increment or gradient exists, an elementary separation step can occur. Anywhere random flow currents exist, separation is dissipated. Thus random flow currents are parasitic in regions where incremental chemical potential is used for separation. These currents should thus be eliminated, insofar as possible, in regions where electrical, sedimentation, and other continuous (c) fields are generating separations. Likewise, they should not be allowed to transport matter over discontinuous (d) separative interfaces such as phase boundaries or membrane surfaces. However, they are nonparasitic in bulk phases (removed from the separative interface) where only diffusion occurs. Here, in fact, they aid diffusion and speed the approach to equilibrium. This positive role is recognized in the following category of flow. 

In comparing separation techniques, we generally find a striking difference in methods based on continuous (c) chemical potential profiles and those involving discontinuous (d or cd) profiles. There is, for example, a glaring contrast in instrumentation, applications, experimental techniques, and the capability for multicomponent separations between the two basic static systems, Sc (e.g., electrophoresis) and Sd (e.g., extraction). Similarly, there... 

The Ehrenfest17 classification of phase transitions (first-order, second-order, and lambda point) assumes that at a first-order phase transition temperature there are finite changes AV 0, Aft 0, AS VO, and ACp VO, but hi,lower t = hi,higher t and changes in slope of the chemical potential /i, with respect to temperature (in other words (d ijdT)lowerT V ((9/i,7i9T)higherT). At a second-order phase transition AV = 0, Aft = 0, AS = 0, and ACp = 0, but there are discontinuous slopes in (dV/dT), (dH/<)T), (OS / <)T), a saddle point in and a discontinuity in Cp. A lambda point exhibits a delta-function discontinuity in Cp. 

Many early attempts were made to correct the Kelvin equation (see Brunauer, 1945). As already indicated, when the Kelvin equation is applied to capillary condensation it is normally assumed that the reduction in chemical potential is entirely dependent on the curvature of the meniscus. This assumption implies a sharp discontinuity between the state of the adsorbed layer and the condensate. However, as Detjaguin first pointed out (1957), the transition is more likely to be a gradual one. This problem was also discussed by Everett and Haynes (1973). 

The application of the Nemst equation has no basis. However, because the underlying idea of a discontinuous change in chemical potential near the macroion surface is so prevalent, I wish to prove that it leads to the system not being at thermal equilibrium. 

A partitioning function for a system of rigid rod-like particles with partial orientation around an axis is derived from the use of a modified lattice model. The free energy of mixing is shown as a function of the mole numbers, the axis ratio of the solute particles and a disorientation parameter this function passes through a minimum with increase in the disorientation parameter. The chemical potentials display discontinuities at the concentration at which the minimum appears and then separation into an isotropic phase and a somewhat more concentrated anisotropic phase arises. The critical concentration, v, is given in the form 13) ... 

The Two-film theory enables the difficultly accessible chemical potential difference A/z to be replaced by the concentrations of the gas in the gas phase, liquid phase and in the phase boundary c, only then the concentration change from phase to phase is discontinuous but makes a jump, as demonstrated on the left of Fig. 4.1. 

The physical basis of mass transfer in the solid/liquid (S/L) system is the leveling out of the chemical potentials /z between the two phases. In the present case, the change from A/z to Ac is simple to understand, see Fig. 5.22. The change of the concentration of the solid material cs to its saturation concentration Cs at the interface remains discontinuous cs Cj. 

Figure 9.9 Adsorption isotherms for a lattice gas model consisting of shell and axial sites at the indicated temperatures (reduced by the pair interaction well depth) and various values of the reduced chemical potential. While mean field (MF) results exhibit a discontinuous shell-filling transition at T = 1, essentially exact Monte Carlo (MC) results show a near discontinuity there. (Adapted from Ref. [31, 32])...

For most systems, the mean temperature of a phase change shifts toward low temperatures with the reduction of the cluster size [8]. (Recently, exceptions to this general behavior have been observed [9-11].) The phase changes of finite-size systems are rounded-off and occur smoothly through the points of equal chemical potentials, even though they are sharp and effectively discontinuous in... 

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