Chemical potential concentration dependence

In the case of ions in solution, and of gases, the chemical potential will depend upon concentration and pressure, respectively. For ions in solution the standard chemical potential of the hydrogen ion, at the temperature and pressure under consideration, is given an arbitrary value of zero at a specified concentration... 

Thus, in principle, we could determine the adsorption excess of one of the components from surface tension measurements, if we could vary ii independently of l2. But the latter appears not to be possible, because the chemical potentials are dependent on the concentration of each component. However, for dilute solutions the change in p for the solvent is negligible compared with that of the solute. Hence, the change for the solvent can be ignored and we obtain the simple result that... 

For example, if a Pt rod containing some Zn is placed in contact with an aqueous ZnCl2 solution, Zn+2 ions are transferred between the solution and the metal, until Eq. (36) holds. It takes only a minuscule transfer of Zn+2 to build up a very appreciable electrostatic potential, because electrons from the rod cannot transfer into the solution. Chemical potentials, which depend on concentrations, can therefore be calculated neglecting the ionic transfer. If the electrons could also... 

The constancy of phenomenological coefficients L may be maintained by applying appropriate constraints to vary the force X in the relationship J=LX. The values of L reflect the nature of the membrane, and can control the force X. If a thin homogeneous membrane is exposed to the same concentrations at each surface, flow is induced solely by the electric potential difference, and L is constant with the variation of X. However, if X is the chemical potential difference, dependent upon the bath solute concentrations, then L becomes... 

Note In principle, the ion activity coefficient of a salt (y+) can be determined, but not those of individual ions (y and y,), because their concentrations cannot be varied independently. Nevertheless, the activity coefficients of individual ions are very useful, and as mentioned, one tries to calculate them from theory. Ion-selective electrodes measure chemical potentials (which depend on activities, not concentrations), but the standard potential is unknown. For the measurement of pH, which is the negative logarithm of the hydrogen ion activity, one has therefore arbitrarily chosen a reference potential for a certain buffer, which potential is of course as close to the real one as theory permits it to be calculated. 

Chemical potentials also depend upon the concentrations of substances, and in many cases, these dependencies are well known. Therefore, it is possible to utilize the measured voltages to find ion concentrations, especially solubilities and pH values (see Sect. 22.7). This method is calledpotentiometry and is widely applied in analytic chemistry. 

The standard chemical potential, pf, depends only on the pressure and the temperature, while the logarithmic term includes the activity, Oj, which depends on the concentration as will be discussed later in this chapter. 

As before, if the derivatives are evaluated at the state of equilibrium, the chemical potentials of the two parts must be equal. Hence the first term vanishes. Furthermore, if system 1 is small compared to system 2, the change in the chemical potential (which depends on the concentrations) with respect to Nk of system 2 will be small compared to the corresponding change in system 1. That... 

The real behavior of systems is described by the activity coefficient y,. Instead of the concentration C of a dissolved species, one uses the activity a, = c y,. In the light of the Debye-Hiickel theory, y takes care of the electrostatic interactions of the ions. This is the main interaction for charged species in comparison with the smaller dipole and Van der Waals forces, which may be important in the case of uncharged species, but which are not included in the Debye-Huckel theory. The chemical potential p depends on the concentration according to Equation 1.37. 

The first term describes contributions to the chemical potentials not dependent on ion concentration. The concentration dependence here enters through an ideal gas... 

Analytic teclmiques often use a time-dependent generalization of Landau-Ginzburg ffee-energy fiinctionals. The different universal dynamic behaviours have been classified by Hohenberg and Halperin [94]. In the simple example of a binary fluid (model B) the concentration difference can be used as an order parameter m.. A gradient in the local chemical potential p(r) = 8F/ m(r) gives rise to a current j... 

As noted above, all of the partial molar quantities are concentration dependent. It is convenient to define a thermodynamic concentration called the activity aj in terms of which the chemical potential is correctly given by the relationship... 

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

Formula for the chemical potentials have been derived in terms of the formation energy of the four point defects. In the process the conceptual basis for calculating point defect energies in ordered alloys and the dependence of point defect concentrations on them has been clarified. The statistical physics of point defects in ordered alloys has been well described before [13], but the present work represents a generalisation in the sense that it is not dependent on any particular model, such as the Bragg-Williams approach with nearest neighbour bond energies. It is hoped that the results will be of use to theoreticians as well as... 

The chemical potential of one half-cell depends on the concentration ct of the compounds which react at the electrode ... 

It will be observed that the chemical potentials depend appreciably on the chain length, represented by x only at low concentrations. The influence of the chain length vanishes with increase in concentration of... 

For snfficiently dilute solutions the concentration dependence of chemical potential is given similarly by... 

Solntions in which the concentration dependence of chemical potential obeys Eq. (3.6), as in the case of ideal gases, have been called ideal solutions. In nonideal solntions (or in other systems of variable composition) the concentration dependence of chemical potential is more complicated. In phases of variable composition, the valnes of the Gibbs energy are determined by the eqnation... 

The departure of a system from the ideal state is due to interaction forces between the individual particles contained in the system. The dependence of chemical potential of a species on its concentration can be written as... 

FIG. 8 Dependence of the adsorbed amount of C12E4 on the applied potential. Concentration of C12E4 in nitrobenzene is 50 (curve 1), 20 (curve 2), 10 (curve 3), and 5 (curve 4)mmoldm (From Ref. 47, reproduced by permission. The Chemical Society of Japan.)... 

The next problem is to find an expression for Asg. This entropy difference is a function of the particle volume fractions in the dispersion ( ) and in the floe (<(> ). As a first approximation, we assume that Ass is independent of the concentration and chain length of free polymer. This assumption is not necessarily true the floe structure, and thus < >f, may depend on the latter parameters because also the solvent chemical potential in the solution (affected by the presence of polymer) should be the same as that in the floe phase (determined by the high particle concentration). However, we assume that these effects will be small, and we take as a constant. 

Situations that depart from thermodynamic equilibrium in general do so in two ways the relative concentrations of different species that can interconvert are not equilibrated at a given position in space, and the various chemical potentials are spatially nonuniform. In this section we shall consider the first type of nonequilibrium by itself, and examine how the rates of the various possible reactions depend on the various concentrations and the lattice temperature. 

The first term in parentheses has the following meaning If a reference electrode is used whose potential is determined by a simple exchange reaction involving the anion A, the electrode potential A with respect to this reference will depend on the concentration of the anion, and d(pA — d — dp a- / eo- The term dpB+ + dpA- denotes the change in the chemical potential of the uncharged species AB, and is determined by the change in the mean activity 2RT d In a . Hence ... 

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