Chemical potential change equation

A cell may be considered as a heterogenous system at equilibrium with restrictions. In most cells the pressure on each phase is the same and a change of pressure of the system would cause the same change of pressure on all phases. However, it is possible to construct a cell so that the various phases may have different pressures. Then the pressures of some phases may be held constant while the pressures of other phases are changed. In such cases some of the derivatives of the chemical potentials in Equation (12.86) would be zero unless matter would have to be transported across the boundary between phases in order to maintain the equilibrium conditions with a change of pressure. 

Hence, to analyze the physical significance of the activity coefficient term in Eq. (3.57), it is necessary to compare this equation with Eq. (3.52). It is obvious that when Eq. (3 52) is subtracted from Eq. (3.57), the difference [i.e., /r,- (real) - fij (ideal)] is the chemical-potential change arising from interactions between the solute particles (ions in the case of electrolyte solutions). That is. 

In order to test these predictions, attention was drawn to an empirical treatment of ionic solutions. For solutions of noninteracting particles, the chemical-potential change in going from a solution of unit concentration to one of concentration X/ is described by the equation... 

Equilibrium between a pseudobinary III—V solid solution and a ternary liquid solution is described by Equations 3 and 4. By the methods presented in the previous section, the determination of the reduced standard state chemical potential change, 0jq, can proceed in a reliable manner. The other term contained in Equations 3 and 4 is and its determination is discussed here. 

Four different methods were presented to determine the reduced standard state chemical potential change and applied to the Ga-Sb system. It is common practice to use Equation 7 and a solution model representing the stoichiometric liquid activities to determine 0. The solution model parameters are then estimated from a fit of the binary phase diagram. It has been shown that this procedure can lead to large errors in the value of 0. The use of Equation 9, however, gave the correct temperature dependence of 0 and the inclusion of activity measurements in the data base replicated the recommended values of 0Tp. 

Here p is the chemical potential just as the pressure is a mechanical potential and the temperature Jis a thennal potential. A difference in chemical potential Ap is a driving force that results in the transfer of molecules tlnough a penneable wall, just as a pressure difference Ap results in a change in position of a movable wall and a temperaPire difference AT produces a transfer of energy in the fonn of heat across a diathennic wall. Similarly equilibrium between two systems separated by a penneable wall must require equality of tire chemical potential on the two sides. For a multicomponent system, the obvious extension of equation (A2.1.22) can be written... 

At the junction of the adsorbed film and the liquid meniscus the chemical potential of the adsorbate must be the resultant of the joint action of the wall and the curvature of the meniscus. As Derjaguin pointed out, the conventional treatment involves the tacit assumption that the curvature falls jumpwise from 2/r to zero at the junction, whereas the change must actually be a continuous one. Derjaguin put forward a corrected Kelvin equation to take this state of affairs into account but it contains a term which is difficult to evaluate numerically, and has aroused little practical interest. 

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

The Gibbs equation applied to each component relates the change in chemical potential to the change in temperature and pressure. 

In equation (5.27), we used the Gibbs-Duhem equation to relate changes in the chemical potentials of the two components in a binary system as the composition is changed at constant temperature and pressure. The relationship is... 

These simple expressions may also be obtained from the chemical potentials according to Eqs. (XII-26) and (XII-32) by appropriately changing subscripts and recalling that x in these equations represents the ratio of the molar volumes, which in the present case is unity. Owing to the identity of volume fractions with mole fractions in this case, Eqs. (18) and (19) are none other than the chemical potentials for a regular binary solution in which the heat of dilution can be expressed in the van Laar form. The critical conditions (see Eqs. 2)... 

The elementary step of ion transfer is considered to take place between positions x and X2, and therefore the electrical potential drop affecting this transfer is Ao02- The ion transfer involves the renewal of the solvation shell. The change in standard chemical potential Ao f associated with this process takes place over very short distances in the interfacial region [51] and can be assumed to occur between positions X2 and x - Thus, the BV equation for the flux density /, of an ionic species i is [52]... 

Let now consider a system where in addition to the diffusion flux due to the chemical potential differences, there is also a certain flow field v(r, t). The equation for the temporal change of the order parameter field in this case is [1,4,157]... 

Peppas and Merrill (1977) modified the original Flory-Rehner theory for hydrogels prepared in the presence of water. The presence of water effectively modifies the change of chemical potential due to the elastic forces. This term must now account for the volume fraction density of the chains during crosslinking. Equation (4) predicts the molecular weight between crosslinks in a neutral hydrogel prepared in the presence of water. 

The change in Gibbs free energy, AG, for this process is the difference between the solution and gas-phase chemical potentials. Using the chemical potential of each of the species in the gas phase and in solution and Equation 8.16 gives ... 

The first equation simply states the balance in chemical potentials inside and outside of the cell. The expression for the chemical potential inside the protocell separates into a term involving the mole fraction and the chemical potential associated with the pressure difference. The work done by the cell in opposing the pressure change, assuming that the cell remains at constant volume, is given below, where the change in pressure is from p to p + tv. 

If the species is neutral, its chemical potential p% can be varied by changing its concentration and hence its activity ay. dpt — RT d nat. In this case the determination of the surface excesses offers no difficulty in principle. However, if a species is charged, its concentration cannot be varied independently from that of a counterion, since the solution must be electrically neutral. To be specific, we consider the case of a 1-1 electrolyte composed of monovalent ions A and D+. The electro capillary equation then takes the form ... 

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