Replacement Chemical Potential

1 Replacement Chemical Potential As x exceeds 1/2 and increases further, A2 becomes negative and its absolute value increases. The unfavorable polymer-solvent interaction can be sufficiently strong to cause the solution to separate into two phases. We will examine the phase diagram of the solution in the mean-field theory for a system of a fixed volume. 

This Aptfep is also calculated directly from AAnijx (Eq. 2.7) using 

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To find thermodynamic relationships for charged systems, we just replace chemical potentials in equations for uncharged systems by electrochemical potentials. For example, for equilibrium of a charged particle at a phase boundary, we have, in analogy to Eq. (24) of Chapter 6,... 

Replace chemical potential with fugacity and find out how fugacity depends on the species and the state of the mixture. This the subject of Sections 4.2 and 4.3. 

Figure 2.9. Replacement chemical potential (k T), plotted as a function of for...

Both convention and convenience suggest use of the fugacity in practical calculations in place of the chemical potential ]1. Equation 218 is then replaced by the equal fugacity criterion which follows directiy from equation 160 ... 

Determining the cell potential requites knowledge of the thermodynamic and transport properties of the system. The analysis of the thermodynamics of electrochemical systems is analogous to that of neutral systems. Eor ionic species, however, the electrochemical potential replaces the chemical potential (1). 

One criterion for the anode material is that the chemical potential of lithium in the anode host should be close to that of lithium metal. Carbonaceous materials are therefore good candidates for replacing metallic lithium because of their low cost, low potential versus lithium, and wonderful cycling performance. Practical cells with LiCoOj and carbon electrodes are now commercially available. Finding the best carbon for the anode material in the lithium-ion battery remains an active research topic. 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

There is one other three-phase equilibrium involving clathrates which is of considerable practical importance, namely that between a solution of Q, the clathrate, and gaseous A. For this equilibrium the previous formulas and many of the following conclusions also hold when replacing fiQa by fiQL, the chemical potential of Q in the liquid phase. But a complication then arises since yqL and the difference are not only... 

When the adsorbed components are electrically charged, then the partial molar Gibbs energy of the charged component depends on the charge of the given phase, and thus the chemical potentials in the above relationships must be replaced by the electrochemical potentials. The Gibbs adsorption isotherm then has the form... 

Since the energy of electrons in a material is specified by the Fermi level, ep, the flow of electrons across an interface must likewise depend on the relative Fermi levels of the materials in contact. Redox properties are therefore predicted to be a function of the Fermi energy and one anticipates a simple relationship between the Fermi level and redox potential. In fact, the Fermi level is the same as the chemical potential of an electron. Clearly when dealing with charged particles, the local energy levels e are increased by qV, where q is the charge on the particle and V is the local electrostatic potential. The e, should therefore be replaced by e,- + qV and so... 

The Fukui function is primarily associated with the response of the density function of a system to a change in number of electrons (N) under the constraint of a constant external potential [v(r)]. To probe the more global reactivity, indicators in the grand canonical ensemble are often obtained by replacing derivatives with respect to N, by derivatives with respect to the chemical potential /x. As a consequence, in the grand canonical ensemble, the local softness sir) replaces the Fukui function/(r). Both quantities are thus mutually related and can be written as follows ... 

In everyday chemical usage, the word equilibrium means that a reaction has stopped, e.g. because it has reached its position of minimum chemical potential or because one reactant has been consumed completely. In this electroanalytical context, however, we say that we are making a measurement of potential at equilibrium , yet the system has clearly not reached a true equilibrium because as soon as the voltmeter is replaced with a connection having zero resistance, a cell reaction could commence. What then do we mean by equilibrium in this electroanalytical context ... 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

It should be noted that the condition of a dilute solution was introduced into the considerations for two reasons primarily, in order that it would be possible to replace the activities by concentrations and thus determine the equilibrium concentrations on the basis of (2.3.3) and, secondarily, in order for it to be possible to neglect the effect of pressure on the chemical potentials of the components whose electrochemical potentials appear in (2.3.2). Because of the differing ionic concentrations in solutions 1 and 2, the osmotic pressures in these solutions are not identical and this difference must be compensated by external pressure. A derivation considering the effect of pressure can be found, for example in [9] or p. 191 of [18]. 

In multicomponent systems, the single diffusivity is replaced by a multicomponent diffusion matrix. By going through similar steps, it can be shown that the [D] matrix must have positive eigenvalues if the phase is stable. In a multicomponent system, the diffusive flux of a component can be up against its chemical potential gradient except for eigencomponents. 

For the ionic surfactants (1-1 type), we should take account of the electrically charged species and the possibility of doing electrical work. The micelle may be regarded as a charged pseudo-phase, and the chemical potential is replaced by the electrochemical potential (12). The effective electrical work in micelle formation is... 

For a polydisperse polymer, analysis of sedimentation equilibrium data becomes complex, because the molecular weight distribution significantly affects the solute distribution. In 1970, Scholte [62] made a thermodynamic analysis of sedimentation equilibrium for polydisperse flexible polymer solutions on the basis of Flory and Huggins chemical potential equations. From a similar thermodynamic analysis for stiff polymer solutions with Eqs. (27) for IT and (28) for the polymer chemical potential, we can show that the right-hand side of Eq. (29) for the isotropic solution of a polydisperse polymer is given, in a good approximation, by Eq. (30) if M is replaced by Mw [41],... 

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