Chemical potential, defined

The argument is easily extended to the grand canonical formalism. We simply absorb the prefactor into the chemical potential, defining... 

Xv being a phenomenological interaction parameter for the noncombinatorial part of the solute (polymer) chemical potential, defined on a volume fraction basis. Equations similar to equation (1.9) and (1.12) serve to define x on a segment fraction basis, x -... 

Equation (4) shows that the chemical potential, defined as the partial derivative of f with respect to n, at constant A, temperature, and nj (j t i), equals the chemical potential u -j defined on the basis of f0. This arises because Af depends only on ip and r and does not depend separately on n-j. A microemulsion is in equilibrium with an excess dispersed phase when the chemical potentials pf are equal to those in the excess dispersed phase. 

The local balance equation (45) can then be reformulated with the generalized (or gradient-dependent) chemical potential defined by Eq. (50) into... 

We should not close this section without touching upon some delicate technical point. A priori even for a discrete chain the critical chemical potential defined as chemical potential pet segment of an infinitely long chain, does not exist outside the -expansion. We already encountered this problem in Sect. 7.2, where we found that Tc, proportional to p, for d < 4 in naive perturbation theory suffers from infrared singularities. That problem has been considered in the field theoretic framework. The results, expressed in polymer language, show that the u-expansion is not invalidated, as long as we consider quantities which do not involve explicitly. Almost all the quantities of interest to us are of this type. Only in the equation of state relating Pp n) and Cp n) does an explicit contribution pi n occur. But even... 

Instead of employing the molar chemical potentials it is more convenient in this case to employ the specific chemical potential defined by... 

The chemical potential defined in (A.8) is the partial molar Gibbs energy. 

The AIM chemical potentials defined by the partial functional derivatives of equation (35), calculated for the fixed external potential and the frozen embedding densities pp a r) of the remaining subsystems, are equalized only when the subsystems are mutually open [4,5], This is the case in the global equilibrium state considered in the preceding section. In what follows we shall denote such open subsystem condition by the vertical broken fines in the symbolic representation of the molecular system as a whole, Mg = (a fi y. ..), in the global (g, intersubsystem) equilibrium of the ground-state of an externally open system ... 

Chemical potential Defined in terms of Gibbs free energy, it is a measure of how the Gibbs function for a system changes when a specified amount of component is added to the system. 

The key assumption is that the chemical potentials defined thermodynamically in (92), and phenomenologically in (91), are the same quantity. This is a reasonable approximation so long as equilibrium considerations can be applied locally. The chemical potential follows from the variational derivative in (92),... 

The grouping of the related efforts into a single variable corresponds to the use of an apparent chemical potential defined as a partial derivative ... 

It is clear that insofar as we are interested in the solvation properties of the system, the first term on the rhs of (7.20) carries no relevant information. In fact, it would be the same for different solvents at the same solute density Qg. Therefore, we focus our attention on the pseudo-chemical potential, defined by... 

The key thermodynamic characteristic of a component / in a mixture is its chemical potential, defined as... 

The chemical potential (defined below) is equal, but has an opposite sign, to the Mulliken electronegativity. 

We have studied the double-proton transfer reaction in the adenine-uracil complex. The calculations were performed at the HF/6-311G level, and the REF was obtained by numerical differentiation of the chemical potential defined in terms of the frontier orbital energies, as indicated in Equation 12.6. The double-proton transfer in the complex adenine-uracil is a stepwise reaction in which the two protons are transferred... 

In the [A, A 5,v,vs] representation, the chemical potential /Xjv, which is a generalization of the chemical potential defined earlier in the spin-free description, and the spin potential ns are given by ref. 71... 

By fitting the experimental results to Equation (9.29) both and Oh may be determined at the chemical potentials defined by equilibrium with the reversible electrode. 

At this point, we note that the chemical potential defined from a stepwise insertion procedure as described above can also be written as... 

Figure 6.10 The stable ol-AIFj (0112) surfaces, including terminations derived from the (1x1) and ( 2 x 2) 3F terminations, as a function ofHF and H2O effective chemical potential (defined in Equation (6.9)) and partial pressure and temperature. The terminations derived om the (1 x I) 3F termination are denoted by an asteric. The area within the small rectangle is the accessible region of the phase diagram at 300 K and the region within the large rectangle is the accessible region at 600 K (see text for details)...

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