Chemical potential equalization

The van der Waals p., p. isothenns, calculated using equation (A2.5.3), are shown in figure A2.5.8. It is innnediately obvious that these are much more nearly antisynnnettic around the critical point than are the conespondingp, F isothenns in figure A2.5.6 (of course, this is mainly due to the finite range of p from 0 to 3). The synnnetry is not exact, however, as a carefiil examination of the figure will show. This choice of variables also satisfies the equal-area condition for coexistent phases here the horizontal tie-line makes the chemical potentials equal and the equal-area constniction makes the pressures equal. 

Setting the chemical potentials equal, /[Pg.116]

Chelli R, Procacci P (2002) A transferable polarizable electrostatic force field for molecular mechanics based on the chemical potential equalization principle. J Chem Phys 117(20) 9175—9189... 

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Coupled with the fact that the chemical potential equals the ionization energy of a system, Equation 7.6 implies that the eigenvalue emax will be equal [4] to the negative of the ionization potential I of a system, i.e.,... 

If all values of y are decreased, then differences between the activities also decrease - it is these differences in activity that cause the diffusion in the first place. Accordingly, after addition of a swamping electrolyte, fewer ions diffuse and so the chemical potentials equalize, with a smaller junction potential being formed. 

The chemical potential is defined as the increase in free energy of a system on adding an infinitesimal amount of a component (per unit number of molecules of that component added) when T, p and the composition of all other components are held constant. Clearly, from this definition, if a component i in phase A has a higher chemical potential than in phase B (that is, xf > pf) then the total free energy will be lowered if molecules are transferred from phase A to B and this will occur in a spontaneous process until the chemical potentials equalize, at equilibrium. It is easy to see from this why the chemical potential is... 

Let us similarly consider Zf to be in equilibrium with an external electron reservoir, characterized by its chemical potential pr, (Zf2 Zf2 r), which implies the chemical potential equalization pf = pf = p = pr. The virtual flow of electrons between Zf and r, dN = — d/Vr, gives rise to the associated quadratic energy change ... 

Because of the problem associated with Teller s theorem, discussed in Section 11, let us again examine the predictions of the central field model of molecules of Sections 9 and 10. From this model stemmed the energy relations (96)—(98). Equation (81) is again the complete expression for the sum of the eigenvalues in this simplest density description. Using equation (93), with the chemical potential equal to zero, as was demonstrated to be so for neutral molecules in the central field model, one can eliminate Fen + 2Fee by subtracting equations (81) and (93), to obtain... 

For the undissociated part of the electrolyte it is convenient to define the standard state in such a way as to make its chemical potential equal to the sum of the values for the ions in their standard states. Consider, for example, the electrolyte which ionizes thus... 

This new perspective has enriched the theory of electronic structure and chemical reactivity by both rationalizing and quantifying basic, classical ideas and rules of chemistry, e.g., the electronegativity/chemical potential equalization of Sanderson [104] or the hard-soft acids and bases (HSAB) principle of Pearson [95,105], bringing about a deeper understanding of the nature of chemical bonds and variety of reactivity preferences [3-5,11,117]. 

The equalization principle for the electronic chemical potential (equivalently, the electronegativity equalization principle) may be couched in a form reminiscent of the argument from classical thermodynamics [4], However, the chemical potential equalization principle follows most directly from the variational principle and, in particular, Eq. (32). First, define the local chemical potential by... 

Swope and Andersen describe how this formalism can be used to construct a simulation protocol for study of crystalline phases. In essence, it is necessary to explore values of v and h in search of values that make the lattice-site chemical potential equal to zero. The pressure Pc and chemical potential Pc can be evaluated using existing techniques (although p is tricky for the... 

The swelling equilibrium occurs when these two opposing tendencies balance out each other. The difference of the above introduced difference of chemical potential equals to zero. 

The symmetry is not exact, however, as a careful examination of the figure will show. This choice of variables also satisfies the equal-area condition for coexistent phases here the horizontal tie-line makes the chemical potentials equal and the equal-area construction makes the pressures equal. 

These workers have since gone on [73,74] to conduct phase-coexistence calculations in pores using these ideas. These recent studies have many notable features, including the use of free-energy perturbation formulas in lieu of thermodynamic integration to maintain the chemical potential equality, and the introduction of a restraining field to eliminate unilateral or bilateral phase changes. 

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