Hardness chemical potential

Here we review our work aimed at correlating the reactivity of a series of M(II) N4-ligands, see Figure 12.1, (M = Co, Fe, Mn and N4 = porphyrin (P), phthalocyanine (Pc), teraphenylporphyrin (TPP), tetrabenzoporphyrin (TBP) and tetraazaporphyrin (TAP)) towards the electrocatalytic oxidation of 2-mercaptoethanol. Different effects will be analysed, namely the role of the metal atoms, the role of the N4 functionalisation, solvent and the impact of the adsorption on the electrode on the electrochemical activity. The whole machinery of DFT and the notions of hardness, chemical potential, intramolecular hardness and elec-trophilicity are used to better quantify these effects and discriminate between the examined molecular complexes. 

Chattaraj PK, Nath S, Sannigrahi AB (1994) Hardness, chemical potential, and valency profiles of molecules under internal rotations. J Phys Chem 98(37) 9143-9148... 

Attard P 1993 Simulation of the chemical potential and the cavity free energy of dense hard-sphere fluids J. Chem. Phys. 98 2225-31... 

The hard-sphere excess chemical potential p that appears on the left-hand side of the hierarchy is related to the compressibility factor Z = pV/pkT,... 

Let us consider an V-component fluid in a volume V, at temperature T, and at chemical potentials /r = mi, > Mv - The fluid is in contact with an impermeable solid surface. We assume that the fluid particles interact between themselves via the pair potential denoted by u pir), and interact with the confining surface via the potential (a,f3= 1,2,. ..,V). The potential v ir) contains a hard-wall term to ensure that the solid surface is impermeable. For the sake of convenience, the hard-wall term is assumed to extend into the bulk of the solid [46,47], such that the Boltzman factor (r), and the local density Pa r) are cutoff at a certain distance z = z, ... 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

Finally, let us discuss the adsorption isotherms. The chemical potential is more difficult to evaluate adequately from integral equations than the structural properties. It appears, however, that the ROZ-PY theory reflects trends observed in simulation perfectly well. The results for the adsorption isotherms for a hard sphere fluid in permeable multiple membranes, following from the ROZ-PY theory and simulations for a matrix at p = 0.6, are shown in Fig. 4. The agreement between the theoretical results and compu-... 

Now, we would like to investigate adsorption of another fluid of species / in the pore filled by the matrix. The fluid/ outside the pore has the chemical potential at equilibrium the adsorbed fluid / reaches the density distribution pf z). The pair distribution of / particles is characterized by the inhomogeneous correlation function /z (l,2). The matrix and fluid species are denoted by 0 and 1. We assume the simplest form of the interactions between particles and between particles and pore walls, choosing both species as hard spheres of unit diameter... 

Our first example of an application of the potential distribution formula [Eq. (5)] is in the calculation of the excess chemical potential of a simple hydrophobic solute dissolved in water (Hummer et al., 1998a, 2000), which we consider to be a hard-core particle that excludes all water molecules from its molecular volume. We note that these purely... 

Fig. 6. Excess chemical potential of hard-sphere solutes in SPC water as a function of the exclusion radius d. The symbols are simulation results, compared with the IT prediction using the flat default model (solid line). (Hummer et al., 1998a)...

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

Chemical hardness measures the resistance to a chemical change. This may be seen by considering a reaction between two molecules C and D in which electrons are transferred from D to C. Initially the two chemical potentials are ... 

To get an approximate expression for the chemical hardness, start with an expression for the electronic chemical potential. Let a hypothetical atom have an energy, UQ. Subtract one electron from it. This costs I = ionization energy. Alternatively, add one electron to it. This yields A = electron affinity. The derivative = electronic chemical potential = p = AU/AN = (I + A)/2. The hardness is the derivative of the chemical potential = r = Ap/AN = (I - A)/2. 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

A remarkable fact is that Equation 2.19 is the same as Equation 2.14, and Equation 2.20 is the same as Equation 2.17 if one ignores the Dirac delta function, although Equations 2.14 and 2.17 result from the ensemble approach, while Equations 2.19 and 2.20 result from the smooth quadratic interpolation. Thus, the expressions given by Equations 2.19 and 2.20 are fundamental to evaluate the chemical potential (electronegativity) and the chemical hardness. 

For the chemical potential [28] and the chemical hardness [29,30] one finds that... 

It is important to mention that the chemical potential and the hardness, p, and 17, are global-type response functions that characterize the molecule as a whole, while the electronic density p(r), the Fukui function fir), and the dual descriptor A/(r) are local-type response functions whose values depend upon the position within the molecule. 

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