Excess chemical potentials

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

This is Kirkwood s expression for the chemical potential. To use it, one needs the pair correlation fimction as a fimction of the coupling parameter A as well as its spatial dependence. For instance, if A is the charge on a selected ion in an electrolyte, the excess chemical potential follows from a theory that provides the dependence of g(i 2, A) on the charge and the distance r 2- This method of calculating the chemical potential is known as the Gimtelburg charging process, after Guntelburg who applied it to electrolytes. 

The excess chemical potential, that is the difference between the actual value and that of equivalent ideal gas system, is given by ... 

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,... 

The definition of the terms can be found in Refs. 91, 92. The excess chemical potential, computed from the direct correlation function of the... 

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]... 

Abbreviations N is the total number of particles Pdim is the average density of dimers in each of the parts I and III is the average density of monomers in each of the parts II and IV p is the average density at the middle of parts II and IV this value has been used to calculate the excess chemical potential from Eq. (148). All remaining symbols are explained in the text. 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

The term 6 is important it has the same units as temperature and at critical value (0 = T) causes the excess chemical potential to disappear. This point is known as the 6 temperature and at it the polymer solution behaves in a thermodynamically ideal way. 

In a poor solvent, where both ki and Ki/rpi generally are positive, also will be positive. At the temperature T = , the chemical potential due to segment-solvent interactions is zero according to Eq. (430- Hence the temperature is that at which the excess chemical potential is zero and deviations from ideality vanish. The free energy of interaction of the segments within a volume element is therefore zero also. 

Other thermodynamic functions may be derived from the partition function Q, or from the expression for the osmotic pressure. The chemical potential of the solvent in the solution (not to be confused with the excess chemical potential (mi —within a region of uniform segment expectancy, or density) is given, of course, by ... 

In metal deposition, the primary products form adsorbates on the electrode surface rather than a supersaturated solution. Their excess chemical potential is directly related to polarization and given by nFAE. The total excess surface energy = 2 S,o,. Otherwise, all the results described above remain valid. 

Equations (2) and (3) relate intermolecular interactions to measurable solution thermodynamic properties. Several features of these two relations are worth noting. The first is the test-particle method, an implementation of the potential distribution theorem now widely used in molecular simulations (Frenkel and Smit, 1996). In the test-particle method, the excess chemical potential of a solute is evaluated by generating an ensemble of microscopic configurations for the solvent molecules alone. The solute is then superposed onto each configuration and the solute-solvent interaction potential energy calculated to give the probability distribution, Po(AU/kT), illustrated in Figure 3. The excess... 

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... 

Figure 5 shows pn distributions for spherical observation volumes calculated from computer simulations of SPC water. For the range of solute sizes studied, the In pn values are found to be closely parabolic in n. This result would be predicted from the flat default model, as shown in Figure 5 with the corresponding results. The corresponding excess chemical potentials of hydration of those solutes, calculated using Eq. (7), are shown in Figure 6. As expected, /x x increases with increasing cavity radius. The agreement between IT predictions and computer simulation results is excellent over the entire range d < 0.36 nm that is accessible to direct determinations of po from simulation.

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)...

Note that the additional factor within the average, the n j (1 — bj), would be zero for any solvent configuration in which a solvent molecule is found in the inner shell. Thus, this expression involves a potential distribution average under the constraint that no binding in the inner shell is permitted. We can formally write the full expression for the excess chemical potential as... 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

Computing thermodynamic properties is the most important validation of simulations of solutions and biophysical materials. The potential distribution theorem (PDT) presents a partition function to be evaluated for the excess chemical potential of a molecular component which is part of a general thermodynamic system. The excess chemical potential of a component a is that part of the chemical potential of Gibbs which would vanish if the intermolecular interactions were to vanish. Therefore, it is just the part of that chemical potential that is interesting for consideration of a complex solution from a molecular basis. Since the excess chemical potential is measurable, it also serves the purpose of validating molecular simulations. 

The second necessary ingredient in the primitive quasichemical formulation is the excess chemical potential of the metal-water clusters and of water by itself. These quantities p Wm — can typically be obtained from widely available computational packages for molecular simulation [52], In hydration problems where electrostatic interactions dominate, dielectric models of those hydration free energies are usually satisfactory. The combination /t xWm — m//, wx is typically insensitive to computational approximations because the water molecules coat the surface of the awm complex, and computational errors can compensate between the bound and free ligands. 

The final ingredient that enters the calculation is the density factor pw. This is the actual density of water appropriate to the thermodynamic state intended in the calculation. For the usual case of 1 atm. pressure and 298K, this is I gem 3. The reference density in the electronic structure calculations is p° = 1 atm//entropic cost of sequestering water in the metal-water complexes, the free energies should be adjusted by —mRT In (pi 2o/p ) = —mRTIn (1354). With these inputs the excess chemical potential is readily composed as per (9.50), provided the optimal value of m is known. This is found by composing the excess chemical potential for different assumed m values and identifying the most stable case. For the dication transition metals studied, this is found to be six, consistent with experiment [12]. 

The above procedure readily yields Fig. 9.2. For estimating the excess chemical potential in the protein, again we need to know the ligation state of the metal ion. This is well known for the zinc-finger case. So we followed the above procedure, deciding what clusters to study quantum mechanically, and then composing the free energies as above. 

Pohorille, A. Wilson, M. A., Excess chemical potential of small solutes across water-membrane and water-hexane interfaces, J. Chem. Phys. 1996,104, 3760-3773... 

At equilibrium, the chemical potential for a given molecular species is constant throughout the system. The two terms on the right-hand side of (11.4) can vary in space, however, so as to add up to a constant. In an inhomogeneous system, the number density and excess chemical potential adjust so as to yield the same constant chemical potential. Due to the local nature of the excess chemical potential, it is reasonable to define an excess chemical potential at a single point in space and/or for a single molecular conformation [29]. That excess chemical potential then determines... 

The calculation of the quantum correction to the excess chemical potential is of the form... 

We can view obtaining the QFH correction to the excess chemical potential in two ways. If we simply insert (11.27) back into (11.22), this suggests that we first compute the classical excess chemical potential and then insert the classical solute into the system and evaluate... 

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