Calculating the Chemical Potential

By substituting unity in the form it is possible to show that 

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

The excess chemical potential is thus determined from the average of exp[— (r )/fcBr]-In ensembles other than the canonical ensemble the expressions for the excess chemical potential are slightly different. The ghost particle does not remain in the system and so the system is unaffected by the procedure. To achieve statistically significant results many Widom insertion moves may be required. However, practical difficulties are encountered when applying the Widom insertion method to dense fluids and/or to systems containing molecules, because the proportion of insertions that give rise to low values of i (r ) falls dramatically. This is because it is difficult to find a hole of the appropriate size and shape. 

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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. 

In the case of a Lennard-Jones fluid, the knowledge of the bulk density in the nonreactive part is all that is needed to calculate the chemical potential. Actually, one can use the equation of state of Nicolas et al. [115] (or the... 

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... 

Eqs. (1,4,5) show that to determine the equilibrium properties of an adsorbate and also the adsorption-desorption and dissociation kinetics under quasi-equilibrium conditions we need to calculate the chemical potential as a function of coverage and temperature. We illustrate this by considering a single-component adsorbate. The case of dissociative equilibrium with both atoms and molecules present on the surface has recently been given elsewhere [11]. 

As a result of Eq. (11) we are able to calculate the chemical potential of any molecule X in any liquid system S, relative to the chemical potential in a conductor, i.e. at the North Pole. Hence, COSMO-RS provides us with a vehicle that allows us to bring any molecule from its Uquid state island to the North Pole and from there to any other liquid state, e.g. to aqueous solution. Thus, given a liquid, or a reasonable estimate of AGjis of a soUd, COSMO-RS is able to predict the solubility of the compound in any solvent, not only in water. The accuracy of the predicted AG of transfer of molecules between different Uquid states is roughly 0.3 log units (RMSE) [19, 22] with the exception of amine systems, for which larger errors occur [16, 19]. Quantitative comparisons with other methods will be presented later in this article. 

Usnally, only very dilute solutions can be considered ideal. In most aqueous solutions, ions are stabilized because they are solvated by water molecules. As the ionic strength is increased, ions interact with each other. Thus, when calculating the chemical potential of species i, a term that takes into account the deviation from ideal conditions is added. This term is called an excess term and can be either positive or negative. The term usually is written as 7 riny., where y. is the activity coefficient of component i. The complete expression for the chemical potential of species i then becomes... 

In applications concerning equilibria in the micellar solution it is often preferable to calculate the chemical potentials ju of the components instead of explicitly calculating the electrostatic free energy Ge. The point of using n j s is that these are determined by the calculated ion concentrations at the border of the free micellar volume where the electrostatic potential and field are zero302. For an ion... 

Assuming the free energy of pure A at temperature 600K is set to zero, calculate the chemical potential of A in the solution of NA - 0.6. 

Let us consider a solution composed of hard-sphere droplets of a single size g present in a multicomponent solvent. The expression for the osmotic pressure due to the hard spheres allows us to calculate the chemical potentials of the components in the mixed solvent. Subsequently, the Gibbs—Duhem equation is used to calculate the chemical potentials of the hard-sphere droplets.26 The mole fraction X, of the components and their volume fraction d> are related via the expressions... 

Long ago, Langmuir suggested that the rate of deposition of particles on a surface is proportional to the density of particles in the vicinity of the surface and to the available area on the surface [1], However, the calculation of the available area is still an open problem. In a first approximation, one can assume that the available area is the total area of the surface minus the area already occupied by the adsorbed particles [1]. A better approximation can be obtained if the adsorbed particles, assumed to have the shape of a disk, are in thermal equilibrium on the surface, either because of surface diffusion and/or of adsorption/desorption kinetics. In this case, one can use one of the empirical equations available for the compressibility of a 2D gas of hard disks, calculate the chemical potential in excess to that of an ideal gas [2] and then use the Widom relation between the area available to one particle and its excess chemical potential on the surface (the particle insertion method) [3], The method is accurate at low densities of adsorbed particles, where the equations of state are accurate, but, in general, poor at high concentrations. The equations of state for hard disks are based on the virial expansion and only the first few coefficients of this... 

Thus, the problem of calculating the chemical-potential change due to the interactions between one ionic species and the assembly of all the other ions has been reduced to the following problem On a time average, how are the ions distributed around any specified ion If that distribution became known, it would then be easy to calculate the electrostatic potential of the specified ion due to the other ions and then, by Eq. (3.3), the energy of that interaction. Thus, the task is to develop a model that describes the equilibrium spatial distribution of ions inside an electrolytic solution and then to describe that model mathematically. 

It will be recalled (see Section 3.3.1) that it was the potential at the surface of the reference ion which needed to be known in order to calculate the chemical-potential changearising from the interactions between a particular ionic species / and the rest of the ions of the solution, i.e., one needed to know V cioud Eq. (3.3),... 

Gas Phase At lower pressures (say, less than 50 atm), the partial pressure can be used to calculate the chemical potential of a gaseous species in a mixture. The expression is... 

Finally, we shall calculate the chemical potential as a function of T and p. Since h and s have already been evaluated this is easily done since yi = h-Ts, We obtain, using (12.23) and (12.26),... 

For calculating the chemical potential of water in the liquid solution, or ice phase. Holder, Corbin, Papadopoulos generated chemical potential, enthalpy, and heat capacity functions for gas hydrates at temperatures between 150 and 300 K and derived... 

In (3.69) we have an explicit expression for the activity coefficient yldeal gas, which measures the extent of deviation of the chemical potential from the ideal-gas form. The quantity pg(R, ), is the local density of particles around a given particle that is coupled to the extent of to the rest of the system. Note that (3.67) is not a simple integral involving g(R). A more detailed knowledge of the function g(R, ) is required to calculate the chemical potential. 

Another difference from PR2 is that the subroutine SLCHEMPOT calculates die chemical potentials of both components in both phases rather than the fugacity coefficients. Also, SLCHEMPOT calls CONVERT to change the weight to mole fractions to calculate the chemical potential, and it calls CONVERT again to change the fractions back before returning to the main program. 

Subroutine SLCHEMPOT is used to calculate the chemical potentials of the components with the Sanchez-Lacombe EOS. The line numbers start from 1 in this subroutine. 

SL3LV is very similar to SL2 and PR3LV. In this program weight fractions are entered and are converted to mole fractions in the subroutines that are used to calculate the chemical potential. 

Consequently, such an expression does not allow us to calculate the chemical potentials directly... 

Calculate the chemical potential of sulfuric acid as a function of the chemical potentials of nitric acid and the two solids. 

In the preceding sections, we have considered only particle insertions for the estimation of the chemical potential. However, in principle, particle removals could also be used to calculate the chemical potential according to... 

The Gibbs-Duhem equation can be used to calculate the chemical potential of the solute from that of the solvent in a binary ideal system. The Gibbs-Duhem equation, Eq. (11.96), for a binary system (T,p constant) is... 

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