Chemical potentials in solution

We propose the study of Lennard-Jones (LJ) mixtures that simulate the carbon dioxide-naphthalene system. The LJ fluid is used only as a model, as real CO2 and CioHg are far from LJ particles. The rationale is that supercritical solubility enhancement is common to all fluids exhibiting critical behavior, irrespective of their specific intermolecular forces. Study of simpler models will bring out the salient features without the complications of details. The accurate HMSA integral equation (Ifl) is employed to calculate the pair correlation functions at various conditions characteristic of supercritical solutions. In closely related work reported elsewhere (Pfund, D. M. Lee, L. L. Cochran, H. D. Int. J. Thermophvs. in press and Fluid Phase Equilib. in preparation) we have explored methods of determining chemical potentials in solutions from molecular distribution functions. 

The amount of substance present in the micellar state, cmjc = mnmic / NA may exceed the concentration of it in the molecular solution by several orders of magnitude. The micelles thus play a role of a reservoir (a depot) which allows one to keep the surfactant concentration (and chemical potential) in solution constant, in cases when surfactant is consumed, e.g. in the processes of sol, emulsion and suspension stabilization in detergent formulations, etc. (see Chapter VIII). A combination of high surface activity with the possibility for one to prepare micellar surfactant solutions with high substance content (despite the low true solubility of surfactants) allows for a the broad use of micelle-forming surfactants in various applications. 

We now consider chemical reactions in which one or more of the products or reactants is a solid or liquid. In the present chapter the discussion will be limited to cases where each of these solids or liquids is present in the system as a pure phase, i.e. when they do not take into solution appreciable amounts of the other components. Under these conditions the free energy of mixing, which has been shown to be an important part of the driving force of reaction, is limited to the gaseous phase. (The discussion of the case where there is an additional free energy of mixing in the condensed phases depends on a knowledge of the chemical potentials in solutions and will be deferred to Chapter 10.)... 

The form of Equation 13.16 is familiar from the solution chemistry of neutral species. The chemical potential in solution is separated in a contribution from solvation (the excess chemical potential) and a gas-phase term describing the chemical reactivity of the species. The chemical reference state for hydrogen is the hydrogen molecule. p,f ° in Equation 13.16 is therefore the free energy for the formation of a proton from H2 in the gas phase, i.e.. 

It follows that, because phase equilibrium requires that the chemical potential p. be the same in the solution as in the gas phase, one may write for the chemical potential in the solution ... 

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... 

The chemical potential in the dyebath solution, is defined in equation 1 where is the standard chemical potential in the solution, R is the gas constant, Tis temperature in K, and a is activity. 

By assuming that fi o the same as that for pure water, which is unlikely in this concentrated solution, and by substituting for the chemical potentials in equation 20.228, Pourbaix has calculated that... 

The chemical potential in the solution is related to the activity of the solvent a, by... 

The first qualitatively correct attempt to model the relevant chemical potentials in a polymer solution was made independently by Huggins (4, ) and Flory [6). Their models, which are similar except for nomenclature, are now usually called the Flory-Huggins model ( ). 

In the state of equilibrium between both phases, i.e. the solution phase eontaining the M" species and the solid metal phase, the sum of the chemical potentials in both phases are equal. Sinee charged speeies are involved, the usual chemical potential jUi has to be extended by a term representing the work neeessary to bring one mol of charged species with a charge of Zj e into a phase where an eleetrostatie potential E is present... 

These expressions comprise the nonideal terms in the previous equations for the chemical potential, Eqs. (30) and (31 ). They may therefore be regarded as the excess relative partial molar free energy, or chemical potential, frequently used in the treatment of solutions of nonelectrolytesi.e, the chemical potential in excess (algebraically) of the ideal contribution, which is —RTV2/M in dilute solutions. 

The temperature at which this condition is satisfied may be referred to as the melting point Tm, which will depend, of course, on the composition of the liquid phase. If a diluent is present in the liquid phase, Tm may be regarded alternatively as the temperature at which the specified composition is that of a saturated solution. If the liquid polymer is pure, /Xn —mS where mS represents the chemical potential in the standard state, which, in accordance with custom in the treatment of solutions, we take to be the pure liquid at the same temperature and pressure. At the melting point T of the pure polymer, therefore, /x2 = /xt- To the extent that the polymer contains impurities (e.g., solvents, or copolymerized units), ixu will be less than juJ. Hence fXu after the addition of a diluent to the polymer at the temperature T will be less than and in order to re-establish the condition of equilibrium = a lower temperature Tm is required. 

In essence, we assume that the gel solution is sufficiently dilute to justify the assumption that the first two contributions enter additively. The first and third are given by Eq. (38). Proceeding at once to the case of swelling equilibrium, we observe that fulfillment of the condition jLti = Ml is required, where mi is the chemical potential in the external solution. Inserting this condition in Eq. (B-1) and writing (Ami )z for Mi -... 

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. 

The algorithms used by module section GQUAL are, again, based on those incorporated in the SERATRA model. The chemical forms which it can handle and the processes included are shown schematically in Figure 8. In this section of the module it is assumed that all chemicals exist in solution and are, thus, potentially subject to the processes shown on the left side of the figure. These include ... 

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... 

The next problem is to find an expression for Asg. This entropy difference is a function of the particle volume fractions in the dispersion ( ) and in the floe (<(> ). As a first approximation, we assume that Ass is independent of the concentration and chain length of free polymer. This assumption is not necessarily true the floe structure, and thus < >f, may depend on the latter parameters because also the solvent chemical potential in the solution (affected by the presence of polymer) should be the same as that in the floe phase (determined by the high particle concentration). However, we assume that these effects will be small, and we take as a constant. 

The first sum is over all ionic species in the solution, the second sum over all neutral species except the metal atoms. For a pure metal the concentration of the metal atoms is constant so the differential of the chemical potential of the metal atoms vanishes dpM = 0 we note in passing that complications can arise for amalgams, if the surface concentration of the metal atoms changes. All chemical potentials in Eq. (16.11) refer to the solution. 

Beyond the clusters, to microscopically model a reaction in solution, we need to include a very big number of solvent molecules in the system to represent the bulk. The problem stems from the fact that it is computationally impossible, with our current capabilities, to locate the transition state structure of the reaction on the complete quantum mechanical potential energy hypersurface, if all the degrees of freedom are explicitly included. Moreover, the effect of thermal statistical averaging should be incorporated. Then, classical mechanical computer simulation techniques (Monte Carlo or Molecular Dynamics) appear to be the most suitable procedures to attack the above problems. In short, and applied to the computer simulation of chemical reactions in solution, the Monte Carlo [18-21] technique is a numerical method in the frame of the classical Statistical Mechanics, which allows to generate a set of system configurations... 

In order to apply the Monte Carlo method to a chemical reaction in solution, two general problems immediately appear. Firstly, how do the configurational space have to be sampled That is, which configurations are considered and what kind of chemical information can be extracted from them. Second, how is the potential energy of each configuration evaluated The discussion of this last point will be delayed until section 6. 

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