Chemical Potential of Solutions

Atmospheric aerosols at high relative humidities are aqueous solutions of species such as ammonium, nitrate, sulfate, chloride, and sodium. Cloud droplets, rain, and so on are also aqueous solutions of a variety of chemical compounds. 

1 For a discussion of non-ideal-gas mixtures, the reader is referred to Denbigh (1981). Discussion of these mixtures is not necessary here, as the behavior of all gases in the atmosphere can be considered ideal for all practical purposes. 

Ideal Solutions A solution is defined as ideal if the chemical potential of every component is a linear function of the logarithm of its aqueous mole fraction xi according to the relation 

A multicomponent solution is ideal only if (10.43) is satisfied by every component. A solution, in general, approaches ideality as it becomes more and more dilute in all but one component (the solvent). The standard chemical potential p is the chemical potential of pure species i(jq = 1) at the same temperature and pressure as the solution under discussion. Note that in general p is a function of both T and p but does not depend on the chemical composition of the solution. 

Let us discuss the relationship of the preceding definition with Henry s and Raoult s laws, which are often used to define ideal solutions. Assuming that an ideal solution of i is in equilibrium with an ideal gas mixture, we have 

The standard chemical potentials p and p° are only functions of temperature and pressure and therefore the constant A, is independent of the solution s composition. 

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Here /i represents the chemical potential of solute i at unit molality and yi = 1. 

Equation 5.21 shows the interrelationship among the chemical potentials of the constituent substances in a homogeneous mixture and is often used for the determination of the chemical potential of solute constituents in solutions. 

In this case the unitary value of the chemical potential of solute substance i can be estimated, as mentioned above, by extrapolating the chemical potential of dilute constituent i to xt = 1 from the dilute concentration range in which the linear relation of Eq. 5.22 holds. 

Considering solute molecules at the density of ps g in the ideal gas phase in equilibrium with solute molecule in the liquid phase at a density of ps and considering the fact that at equilibrium the chemical potentials of solute in both phases are equal, we can write... 

Thus, if the partial molar volume of solute in aqueous solution is greater than the molar volume of solid solute, an increase in pressure will increase the chemical potential of solute in solution relative to that in the solid phase solute will then leave the solution phase until a lower, equilibrium solubility is attained. Conversely, if the partial molar volume in the solution is less than that in the solid, the solubility will increase with pressure. 

Standard chemical potential of solute i standard chemical potential of solute i at infinite dilution refractive index (at sodium D line)... 

In thermodynamic reasoning, there has to be a standard state. The standard state for the solid crystal is the substance in its pure state at 298 K. It follows that the standard chemical potential of solution is ... 

There is, however, another unifying feature that is achieved by referring the chemical potentials of solutions to the chemical potentials of the pure constituents under standard conditions, P — bar, as follows ... 

Molalities or molarities of ions and non-ionic compounds, mol/kg or mol/L Fluid viscosity. Pa s, or chemical potential, J/mol Chemical potential of solute, J/mol Chemical potential of solvent, J/mol Chemical potential of the /th component, J/mol Number of pores Water flux, kg-water/s m ... 

According to Rehbinder, we choose the chemical potential of solute p as an independent variable. The differentiation of the above equation with... 

When the system is at thermodynamic equilibrium, the chemical potential of any component (including the adsorbed one) is the same in all phases in contact, as well as within the interfacial layer. If p is the chemical potential of solute in the bulk, one can write... 

This equation suggests that for spontaneous physical and/or chemical change to occur, the process proceeds with a decrease in free energy. As applied to phase distribution, equilibrium is reached when the infinitesimal increase in G per infinitesimal increase in the number of moles of solute i added to each phase becomes equal. Hence, the chemical potential of solute i is defined as... 

The chemical potential of solutions of nonelectrolytes can always be written in terms of a series of positive integral powers of the concentration... 

The excess chemical potential of solute, or the solvation free energy , at infinite dilution is of particular interest, because it is the quantity which measures the stability of solute in solvent, and because all other excess thermodynamic quantities are derived from the free energy. The excess chemical potential, which is defined as an excess from the ideal gas, can be expressed in terms of the so called Kirkwood coupling parameter. The excess chemical potential is defined as the free energy change associated with a process in which a solute molecule is coupled into solvent [41]. The coupling procedure can be expressed by. 

Figure 1 Solvent effects on chemical potentials of solutes.

An important feature of ion-exchange membranes is their permeabihty to counterions and their impermeabihty to co-ions (ions with a charge like that of the fixed charge). But they are not perfectly impermeable to co-ions. As solvent penetrates ion-exchange material some co-ions penetrate the membrane as well, driven by the difference of chemical potentials of solution and membrane. To foUow the law of electric neutrality, an additional amount of counterions equivalent to the amount of co-ions must move into the ion-exchange membrane. This quantity of electrolyte absorbed by this mechanism may be estimated by the Dorman equation, which is based on the law of equity of chemical potentials in both hquid and soHd phases. In the case of an electrolyte in which both the an-... 

With the functional form of intermolecular interactions at hand, they have now to be evaluated for a liquid ensemble of interacting segments. A detailed derivation of the necessary statistical thermodynamics and a proof of thermodynamic consistency have been given by Klamt et al. [12], This yields the chanical potential as a function of the polarization charge density ]i (o) and one finally obtains the chemical potential of solute X in solvent S by integrating the solvent chemical potential over the binned solute surface weighted by the o-profile ... 

In an ideal dilute solution, that is, one that obeys Hemy s Law, each solute particle A is fully solvated, and there is no aggregation occurring that could otherwise influence the behavior of the solution. In such cases, the chemical potential of solute A is given by (3) ... 

Plot of the total GFE as a function of the activity of a solute A. The slope at each point in the curve is the chemical potential of solute A (//a)- There is one specific slope that is defined as the reference point. This is the slope for the activity of A when [A] = 1M(//a ). 

Consider a dilute binary nonelectrolyte solution in which the dependence of the chemical potential of solute B on composition is given by... 

Some of the MD codes mentioned above (such as DL POLY, AMBER, and NAMD) can perform thermodynamic integration and various free energy perturbation calculations. These techniques can be used to compute excess chemical potentials of solutes in ionic liquids and thereby obtain information on solvation. We believe MC methods are better suited for these types of calculations, however. 

SCFs are potentially very attractive media for conducting chemical transformations (4), primarily because the solvent and transport properties of a single solution can be varied appreciably and continuously with relatively minor changes in temperature or pressure. The density variation in an SCF also influences the chemical potential of solutes and, thus, reaction rates and equilibrium constants (59). Therefore, the solvent environment can be optimized for a specific reac-... 

Consider now equation (3.3.39b). It relates the interfacial concentration of solute 1 to the bulk concentration of solute 1 through the dependence of interfacial tension on the bulk chemical potential of solute 1. Remember, we need the equilibrium criterion so that we can relate the... 

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