Dilute solution chemical potential

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. 

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

The pf is the chemical potential the solute would have in a 1 molal solution if that solution behaved according to the ideal dilute rule. This standard state is called the ideal solution of unit molality. It is a hypothetical state of a system. According to Eq. (16.20) the practical activity measures the chemical potential of the substance relative to the chemical potential in this hypothetical ideal solution of unit molality. Equation (16.20) is applicable to either volatile or in volatile solutes. 

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. 

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) ... 

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

To proceed fiirther, to evaluate the standard free energy AG , we need infonnation (experimental or theoretical) about the particular reaction. One source of infonnation is the equilibrium constant for a chemical reaction involving gases. Previous sections have shown how the chemical potential for a species in a gaseous mixture or in a dilute solution (and the corresponding activities) can be defined and measured. Thus, if one can detennine (by some kind of analysis)... 

The McMillan-Mayer theory allows us to develop a fomialism similar to that of a dilute interacting fluid for solute dispersed in the solvent provided that a sensible description of W can be given. At the Ihnit of dilution, when intersolute interactions can be neglected, we know that the chemical potential of a can be written as = W (a s) + IcT In where W(a s) is the potential of mean force for the interaction of a solute... 

The McMillan-Mayer theory offers the most usefiil starting point for an elementary theory of ionic interactions, since at high dilution we can incorporate all ion-solvent interactions into a limitmg chemical potential, and deviations from solution ideality can then be explicitly coimected with ion-ion interactions only. Furthemiore, we may assume that, at high dilution, the interaction energy between two ions (assuming only two are present in the solution) will be of the fomi... 

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

At finite temperature the chemical potentials can be calculated as follows. In the dilute solution approximation, the Gibbs free energy is given by ... 

The subscripts 1,2,3 refer to the main solvent, the polymer, and the solvent added, respectively. The meanings of the other symbols are n refractive index m molarity of respective component in solvent 1 C the concentration in g cm"3 of the solution V the partial specific volume p the chemical potential M molecular weight (for the polymer per residue). The surscript ° indicates infinite dilution of the polymer. 

Equation 13 has an important implication a clathrate behaves as an ideally dilute solution insofar as the chemical potential of the solvent is independent of the nature of the solutes and is uniquely determined by the total solute concentrations 2K yK1.. . 2x yKn in the different types of cavities. For a clathrate with one type of cavity the reverse is also true for a given value of fjiq (e.g. given concentration of Q in a liquid solution from which the clathrate is being crystallized) the fraction of cavities occupied 2kVk s uniquely determined by Eq. 13. When there are several types of cavities, however, this is no longer so since the individual occupation numbers 2k2/ki . ..,2k yKn, and hence the total solute concentration... 

It is difficult to point to the basic reason why the average-potential model is not better applicable to metallic solutions. Shimoji60 believes that a Lennard-Jones 6-12 potential is not adequate for metals and that a Morse potential would give better results when incorporated in the same kind of model. On the other hand, it is possible that the main trouble is that metal solutions do not obey a theorem of corresponding states. More specifically, the interaction eAB(r) may not be expressible by the same function as for the pure components because the solute is so strongly modified by the solvent. This point of view is supported by considerations of the electronic models of metal solutions.46 The idea that the solvent strongly modifies the solute metal is reached also through a consideration of the quasi-chemical theory applied to dilute solutions. This is the topic that we consider next. 

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. 

These simple expressions may also be obtained from the chemical potentials according to Eqs. (XII-26) and (XII-32) by appropriately changing subscripts and recalling that x in these equations represents the ratio of the molar volumes, which in the present case is unity. Owing to the identity of volume fractions with mole fractions in this case, Eqs. (18) and (19) are none other than the chemical potentials for a regular binary solution in which the heat of dilution can be expressed in the van Laar form. The critical conditions (see Eqs. 2)... 

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 -... 

Introducing Eq. (XII-43 ) (which merely stipulates the inevitable proportionality between this chemical potential and the square of the concentration in dilute solutions)... 

For snfficiently dilute solutions the concentration dependence of chemical potential is given similarly by... 

The ions in solution are subject to two types of forces those of interaction with the solvent (solvation) and those of electrostatic interaction with other ions. The interionic forces decrease as the solution is made more dilute and the mean distance between the ions increases in highly dilute solutions their contribution is small. However, solvation occurs even in highly dilute solutions, since each ion is always surrounded by solvent molecules. This implies that the solvation energy, which to a first approximation is independent of concentration, is included in the standard chemical potential and has no influence on the activity. 

In equilibrium the chemical potential must be equal in coexisting phases. The assumption is that the solid phase must consist of one component, water, whereas the liquid phase will be a mixture of water and salt. So the chemical potential for water in the solid phase fis is the chemical potential of the pure substance. However, in the liquid phase the water is diluted with the salt. Therefore the chemical potential of the water in liquid state must be corrected. X refers to the mole fraction of the solute, that is, salt or an organic substance. The equation is valid for small amounts of salt or additives in general ... 

In a general case of a mixture, no component takes preference and the standard state is that of the pure component. In solutions, however, one component, termed the solvent, is treated differently from the others, called solutes. Dilute solutions occupy a special position, as the solvent is present in a large excess. The quantities pertaining to the solvent are denoted by the subscript 0 and those of the solute by the subscript 1. For >0 and x0-+ 1, Po = Po and P — kxxx. Equation (1.1.5) is again valid for the chemical potentials of both components. The standard chemical potential of the solvent is defined in the same way as the standard chemical potential of the component of an ideal mixture, the standard state being that of the pure solvent. The standard chemical potential of the dissolved component jU is the chemical potential of that pure component in the physically unattainable state corresponding to linear extrapolation of the behaviour of this component according to Henry s law up to point xx = 1 at the temperature of the mixture T and at pressure p = kx, which is the proportionality constant of Henry s law. 

For a solution of a non-volatile substance (e.g. a solid) in a liquid the vapour pressure of the solute can be neglected. The reference state for such a substance is usually its very dilute solution—in the limiting case an infinitely dilute solution—which has identical properties with an ideal solution and is thus useful, especially for introducing activity coefficients (see Sections 1.1.4 and 1.3). The standard chemical potential of such a solute is defined as... 

Since AG° can be calculated from the values of the chemical potentials of A, B, C, D, in the standard reference state (given in tables), the stoichiometric equilibrium constant Kc can be calculated. (More accurately we ought to use activities instead of concentrations to take into account the ionic strength of the solution this can be done introducing the corresponding correction factors, but in dilute solutions this correction is normally not necessary - the activities are practically equal to the concentrations and Kc is then a true thermodynamic constant). 

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