Limiting laws solvent

Equation (7.45) is a limiting law expression for 7 , the activity coefficient of the solute. Debye-Htickel theory can also be used to obtain limiting-law expressions for the activity a of the solvent. This is usually done by expressing a in terms of the practical osmotic coefficient

electrolyte solute, it is defined in a general way as... 

A hypothetical solution that obeys Raoult s law exactly at all concentrations is called an ideal solution. In an ideal solution, the interactions between solute and solvent molecules are the same as the interactions between solvent molecules in the pure state and between solute molecules in the pure state. Consequently, the solute molecules mingle freely with the solvent molecules. That is, in an ideal solution, the enthalpy of solution is zero. Solutes that form nearly ideal solutions are often similar in composition and structure to the solvent molecules. For instance, methylbenzene (toluene), C6H5CH, forms nearly ideal solutions with benzene, C6H6. Real solutions do not obey Raoult s law at all concentrations but the lower the solute concentration, the more closely they resemble ideal solutions. Raoult s law is another example of a limiting law (Section 4.4), which in this case becomes increasingly valid as the concentration of the solute approaches zero. A solution that does not obey Raoult s law at a particular solute concentration is called a nonideal solution. Real solutions are approximately ideal at solute concentrations below about 0.1 M for nonelectrolyte solutions and 0.01 M for electrolyte solutions. The greater departure from ideality in electrolyte solutions arises from the interactions between ions, which occur over a long distance and hence have a pronounced effect. Unless stated otherwise, we shall assume that all the solutions that we meet are ideal. 

The main reason why we want to make clear the assumptions involved in Eq. (154) is that, in transport problems, the solvent plays a much more important role because it is largely responsible for the dissipation. We shall thus have to improve these assumptions in order to get a satisfactory description of the non-equilibrium limiting laws. 

On the contrary, Figs. 14b and 14c both give a contribution of the order e6 to the current they have however a completely different behavior in the limit of small wave numbers. Indeed in Fig. 14b, no solvent molecule interacts with both ions as we shall see below, this diagram gives a contribution to the limiting law and has to be retained. On the other hand, Fig. 14c shows a process in which molecule j interacts with both ions a and Due to the short-range of the forces F >0, this kind of term may be neglected, as we now show we have (Latin indices refer to solvent molecules) ... 

Friedman (1962) has used the cluster theory of Mayer (1950) to derive equations which give the thermodynamic properties of electrolyte solutions as the sum of convergent series. The first term in these series is identical to and thus confirms the Debye-Huckel limiting law. The second term is an I2.nl term whose coefficient is, like the coefficient in the Debye-Huckel limiting law equation, a function of the charge type of the salt and the properties of the solvent. From this theory, as well as from others referred to above, a higher order limiting law can be written as... 

These equations are the same as Equation (14.6) and Equation (14.7), statements of Raoult s law thus, the solvent obeys Raoult s law when the solute obeys Henry s law. As Henry s law is a limiting law for the solute in dilute solution, Raoult s law... 

The deviation of a solvent from the limiting-law behavior of Raoult s law is described conveniently by a function called the activity coefficient, which is defined (on a mole fraction scale) as... 

Solvent in Solution. We shall use the pure substance at the same temperature as the solution and at its equilibrium vapor pressure as the reference state for the component of a solution designated as the solvent. This choice of standard state is consistent with the limiting law for the activity of solvent given in Equation (16.2), where the limiting process leads to the solvent at its equilibrium vapor pressure. To relate the standard chemical potential of solvent in solution to the state that we defined for the pure liquid solvent, we need to use the relationship... 

In the limit of zero mole fraction of solute or unit mole fraction of solvent, the condition in which Henry s law is accurate as a limiting law. Equation (16.73) becomes... 

In solution thermodynamics, the concentration (C) of ions is replaced by their activity, a, where a = Cy and y is the activity coefficient that takes into account nonideal behavior due to ion-solvent and ion-ion interactions. The Debye-Hiickel limiting law predicts the relationship between the ionic strength of a solution and y for an ion of charge Z in dilute solutions ... 

The theory of Debye and Hiickel has survived much criticism since the appearance of their celebrated paper (I). This is no doubt because of the simplicity and essential correctness of the limiting laws (2,3,4). Nevertheless, many modifications of their treatment have failed to provide a convincing picture of the interionic effects and structure in the concentration range of practical importance (5, 6). The work presented here was stimulated by the difficulties of extrapolation encountered in a mixed-solvent emf study (7), and contradicts current trends suggesting that the inadequacy of the DH theory for all but very dilute solutions springs solely from the crudity of the original model. The authors propose a more realistic model that allows the ions to be polarizable and leads to markedly different results. 

Solvation Effects. Many previous accounts of the activity coefficients have considered the connections between the solvation of ions and deviations from the DH limiting-laws in a semi-empirical manner, e.g., the Robinson and Stokes equation (3). In the interpretation of results according to our model, the parameter a also relates to the physical reality of a solvated ion, and the effects of polarization on the interionic forces are closely related to the nature of this entity from an electrostatic viewpoint. Without recourse to specific numerical results, we briefly illustrate the usefulness of the model by defining a polarizable cosphere (or primary solvation shell) as that small region within which the solvent responds to the ionic field in nonlinear manner the solvent outside responds linearly through mild Born-type interactions, described adequately with the use of the dielectric constant of the pure solvent. (Our comments here refer largely to activity coefficients in aqueous solution, and we assume complete dissociation of the solute. The polarizability of cations in some solvents, e.g., DMF and acetonitrile, follows a different sequence, and there is probably some ion-association.)... 

DEBYE-HOCKEL LIMITING LAW. The departure from ideal behavior in a given solvent is governed by the ionic strength of the medium and the valences of the ions of the electrolyte, but is independent of their chemical nature. For dilute solutions, the logarithm of the mean activity is proportional to the product of the cation valence, anion valence, and square root of ionic strength giving the equation... 

This equation approximates Raoult s law and becomes equal to it when the molar volume of the pure liquid solvent is negligibly small with respect to the molar volume of the gas. We see, once again, that Raoult s law is a limiting law, strictly valid only when the equation of state for the ideal gas is applicable to the gas phase, and that the molar volume of the pure liquid is negligibly small with respect to that of the gas phase. 

The neglect of activity coefficients y in equations (12) and (13)—which meant that they were strictly valid only in the limit of infinitely dilute solutions—is less serious in equation (14). The interionic mean activity coefficient y for the hydrogen ion and the anion will, at low ionic strengths, be a function of the nature of the solvent only to the extent that the coefficient A in the Debye-Hiickel limiting law for activity coefficients depends on the dielectric constant of the solvent. In view of the similar values of the dielectric constants of H20 and D20 (see p. 261), the resultant difference between activity coefficients in H20 and D20 solutions of the same low ionic strength should be small. (If both Aha and Kn are results extrapolated to zero ionic strength, the problem disappears in its entirety.) (See also Section IVC.)... 

The inclusion of activity coefficients into the simple equations was briefly considered by Purlee (1959) but his discussion fails to draw attention to the distinction between the transfer effect and the activity coefficient (y) which expresses the non-ideal concentration-dependence of the activity of solute species (defined relative to a standard state having the properties of the infinitely dilute solution in a given solvent). This solvent isotope effect on activity coefficients y is a much less important problem than the transfer effect, at least for fairly dilute solutions. For example, we have already mentioned (Section IA) that the nearequality of the dielectric constants of H20 and D20 ensures that mean activity coefficients y of electrolytes are almost the same in the two solvents over the concentration range in which the Debye-Hiickel limiting law applies. For 0-05 m solutions of HC1 the difference is within 0-1% and thus entirely negligible in the present context. Of course, more sizeable differences appear if concentrations are based on the molality scale (Gary et ah, 1964a) (see Section IA). 

The success of the Debye-Hlickel limiting law is no mean achievement. One has only to think of the complex nature of the real system, of the presence of the solvent, which has been recognized only through a dielectric constant, of the simplicity of the Coulomb force law used, and, finally, of the fact that the ions are not point charges, to realize (Table 3.7) that the simple ionic cloud model has been brilliantly successful—almost unexpectedly so. It has grasped the essential truth about electrolytic... 

Electrolytes for which the concentration is less than lO Mcan usually be dealt with by the Debye-Huckel limiting law. Utilize the Debye-Huckel theory extended by allowance for ion size and also for removal of some of the active solvent into the ion s primary solvation shell to calculate the activity coefficient of 5 M NaCland 1M LaClj solutions (neglecting ion association or complexing). Take the total hydration number at the 5 M solution as 3 and at the 1 M solution as 5. Take r,- as 320 pm. 

Use the Debye-Hiickel limiting law to derive an expression for the solvent activity in dilute solutions. (Xu)... 

Rb, and Cs. The heats of hydration of these ions vary in the same order. Thus it seems that the deviations from the limiting law are related to the intensity of the interaction energy between the dissolved ions and the solvent. A similar conclusion is suggested by Stewart s f observation that the variation of the partial molar volume of ions with concentration is closely linked with the change of the structure of water in the same solutions, as revealed by X-ray studies. Unfortunately there are no satisfactory quantitative theories to account for these complex phenomena. 

The Nernst-Planck model is based on limiting laws for ideal systems. It accounts only for diffusion and electric transference of ions, not for electroosmotic solvent transfer in the ion-exchanger phase, swelling or shrinking of the ion-exchange material, variations of activity coefficients and diffu-sivities, and possible slow structural relaxation of the exchanger matrix. It also postulates the existence of individual diffusion coefficients for ions. 

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