Hiickel’s law

Nevertheless, let us remember that Debye and Hiickel s law supposes the ions are spherical, which can be accepted for sufficiently dilute solutions in which the radius of the ionic atmosphere is larger than a and therefore in which the ions are relatively far removed from one another. If the ions come closer together, the hypothesis of sphericity becomes trickier to accept for a large number of t5q)es of ions - particularly for polyatomic ions. 

The limits exhibited by Debye and Hiickel s law in its application to real-world cases have led many researchers to attempt to improve it. These improvements were first made in the strict fimnework of the law [4.82], before branching out in a new direction, integrating the molecular interactions that we shall see in sections 4.3 and 4.4. 

For non-ideal mixtures, a correction factor, the activity coefficient 7 is introduced into a further description. This factor formally states the deviation from ideality as follows (see Mathematical appendix, subchapter A9, Debye-Hiickel s law)... 

Thermodynamic derivations that support and expand the theoretical sections in the text (Maxwell s relations, Debye Hiickel s law, etc.). 

Once a value of E° is obtained by extrapolation, the 7 corresponding to each molality can be obtained from equation (9.104). Figure 9.6 is a graph of ln7 calculated from Linhart s results plotted against m1/2. The Debye-Hiickel limiting law value is shown as the dashed line. The agreement is excellent below m1 2 = 0.10 (m = 0.03), which attests to the reliability of Linhart s work and the validity of the Debye-Hiickel limiting law. 

Again we wish to emphasize that it is the modified moments (aWtU ) and (V5(D) Moi) arising from the use of Onsager s theory of the moments in liquids that are important. With reasonable values of internal refractive index, the moments in vacuum can be reduced. Furthermore, our theory does account for values of j greater than those predicted by the Debye-Hiickel limiting law if the ions can be assumed to have dipole characteristics. 

Figure 3.48 shows two ways of expressing the results of Mayer s viriai coefficient approach using the osmotic pressure of an ionic solution as the test quantity. Two versions of the Mayer theory are indicated. In the one marked DHLL + B2, the authors have taken the Debye-Hiickel limiting-law theory, redone for osmotic pressure instead of activity coefficient, and then added to it the results of Mayer s calculation of the second viriai coefficient, B. In the upper curve of Fig. 3.48, the approximation within the Mayer theory used in summing integrals (the one called hypernetted chain or HNC) is indicated. The former replicates experiment better than the latter. The two approxi-... 

The first postulate of the Debye-Hiickel theory is that if the ions of an electrolyte lost their charges and became neutral particles, the solution would behave like a dilute solution obeying Henry s law ( 36a). The departure from ideal behavior is then attributed to the mutual interaction of the electrical charges carried by the ions. Hence, in writing the chemical potential (partial molar free energy) of an ion i in the form... 

The activity coefficients given by the Debye-Hiickel treatment presumably represent deviations from the dilute solution behavior, i.e., from Henry s law, and are consequently based on the standard state which makes the activity of an ion equal to its mole fraction at infinite dilution ( 37b, III B). In the experimental determination of activity coefficients, however, it is almost invariably the practice to take the activity as equal to the molarity or the molality at infinite dilution. The requisite corrections can be made by means of equation (39.13), but this is unnecessary, for in solutions that are sufficiently dilute for the Debye-Hackel limiting law to be applicable, the difference between the various activity coefficients is negligible. The equations derived above may thus be regarded as being independent of the standard state chosen for the ions, provided only that the activity coefficients are defined as being unity at infinite dilution. 

The type of construction given in Fig. 10a, which utilizes the Hess s law of constant heat summation, can serve as a means of quantitatively analyzing the thermodynamics of solvation. Further, this view of the solvation process provides a method for considering different standard states. For nonionic species a commonly used standard state is infinite dilution. Although activities become infinite for ions in this limit, it is still a useful reference state because the analytic Debye-Hiickel limiting law is valid in this regime.168... 

Base-lines for theoretical predictions about the behaviour expected for a solution consisting of free ions only, Debye-Hiickel and Fuoss-Onsager theories and the use of Beer s Law... 

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