Debye-Huckel limit

If the coefficients dy vanish, dy = 28y, we recover the exact Debye-Huckel limiting law and its dependence on the power 3/2 of the ionic densities. This non-analytic behavior is the result of the functional integration which introduces a sophisticated coupling between the ideal entropy and the coulomb interaction. In this case the conditions (33) and (34) are verified and the... 

Figure 9.6 Mean ionic activity coefficients for HCl(aq) at T = 298.15 K obtained from the emf results of G. A. Linhart, J. Am. Chem. Soc.. 41, 1175-1180 (1919). The dashed line is the Debye-Huckel limiting law prediction.

In this one-dimensional flat case the Laplace operator is simpler than in the case with spherical symmetry arising when deriving the Debye-Huckel limiting law. Therefore, the differential equation (B.5) can be solved without the simplification (of replacing the exponential factors by two terms of their series expansion) that would reduce its accuracy. We shall employ the mathematical identity... 

In fact, the symbol Ic should be used, as the molality ionic strength Im can be defined analogously in dilute aqueous solutions, however, values of c and m, and thus also Ic and Im, become identical.) Equation (1.1.21) was later derived theoretically and is called the Debye-Huckel limiting law. It will be discussed in greater detail in Section 1.3.1. 

In 1923, the Debye-Huckel limiting law was derived and it has served as an excellent model for simple salts at very low concentrations. The limiting form of this theory can be derived in several ways which should also give correct results at moderate concentrations. The mathematics involved in proceeding beyond the... 

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... 

In single electrolyte solutions the Debye-Huckel limiting equation works up to 10-3 m. Its well known extended form used at m > 10"3 is... 

The only three methods which do not require curve-fitting at present are the use of the Debye-Huckel limiting law and of the equations of Guntelberg and of Davies. Unfortunately these equations are of value only in very dilute and simple solutions. 

Strategy. We will assume that the only solutes present are ZnS04 and H2SO4, and first calculate the ionic strength of the solution. (For the purposes of this calculation, we can safely assume that the amounts of zinc sulfate are negligible when compared with the amounts of sulfuric acid.) We will then calculate y for the zinc cation from I (and, for simplicity, obtain y from the Debye-Huckel limiting law). 

In this appendix, we summarize the coefficients needed to calculate the thermodynamic properties for a number of solutes in an electrolyte solution from Pitzer s equations.3 Table A7.1 summarizes the Debye-Huckel parameters for water solutions as a function of temperature. They provide the leading terms for Pitzer s equations, and can also be used to calculate the Debye-Huckel limiting law values from the equations... 

In the lowest level of the theory, the B term in the denominator of Eq. (18) is dropped, to give the Debye—Huckel limiting law (DHLL),... 

Figure 2 Comparison of measured mean ionic activity coefficients with those predicted by the Debye-Huckel limiting law. (Data from Handbook of Chemistry and Physics, 77th ed. A James, M Lord. VNR Index of Chemical Physical Data. New York Van Nostrand Reinhold, 1992.

Figure 15. Excess thermodynamic functions for various 1 1 salts in water at 298 Kj mj = molality of salt and the dotted lines indicate the behaviour predicted by the Debye-Huckel limiting law (Fortier et al., 1974).

Figure 22. Mean ionic activity coefficients, y , for various salts in water, mj = 0-2 at 298 K the dotted line indicates the value required by the Debye-Huckel limiting law (Desnoyers and Jolicoeur, 1969).

University in Ithaca. Nobel Prize in 1936 for contributions to the knowledge of molecular structure based on his research on dipole moments, X-ray diffraction (Debye-Scherrer method), and electrons in gases. His investigations of the interaction between ions and electric fields resulted in the - Debye-Huckel theory. See also -> Debye-Falkenhagen effect, - Debye-Huckel limiting law, - Debye-Huckel length, - Debye relaxation time. 

See also -> Huckel equation of electrophoretic mobility, -> Debye-Huckel approximation, -> Debye-Huckel length, -> Debye-Huckel limiting law, -> Debye-Huckel-Onsager theory, -> Debye-Huckel parameter. 

Measurements of emf (electromotive force) are to be made with this cell under reversible conditions at a number of concentrations c of HCl. From these measurements relative values of activity coefficients at different concentrations can be derived. To obtain the activity coefficients on such a scale that the activity coefficient is unity for the reference state of zero concentration, an extrapolation procedure based on the Debye-Huckel limiting law is used. By this means, the standard electrode emf of the silver-silver chloride electrode is determined, and activity coefficients are determined for all concentrations studied. 

Fig. 3.23. The comparison of the experimentally observed mean activity coefficients of HCI and those that are calculated from the Debye-Huckel limiting law.

What then are the inadequaeies of the Debye-Hiiekel limiting law One does not have to look far. If one examines the experimental log versus I curve, not just in the extreme dilution regions, but at higher concentrations, it turns out that the simple Debye-Huckel limiting law falters. The plot of log versus / is a curve (Fig. 3.27 and Table 3.8) and not a straight line as promised by Eq. (3.90). Further, the curves depend not only on valence type (e.g., 1 1 or 2 2) but also (Fig. 3.28) on the particular electrolyte (e.g., NaCl or KCl). 

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. 

The Debye-Huckel limiting law gives the mean activity coefficient y of a pair of ions as ... 

Introducing the Debye-Huckel limiting law for log /, it is seen that... 

The factor A in equation (123) is proportional to 1/(D T), as shown on page 150 hence, a further test of this equation is to determine the slope of the plot of log S/So against Vy from Solubility data at different temperatures and in media of different dielectric constants. Such measurements have been made in water at 75° (D = 63.7), in mixtures of water and ethyl alcohol (D = 33.8 to 78.6), in methyl alcohol (D = 30), in acetone (D = 21), and in ethylene chloride D = 10.4). The results have been found in all cases to be in very fair agreement with the requirements of the Debye-Huckel limiting law as may be expected, appreciable discrepancies occur when the saturating salt is of a high valence type, especially in the presence of added ions of high valence. ... 

Calculate the activity coefficients of the silver iodate in the various solutions plot the values of — log / against Vv to see how far the results agree with the Debye-Huckel limiting law. Determine the mean ionic diameter required to account for the deviations from the law at appreciable concentrations. 

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