Debye limiting law

Analogous investigations of the HF+BFs+NaBF system showed that in this case B has to be taken as equal to zero, so that the mean activity coefficient is approximately given by the Debye limiting law (21)... 

Equation (1.69) is also known as the Debye-Hiickel limiting law, and the coefficient A is called its marginal coefficient or the coefficient of Debye limiting law. Value of the constant is determined from equation... 

Consider a solution of a single strong electrolyte whose molality is m. From equations (10 53) and (10 54) it is evident that the Debye limiting law can be written in the form... 

In the extremely dilute solution where the Debye limiting law may be applied, the term on the lefb-hcmd side of this relation is given by equation (10 86). Mctking this substitution we obtain... 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

The theory of strong electrolytes due to Debye and Htickel derives the exact limiting laws for low valence electrolytes and introduces the idea that the Coulomb interactions between ions are screened at finite ion concentrations. 

Wlien KC) < i (i.e. at very low concentrations), we have the Debye-Htickel limiting law distribution fiinction ... 

The Debye-Htickel limiting law predicts a square-root dependence on the ionic strength/= MTLcz of the logarithm of the mean activity coefficient (log y ), tire heat of dilution (E /VI) and the excess volume it is considered to be an exact expression for the behaviour of an electrolyte at infinite dilution. Some experimental results for the activity coefficients and heats of dilution are shown in figure A2.3.11 for aqueous solutions of NaCl and ZnSO at 25°C the results are typical of the observations for 1-1 (e.g.NaCl) and 2-2 (e.g. ZnSO ) aqueous electrolyte solutions at this temperature. 

Figure A2.3.11 The mean aetivity eoeffieients and heats of dilution of NaCl and ZnSO in aqueous solution at 25°C as a fiinotion of z zjV I, where / is the ionie strength. DHLL = Debye-Htiekel limiting law.

In the limit of zero ion size, i.e. as o —> 0, the distribution functions and themiodynamic fiinctions in the MS approximation become identical to the Debye-Htickel limiting law. 

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between ([ly and ((ij is a measure of the accuracy of the approximation. The osmotic coefficients calculated via the energy equation in the MS approximation are comparable in accuracy to the HNC approximation for low valence electrolytes. Figure A2.3.15 shows deviations from the Debye-Htickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

Figure A2.3.15 Deviations (A) of the heat of dilution /7 and the osmotie eoeflfieient ( ) from the Debye-Htiekel limiting law for 1-1 and 2-2 RPM eleetrolytes aeeordmg to the DHLL + B2, HNC and MS approximations.

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... 

At sufficiently low ionic strengths the activity coefficient of each electrolyte in a mixture is given by the Debye-Hiickel limiting law... 

It can be seen from Figure 7.8(b) that the curved lines predicted by the extended form of the Debye-Hiickel equation follow the experimental results to higher ionic strengths than do the limiting law expressions for the (1 1) and (2 1) electrolytes. However, for the (2 2) electrolyte, the prediction is still not very good even at the lowest measured molality.0... 

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

Figure 7.8 Comparison of experimental ln7 for 1 1, 2 1, and 2 2 electrolytes. The symbols indicate the experimental results, with representing HC1 (z+ = 1, z = — 1) representing SrC ( + = 2, r = — 1) and A representing ZnS04 (z+ = 2, z = -2). The lines are the Debye-Huckel predictions, with the solid line giving the prediction for (z+ = 1, z = -1) the dashed line for (z+ = 2, r = -1) and the dashed-dotted line for (z+= 2, z =-2). In (a), In 7- calculated from the limiting law [equation (7.45)] is shown graphed against I 2. In (b). In 7- calculated from the extended form [equation (7.43)] is shown graphed against 7m2.

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... 

Debye-Hiickel limiting law for 7 applies [equation (7.45)], equation (7.54) can be used to derive a limiting law for 0. We start with... 

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. 

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.

The extrapolation of a graph of LHS against I,1/2 (with the aid of the Debye-Hiickel limiting law) to I 2 = 0 gives an intercept with a value RT/F) n Kw. As a final example, consider the cell... 

Intense ion-ion interactions which are characteristic of salt solutions occur in the concentrated aqueous solutions from which AB cements are prepared. As we have seen, in such solutions the simple Debye-Hiickel limiting law that describes the strength goes up so the repulsive force between the ions becomes increasingly important. This is taken account of in the full Debye-Hiickel equation by the inclusion of a parameter related to ionic size and hence distance of closest approach (Marcus, 1988). 

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. 

The Debye-Hiickel limiting law is the least accurate approximation to the actual situation, analogous to the ideal gas law. It is based on the assumption that the ions are material points and that the potential of the ionic atmosphere is distributed from r = 0 to r->oo. Within these limits the last equation is integrated by parts yielding, for constant k, the value ezk/Aite. Potential pk is given by the expression... 

Fig. 1.8 Dependence of the mean activity coefficient y tC of NaCl on the square root of molar concentration c at 25°C. Circles are experimental points. Curve 1 was calculated according to the Debye-Hiickel limiting law (1.3.25), curve 2 according to the approximation aB = 1 (Eq. 1.3.32) curve 3 according to the Debye-Hiickel equation (1.3.31), a = 325nm curve 4 according to the Bates-Guggenheim approximation (1.3.33) curve 5 according to the Bates-Guggenheim approximation + linear term 0.1 C curve 6 according to Eq. (1.3.38) for a = 0.4nm, C = 0.055dm5-mor ...

In the above two equations, the former value is valid for basic SI units and the latter value for / in moles per cubic decimetre and a in nanometres. The parameter a represents one of the difficulties connected with the Debye-Hiickel approach as its direct determination is not possible and is, in most cases, found as an adjustable parameter for the best fit of experimental data in the Eq. (1.3.29). For common ions the values of effective ion radii vary from 0.3 to 0.5. Analogous to the limiting law, the mean activity coefficient can be expressed by the equation... 

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