Debye-Hiickel equation limiting

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... 

The nature of the Debye-Hiickel equation is that the activity coefficient of a salt depends only on the charges and the ionic strength. The effects, at least in the limit of low ionic strengths, are independent of the chemical identities of the constituents. Thus, one could use N(CH3)4C1, FeS04, or any strong electrolyte for this purpose. Actually, the best choices are those that will be inert chemically and least likely to engage in ionic associations. Therefore, monovalent ions are preferred. Anions like CFjSO, CIO, /7-CIC6H4SO3 are usually chosen, accompanied by alkali metal or similar cations. 

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). 

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 ...

Can the species activity coefficients be calculated accurately An activity coefficient relates each dissolved species concentration to its activity. Most commonly, a modeler uses an extended form of the Debye-Hiickel equation to estimate values for the coefficients. Helgeson (1969) correlated the activity coefficients to this equation for dominantly NaCl solutions having concentrations up to 3 molal. The resulting equations are probably reliable for electrolyte solutions of general composition (i.e., those dominated by salts other than NaCl) where ionic strength is less than about 1 molal (Wolery, 1983 see Chapter 8). Calculated activity coefficients are less reliable in more concentrated solutions. As an alternative to the Debye-Hiickel method, the modeler can use virial equations (the Pitzer equations ) designed to predict activity coefficients for electrolyte brines. These equations have their own limitations, however, as discussed in Chapter 8. 

It is interesting to compare the Debye-Hiickel and virial methods, since each has its own advantages and limitations. The Debye-Hiickel equations are simple to apply and readily extensible to include new species in solution, since they require few coefficients specific to either species or solution. The method can be applied as well over the range of temperatures most important to an aqueous geochemist. There is an extensive literature on ion association reactions, so there are few limits to the complexity of the solutions that can be modeled. 

Since we have no direct information about the chemistry of the Fountain fluid, we assume that its composition reflects reaction with minerals in the evaporite strata that lie beneath the Lyons. We take this fluid to be a three molal NaCl solution that has equilibrated with dolomite, anhydrite, magnesite (MgCC>3), and quartz. The choice of NaCl concentration reflects the upper correlation limit of the B-dot (modified Debye-Hiickel) equations (see Chapter 8). To set pH, we assume a CO2 fugacity of 50, which we will show leads to a reasonable interpretation of the isotopic composition of the dolomite cement. 

The Extended Debye-Hiickel Equation. This exercise reminds us that the Debye-Hiickel limiting law is not sufficiently accurate for most physicochemical studies. To estimate the calculated activity coefficient more accurately, one must consider the fact that ions are not point charges. To the contrary, ions are of finite size relative to the distance over which the ions interact electrostatically. This brings us to the extended Debye-Hiickel equation ... 

The standard emf E° of the cell was determined by means of an extrapolation technique involving a function of the measured emf E (which was measured experimentally), taken to the limit of zero ionic strength /. A linear function of I was observed when the Debye-Hiickel equation (in its extended form) (12) was introduced for the activity coefficient of hydrobromic acid over the experimental range of molalities m. With this type of mathematical treatment, the adjustable parameter became a0, the ion-size parameter, and a slope factor / . This procedure is essentially the same as that used in our earlier determinations (7,10) although no corrections of E° for ion association were taken into account (e = 49.5 at 298.15°K). 

Qualitative Verification of the Debye-Hiickel Equations.—The general agreement of the limiting law equation (54) with experiment is shown by the empirical conclusion of Lewis and Randall (p. 140) that the activity coefficient of an electrolyte is the same in all solutions of a given ionic strength. Apart from the valence of the ions constituting the particular electrolyte under consideration, the Debye-Hiickel limiting equation contains no reference to the specific properties of the salts that may be present in the solution. It is of interest to record that the... 

Quantitative Tests of the Debye-Hiickel Limiting Equation.—Although the Debye-Hiickel equations are generally considered as applying to solutions of strong electrolytes, it is important to emphasize that they are by no means restricted to such solutions they are of general applicability and the only point that must be noted is that in the calculation of the ionic strength the actual ionic concentrations must be employed. 

The experimentally determined activity coefficients, based on vapor pressure, freezing-point and electromotive force measurements, for a number of typical electrolytes of different valence types in aqueous solution at 25 , are represented in Fig. 49, in which the values of log / are plotted against the square-root of the ionic strength in these cases the solutions contained no other electrolyte than the one under consideration. Since the Debye-Htickel constant A for water at 25 is seen from Table XXXV to be 0.509, the limiting slopes of the plots in Fig. 49 should be equal to —0.509 the results to be expected theoretically, calculated in this manner, are shown by the dotted lines. It is evident that the experimental results approach the values required by the Debye-Hiickel limiting law as infinite dilution is attained. The influence of valence on the dependence of the activity coefficient on concentration is evidently in agreement with theoretical expectation. Another verification of the valence factor in the Debye-Hiickel equation will be given later (p. 177). 

Fig. 50. Limiting Debye-Hiickel equation at different dielectric constants (Hamed, et ah)...

Derivation of the Limiting Form for the Debye-Hiickel Equation 9.5.1 Fundamentals... 

The particular application of the Debye-Hiickel equation to be described here refers to the determination of the true equilibrium constant K from values of the equilibrium function K at several ionic strengths the necessary data for weak acids and bases can often be obtained from conductance measurements. If the solution of the electrolyte MA is sufficiently dilute for the limiting law to be applicable, it follows from equation (40.12), for the activity coe cient of a single ionic species, that... 

Debye-Hiickel equation An expression that permits calculation of activity coefficients in media with ionic strengths less than 0.1. Debye-Hiickel limiting law A simplified form of the Debye-Hiickel equation, applicable to solutions in which the ionic strength is less than 0.01. 

The Davies equation (Eq. 4.31) generates the positive change in slope with an add-on term, bl, where b is the same constant for all ions. The denominator of the Davies equation equals I + VT, which is equivalent to assigning a constant a -, value of about 3.0 to all ions in the extended DH equation. These simplifications make the Davies equation less accurate than the extended Debye-Hiickel equation at low ionic strengths, and limit its use to ionic strengths below that of seawater (0.7 mol/kg). 

What are the Debye-Huckel limiting law and the extended Debye-Hiickel equation and under what general conditions can they be used to compute ion activity coefficients Discuss the meaning and use of the ion size parameter in the Debye-Hiickel equation. How is it related to the ionic potential ... 

Changes in moisture content affect charged species in foods that are not part of the chemical equation, but that may impart their own effects upon reaction rate. Reactions that involve proton and electron transport, which include hydrolysis, Maillard browning, oxidation, and almost every critical shelf-life-limiting reaction in foods, will be affected by the presence of ions. This is part of the theory behind the Debye-Hiickel equation. This model describes the effect of ionic strength on the reaction rate constant in dilute solutions ... 

The Debye-Hiickel equation can be written in terms of a mean activity coefficient (see Section 10.6.15) log ioK = -A ziZ2 - ///(1 + V7) where ziZ2 reads mod ziZ2 and means that only the charges appear and not the sign. The limiting law can be written similarly. 

Since this is a fairly concentrated solution, then the Debye-Hiickel equation must be used rather than the limiting law ... 

Testing the Debye-Hiickel limiting law, the Debye-Hiickel equation and the extended Debye-Hiickel equation has demanded highly accurate experimental activity coefficient determinations. 

The denominator, 1 + Bay/l, will cause logjo to be less negative than the limiting law value, and so values of logjg y calculated from the Debye-Hiickel equation will lie above values of logjo y calculated from the limiting law. This is because ... 

The first two terms on the right hand side of the equation give the Debye-Hiickel-Onsager limiting law equation ... 

The numbers 0.51 and 0.33 are constants for water at 25°C, and the former includes the —I power of both the dielectric constant of the solvent and the absolute temperature ct/ is the ion size parameter, which is the ejffective diameter of the hydrated ion in angstrom units, A. An angstrom is 100 picometers (pm, 10 ° meter). A limitation of the Debye-Hiickel equation is the accuracy to which a, can be evaluated. For many singly charged ions, a, is generally about 3 A, and for practical purposes Equation 6.19 simplifies to... 

The pH of NIST buffers is determined with a cell without liquid junction (Cell 13.22) and is calculated using Equation 13.24. The activity of Cl" must be calculated from the Debye-Hiickel equation, which limits the accuracy of the an calculated from the measured potential. 

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