Activity Debye-Hiickel equation

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. 

Fig. 2.3 was constructed using a K2-3 value at 250°C extrapolated from high-temperature data by Orville (1963), liyama (1965) and Hemley (1967). Ion activity coefficients were computed using the extended Debye-Hiickel equation of Helgeson (1969). The values of effective ionic radius were taken from Garrels and Christ (1965). In the calculation of ion activity coefficients, ionic strength is regarded as 0.5 im i ++mci-) (= mc -)- The activity ratio, an-f/aAb, is assumed to be unity. 

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. 

In each case, we use program spece8 or react and employ an extended form of the Debye-Hiickel equation for calculating species activity coefficients, as discussed in Chapter 8. In running the programs, you work interactively following the general procedure ... 

Geochemical modelers currently employ two types of methods to estimate activity coefficients (Plummer, 1992 Wolery, 1992b). The first type consists of applying variants of the Debye-Hiickel equation, a simple relationship that treats a species activity coefficient as a function of the species size and the solution s ionic strength. Methods of this type take into account the distribution of species in solution and are easy to use, but can be applied with accuracy to modeling only relatively dilute fluids. 

Activity coefficients can be determined by experimental observations. Since they are functions of ionic strength, temperature and pressure, marine scientists typically estimate values at the environmental conditions of interest from semi-empirical equations. In dilute solutions, the activity coefficient of a monoatomic ion can be calculated from the Debye-Hiickel equation ... 

Figure 8JO Comparison between individual ionic activity coefficients obtained with Debye-Hiickel equation and with mean salt method for various ionic strength values. Reprinted from Garrels and Christ (1965), with kind permission from Jones and Bartlett Publishers Inc., copyright 1990.

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 ... 

This technique uses both direct and back titrations of weak acids and bases. Values of are obtained directly. In purely aqueous media, over the pH range 2-10, the titration of dilute (0.005 to 0.05 M) solutions of weak monovalent acids and bases with a glass electrode can lead to reliable thermodynamic pKs. Over this pH interval, the activity coefficients of the ionic species can be calculated by means of the Debye-Hiickel equation. Also, the activity coefficients of the neutral species remain essentially constant and... 

Just as in aqueous solutions, the activity of solute i (acl) in non-aqueous solutions is related to its (molar) concentration (sj by aCii = yCiiCi, where g is the activity coefficient that is defined unity at infinite dilution. For non-ionic solutes, the activity coefficient remains near unity up to relatively high concentrations ( 1 M). However, for ionic species, it deviates from unity except in very dilute solutions. The deviation can be estimated from the Debye-Hiickel equation, -log yci = Az2 /1/2/ (1+aoBf1 2). Here, I is the ionic strength and / (moll-1), a0 is the ion size parameter... 

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). 

The ionic atmosphere model leads to the extended Debye-Hiickel equation, relating activity coefficients to ionic strength ... 

E3 Extended Debye-Hiickel equation. Use Equation 8-6 to calculate the activity coefficient (-y) as a function of ionic strength... 

Extrapolation of Kc to infinite dilution to give K is usually easy because the activity coefficients of most ionic substances vary in a regular manner with ionic strength and follow the Debye-Hiickel equation (Eq. 6-33) in very dilute solutions (ionic strength < 0.01). 

The Guggenheim extensions of the Debye-Hiickel equations (see Section 18.1b) are used to obtain expressions for the activity coefficients. The result is... 

The Debye-Hiickel equation for very dilute solutions can be used for each activity coefficient ... 

Potentiometry has found extensive application over the past half-century as a means to evaluate various thermodynamic parameters. Although this is not the major application of the technique today, it still provides one of the most convenient and reliable approaches to the evaluation of thermodynamic quantities. In particular, the activity coefficients of electroactive species can be evaluated directly through the use of the Nemst equation (for species that give a reversible electrochemical response). Thus, if an electrochemical system is used without a junction potential and with a reference electrode that has a well-established potential, then potentiometric measurement of the constituent species at a known concentration provides a direct measure of its activity. This provides a direct means for evaluation of the activity coefficient (assuming that the standard potential is known accurately for the constituent half-reaction). If the standard half-reaction potential is not available, it must be evaluated under conditions where the activity coefficient can be determined by the Debye-Hiickel equation. 

When calculating the activity coefficient from the Debye-Hiickel equation the average value 3 X 10 8 cm might be substituted for eq into the equation (V-58) for solutions with concentration not exceeding c — 0.1 with uni-univalent electrolytes (c = 0.05 with bi-univalent ones). Because in aqueous solutions at 25° C A — 0.509 and B — 0.329 X 10-8, the Debye -Hiickel equation for the mentioned type of solutions has quite a simple form ... 

The original form of the Debye-Hiickel equation permits the calculation of the mean activity coefficients of strong electrolytes in solutions defined by their molarity c. Should the value of this coefficient be expressed by molality, whioh is more advantageous in electrochemistry, it will be possible in the case of a sufficiently diluted solution to substitute into the equation (V-58) for = y m (see V-41e) and for molarities of all ions the product of their molalities and the density of the solvent s wqp°, so that ... 

An approximate form of the Debye-Hiickel equation appropriate for estimating the values of activity coefficients of ions in relatively dilute aqueous solutions at 25°C is... 

The mean activity coefficient of the free ions is calculated using the Debye-Hiickel Equation with a distance parameter a = 4.0 A. If we ignore AGt° (mic), the above relation can be rearranged as ... 

Use the data of Table 2.8 to calculate the mean activity coefficient of a 5 MNaCI solution, assuming the total hydration number at this high concentration is <3. Values for A and B of the Debye-Hiickel equation can be recovaed from the text. 

However, to calculate the thermodynamic stability constant, we would need reliable values for all the relevant activity coefficients. As indicated above, these are usually inaccessible directly from experimental measurements and, although some form of the Debye-Hiickel equation... 

It will be seen later (p. 230) that there does not appear to be any experimental method of evaluating the activity coefficient of a single ionic species, so that the Debye-Hiickel equations cannot be tested in the forms given above. It is possible, however, to derive very readily an expression for the mean activity coefficient, this being the quantity that is obtained experimentally. The mean activity coefficient f of an electrolyte is defined by an equation analogous to (30), and... 

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... 

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). 

---