Activity coefficient extended Debye-Hiickel equation

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. 

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 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... 

The difference between the extended Debye-Hiickel equation and the Pitzer equations has to do with how much of the nonideahty of electrostatic interactions is incorporated into mass action expressions and how much into the activity coefficient expression. It is important to remember that the expression for activity coefficients is inexorably bound up with equilibrium constants and they must be consistent with each other in a chemical model. Ion-parr interactions can be quantified in two ways, explicitly through stability constants (lA method) or implicitly through empirical fits with activity coefficient parameters (Pitzer method). Both approaches can be successful with enough effort to achieve consistency. At the present, the Pitzer method works much better for brines, and the lA method works better for... 

Measurements of f are then taken for several HCl solutions at different molalities then a plot of vs. should yield a straight line in the range where the extended Debye-Hiickel equation holds. Extrapolation of this line to m = 0 yields and the slope is proportional to C. One then returns to Eq. (5) to find the r m) values for each set of measured f and m values. Very accurate measurements of the activity coefficient thus become available. 

Figure 2. The Henry constant of oxygen in aqueous solutions of sodium sulfate at 25 °C (O) experimental data (a) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Debye-Hiickel equation (b) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the extended Debye-Hiickel equation (c) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Bromley equation (d) the Henry constant calculated with eq 15.

For the mean activity coefficient of the salt, several expressions have been used, such as the Debye-Hiickel equation, the extended Debye-Hiickel equation, and the Bromley equation. The Bromley equation was selected because of its simplicity and its accuracy of course, other accurate equations are also available. The values of the parameter B for all cases examined are listed in Table 3. 

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 ... 

The Debye-Hiickel Equation Activity coefficients (yu) for ions can be calculated for relatively low concentrations by variations of the Debye-Hiickel equation. The extended Debye-Hiickel equation is... 

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

Fig, 3-4, Activity coefficients of aqueous ions based on the extended DeBye-Hiickel equation (Eq. 3-35) and the Gun-telberg approximation (Eq. 3-36). 

Equation (4.18) along with the extended Debye-Hiickel equation for the mean activity coefficient... 

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 ... 

Edwards et al. (6) made the assumption that was equal to 4>pure a at the same pressure and temperature. Further theyused the virial equation, truncated after the second term to estimate pUre a These assumptions are satisfactory when the total pressure is low or when the mole fraction of the solute in the vapor phase is near unity. For the water, the assumption was made that <(>w, , aw and the exponential term were unity. These assumptions are valid when the solution consists mostly of water and the total pressure is low. The activity coefficient of the electrolyte was calculated using the extended Debye-Hiickel theory ... 

In order to solve the equations completely, the mean activity coefficients f of the ions are required. If one asstunes the extended Debye-Hiickel relationship for the concentration-dependence of the mean activity coefficients... 

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). 

Correlations for the determination of the dissociation equilibrium constants and solubility values for SO2 and CO2 as functions of temperature as well as the equations for activity coefficients are given in Ref. [70], Thermodynamic non-idealities are taken into account depending on whether species are charged, or not. For uncharged species, a simple relationship from Ref. [102] is applied, whereas for individual ions, the extended Debye-Hiickel model is used according to Ref. [103]. 

A more rigid but laborious method, for deriving transference numbers from E.M.p. data, makes use of the fact that the activity coefficient of an electrolyte can be expressed, by means of an extended form of the Debye-Hiickel equation, as a function of the concentration and of two empirical constants.When applied to the same data, however, this procedure gives results which are somewhat different from those obtained by the method just described. Since the values are in better agreement with the transference data derived frorq moving boundary and other measurements, they are probably more reliable. 

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