Debye-Hiickel equation extended

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 ... 

The temperature-dependent form of the extended Debye-Hiickel equation 8-6 is... 

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

The extended Debye-Hiickel equation [138] provides a useful approximation for yi(I) for non-dilute solutions ... 

We next utilize the extended Debye-Hiickel equation, Eq. (4.2.5), in the following form... 

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. 

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 ... 

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... 

However, if the Debye-Huckel equation does not fully cope with non-ideality, then the extended Debye-Hiickel equation is required to account for yoH (aq) ycr(aq) tiot quite cancelling each other out. Also it is possible that for NaOH(aq) and NaCl(aq) are not quite equal because the value of b in the bl term of the extended Debye-Hiickel equation logioKi = + Vi) + hi may not be the same for the two electrolytes. 

Not only is this greater than the values calculated above, but it is also greater than unity. This is a very clear indication that a term opposing the —A z Z2 Vl/ + Vl) term is needed. This is precisely what the b term in the extended Debye-Hiickel equation does logjQPj = —A ziZ2 /7/ (It V ) T bl. 

However, it is recommended that one of the extended Debye-Hiickel equations (see Sections 10.10.1 and 10.10.2) should always be used if the ionic strength range is extended, and especially so when extrapolation of experimental data is needed. 

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

Experimental data shows conclusively that the extended Debye-Hiickel equation must be used for moderate and higher ionic strengths. 

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). 

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