Debye-Hiickel extend terms

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 ... 

The Electrolyte NRTL model " and the Extended UNIQUAC model" are examples of activity coefficient models derived by combining a Debye-Hiickel term with a local composition model. Equation of state models with electrostatic terms for... 

This increase in y values with ionic strength can be modeled by adding positive terms to some form of the extended Debye-Hiickel expression for log y. Simple models using this approach have been proposed by Hiickel (see Harned and Owen 1958), and more recently by others, including... 

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). 

The Debye-Hiickel equation as presented above is often called the extended Debye-Hiickel law (EDHL) because a simpler expression is used for very dilute solutions. When the ionic strength is less than 0.001 M, the term in the denominator of equation (3.8.32) goes to one, and one may write... 

Equation (26.41) predicts to within approximately 10% mean molal activity coefficients for salt concentrations up to 0.1 molal. The more accurate form of the activity coefficient equation [Equation (26.40)] allows the model to be extended to salt concentrations up to 0.5 molal. To expand the applicability of the Debye-Hiickel theory to higher concentrations, additional terms are added to Equation (26.40), such as [4]... 

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. 

Furthermore, this treatment can be extended to cover situations other than the spherical symmetry of the Debye-Hiickel approach, and of the simple treatments outlined above. To do so requires consideration of the distance, r, and the two angles 6 and 0. Formally, this is achieved by presenting the argument in terms of vectors. If, instead of the simpler description... 

For example, Marshall and Slusher (1966) made a detailed evaluation of the solubility of ealeium sulphate in aqueous sodium chloride solution, and suggested that variations in the ion solubility product could be described, for ionic strengths up to around 2 M at temperatures from 0 to 100 °C, by adding another term in an extended Debye Hiickel expression. Above 2 M and below 25 °C, however, further correction factors had to be applied, the abnormal behaviour being attributed to an increase in the complexity of the structure of water under these circumstances. Enthalpies and entropies of solution and specific heat capacity were also reported as functions of ionic strength and temperature. 

With increasing electrolyte concentration, the short-range interactions become more and more dominating. Therefore, in activity coefficient models the Debye-Hiickel term, which describes the long-range interactions, has to be extended by a term describing the short-range interactions. A well-known empirical extension of the Debye-Hiickel theory is the Bromley equation [5] ... 

Rows 11 and 12 are optional. If they are included, the program uses the extended form of the Debye-Hiickel equation. If only one number is entered (in column 1) in row 10, it is intepreted as the A term in the Davies equation, and that is used, as in the original eqbrm. 

Wang et al. introduced an UNIQUAC-model extended by a Pitzer-Debye-Hiickel expression for the long-term electrolytic interactions... 

Fig. B.2 The variation of the mean activity coefficient with ionic strength according to the extended Debye-Hiickel theory, (a) The limiting law for a 1,1-electrolyte, (b) The extended law with B = 0.5. (c) The extended law, extended further by the addition of a term CI-, in this case with C = 0.02. The last form of the law reproduces the observed behavior reasonably well.

Solutions to the Poisson—Boltzmann equation in which the exponential charge distribution around a solute ion is not linearized [15] have shown additional terms, some of which are positive in value, not present in the linear Poisson—Boltzmann equation [28, 29]. From the form of Eq. (62) one can see that whenever the work, q yfy - yfy), of creating the electrostatic screening potential around an ion becomes positive, values in excess of unity are possible for the activity coefficient. Other methods that have been developed to extend the applicable concentration range of the Debye—Hiickel theory include mathematical modifications of the Debye—Hiickel equation [15, 26, 28, 29] and treating solution complexities such as (1) ionic association as proposed by Bjerrum [15,25], and(2) quadrupole and second-order dipole effects estimated by Onsager and Samaras [30], etc. 

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