B-dot model

In the B-dot model, as currently applied (Wolery, 1992b), the activity coefficients of electrically neutral, nonpolar species [B(OH)3, C>2(aq), SiC>2(aq), CH aq), and H2(aq) are calculated from ionic strength using an empirical relationship,... 

In the B-dot model, the osmotic coefficient is taken to be described by a power series,... 

Figure 8.7 shows how concentrations and activities of the calcium and sulfate species vary with NaCl concentration. In the B-dot model, there are three ion pairs (CaCl+, NaSO, and CaS04) in addition to the free ions Ca++ and SO4 . 

The activities of the free ions remain roughly constant with NaCl concentration, and their concentrations increase only moderately, reflecting the decrease in the B-dot activity coefficients with increasing ionic strength (Fig. 8.3). Formation of the complex species CaCl+ and NaSOj drives the general increase in gypsum solubility with NaCl concentration predicted by the B-dot model. 

In the HMW model, in contrast, Ca++ and SO4 are the only calcium or sulfate-bearing species considered. The species maintain equal concentration, as required by electroneutrality, and mirror the solubility curve in Figure 8.6. Unlike the B-dot model, the species activities follow trends dissimilar to their concentrations. The Ca++ activity rises sharply while that of SO4 decreases. In this case, variation in gypsum solubility arises not from the formation of ion pairs, but from changes in the activity coefficients for Ca++ and SO4 as well as in the water activity. The latter value, according to the model, decreases with NaCl concentration from one to about 0.7. 

Figures, l-Alanine.Fits to noisy data Calculations A (experimental noise) and B (10% experimental noise). MaxEnt, deformation and error density profiles along the Cl-01 bond. Solid line Model valence density. Dashed line MaxEnt density A. Dot-dashed line MaxEnt density B. Dotted line valence-shells non-uniform prior.

Helgeson (1969 see also Helgeson and Kirkham, 1974) presented an activity model based on an equation similar in form to the Davies equation. The model, adapted from earlier work (see Pitzer and Brewer, 1961, p. 326, p. 578, and Appendix 4, and references therein), is parameterized from 0°C to 300 °C for solutions of up to 3 molal ionic strength in which NaCl is the dominant solute. The model takes it name from the B-dot equation,... 

To explore the differences between the methods, we use spece8 to calculate at 25 °C the solubility of gypsum (CaSCU 2H2O) as a function of NaCl concentration. We use two datasets thermo.dat, which invokes the B-dot equation, and thermo hmw. dat, based on the hmw model. The log K values for the gypsum dis-... 

Fig. 8.6. Solubility of gypsum (CaSCU 2H2O) at 25 °C as a function of NaCl concentration, calculated according to the Harvie-M0ller-Weare and B-dot (modified Debye-Hiickel) activity models. Circles and squares, respectively, show experimental determinations by Marshall and Slusher (1966) and Block and Waters (1968).

Figure 8.6 shows the calculation results plotted against measured solubilities from laboratory experiments. The HMW calculations closely coincide with the experimental data, reflecting the fact that these same data were used in parameterizing the model (Flarvie and Weare, 1980). The B-dot results coincide closely with the... 

Fig. 8.7. Molal concentrations m, and activities a, of calcium and sulfate species in equilibrium with gypsum at 25 °C as functions of NaCl concentration, calculated using the B-dot equation (left) and the hmw activity model (right).

To model the brine, we set the activity model and enter the chemical composition. The SPECE8 commands debye-huckel and hmw, respectively, prescribe the Debye-I Iiickel (B-dot) and Harvie-Mpller-Weare activity models. The hmw model does not account for bromine, so we must type remove Br- before invoking it. Taking the analysis for well RZ-2 as an example, the procedure in SPECE8 to invoke the Dcbyc-I Iuckcl model is... 

Figure 8.8 shows the resulting saturation indices for halite and anhydrite, calculated for the first four samples in Table 8.8. The Debye-Hiickel (B-dot) method, which of course is not intended to be used to model saline fluids, predicts that the minerals are significantly undersaturated in the brine samples. The Harvie-Mpller-Weare model, on the other hand, predicts that halite and anhydrite are near equilibrium with the brine, as we would expect. As usual, we cannot determine whether the remaining discrepancies result from the analytical error, error in the activity model, or error from other sources. 

Fig. 8.8. Saturation indices of Sebkhat El Melah brine samples with respect to halite (left) and anhydrite (right), calculated using the B-dot (modified Debye-Huckel) and Harvie-Mpller-Weare models.

Figure 2. Diabatic (left) and adiabatic (right) population probabiUties of the C (fuU line), B (dotted line), and X (dashed line) electronic states as obtained for Model II, which represents a three-state five-mode model of the benzene cation. Shown are (A) exact quantum calculations of Ref. 180, as well as mean-field-trajectory results [(B), (E)] and surface-hopping results [(C),(D),(F),(G)]. The latter are obtained either directly from the electronic coefficients [(C),(F)] or from binned coefficients [(D),(G)].

Figure 25. Diabatic and adiabatic population probabilities of the C (fuU line), B (dotted hne), and X (dashed line) electronic states as obtained for a five-state 16-mode model of the benzene cation.

Thirdly, another corollary of the first limitation, is the inconsistency and inadequacy of activity coefficient equations. Some models use the extended Delbye-Huckel equation (EDH), others the extended Debye-Huckel with an additional linear term (B-dot, 78, 79) and others the Davies equation (some with the constant 0.2 and some with 0.3, M). The activity coefficients given in Table VIII for seawater show fair agreement because seawater ionic strength is not far from the range of applicability of the equations. However, the accumulation of errors from the consideration of several ions and complexes could lead to serious discrepancies. Another related problem is the calculation of activity coefficients for neutral complexes. Very little reliable information is available on the activity of neutral ion pairs and since these often comprise the dominant species in aqueous systems their activity coefficients can be an important source of uncertainty. 

Fig. 15. (a) Normalized pure-exchange CODEX intensities E(tm) as a function of tm for the aromatic ternary CH and the quaternary Cquat in Td-G2(-Me),6 dendrimer (T=363K). The fit curve for the ternary carbons is a stretched exponential cxp[—(rln/rcyi with /I = 0.51 and tc = 401 ms. The dotted line indicates the final CODEX exchange intensities, (b) Motional model of the localized, cooperative dynamics in polyphenylene dendrimers, including two-site jumps of all phenyl substituents of a pentaphenyl benzene building block. As indicated by X-ray analysis and computer simulations, the peripheral aromatic rings are inclined by 30° with respect to an axis normal to the face of the central benzene ring. For details, see ref. 44. 

Figure 16. Two-electron three-center bonds, (a) Bonding model in which the solid lines are the connections between the atoms found crystallographically, while the dotted lines are the direction of the orbitals, which are indicative for the real geometry around the metal atoms. (b) A model for the calculation of the geometry around the metal center Cu,.. The angle y between the two orbitals on Cu., which are responsible for bonding to ligand L and the bridging ligand X. is the summation of the angles a and X-Cuc-L.

To calculate the activity coefficient of ions, virtually all geochemical modeling programs today use either a variation of the Debye-Hiickel equation or the Pitzer equations. Two variations of the Debye-Hiickel equation in common use are the Davies equation and the B-dot equation. 

Fig. 15.10. Standard deviation of frequencies

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