The Density Model

Another option for calculating K at high temperatures (and in this case high pressures as well) has been suggested by R. E. Mesmer (1985), who found empirically that the variation of many ionization constants in water along the liquid-vapor coexistence curve is accounted for rather well by an expression incorporating the density of the solvent, water, having the form 

Because hi AT is proportional to ArG°(13.8) and In p is specified at specific values of T and P, it follows that an expression relating In K and In p is logically equivalent to one giving ArG°as a function of T and P. In the next chapter we will see that an equation giving A,.G°as a function of T and P is called a fundamental equation, and that it implicitly contains information on the variation of all thermodynamic parameters with T and P. Therefore there are implicit relationships between the parameters pi, p2, and p3 and all other thermodynamic parameters. Anderson et al. (1991) show that for the heat capacity, this relation is 

These and other relationships that follow from the model are summarized in Table 13.2. This empirical formula apparently works well because (dafdT) as a function of temperature for water is U-shaped, which gives to the expression for ArC° (if AC° is negative) an inverted-U shape with a maximum around 100°C, which is the same shape that the C° of many aqueous ions have (see Chapter 17). Furthermore the variation of AC° at higher pressures is also fairly faithfully modeled by the expression for AC°. 

---

Several application of oxide models have been presented. These included the importance of within-die characterization and prediction for process optimization, and the use of the pattern dependent model in run by ran feedback control. Work has also been done to apply the density model to STI polish. We believe that the generalized framework presented for copper polish is also applicable to the modeling of dishing and erosion in STI CMP. 

AND/CAS] Anderson, G. M., Castet, S., Schott, J., Mesmer, R. E., The density model for estimation of thermodynamic parameters of reactions at high temperatures and pressures, Geochim. Cosmochim. Acta, 55,... 

To use the equation, one needs only the values of In K, AH°, and AC° for the reaction at the reference conditions, (which will frequently be 25° C, 1 bar, but could easily be some other conditions in cases where experimental data at high temperatures or pressures are involved), as well as the density of the solvent, water, at the desired P, T conditions. To obtain estimates of ArU° and ArC° for the reaction at T, P, values of a and P for the solvent are also required. As an example we show in Figure 13.6 the measured and predicted values of log K for the ionization of water to 300°C. Several other examples and additional details are given by Anderson et al. (1991). For reactions for which log if, AH°, and AC° at 25°C are available but little else, the Density Model is one of the best ways of obtaining estimates of log K and other parameters at higher temperatures and pressures. 

Figure 9.9 The standard (infinite diiution) heat capacity of aqueous NaCI. Data from Pitzer et ai. (1984) (iower curve). Upper two curves are from the density model crosses - data from Anderson et al. (1991) diamonds - data from program steam, Harvey et al. (2000). The Pitzer et al. data are also shown in Figure 10.12.

For reactions for which log K, AH°, and ACp at 25 °C are available but little else, the density model is one of the best ways of obtaining estimates of log T and other parameters at higher temperatures and pressures. 

Obviously both the isocoulombic and the density model methods are suitable only for reactions having only aqueous species, because solids, liquids and gases... 

For reactions in which the aqueous species are isocoulombic but minerals or other phases are also present, one simply uses the Maier-Kelley expression for log A, Equation (9.25), whether or not the pure phases in the reaction are compositionally balanced. For the density model, two expressions for the variation of log AT are combined, giving... 

---