Ideal Solution Approximation

The accuracy of the approximation given by Eq. (8.23) in relation to the numerical solution of Eq. (8.22) is illustrated in Figs. 8.4 and 8.5. Solid curves in Fig. 8.4 show the results of calculations of jw and 7hp based on function Y(X) obtained by direct integration of Eq. (8.22) along isotherm T = 523 K. Dashed curves in Fig. 8.4 correspond to the approximation given by Eq. (8.23). The solid curve in Fig. 8.5 shows the results of direct integration of Eq. (8.22). The dotted curve in Fig. 8.5 corresponds to the approximation given by Eq. (8.23) the dashed curve corresponds to ideal solution approximation. 

It follows from Fig. 8.4 that at x 0.02, mean and maximum discrepancies A7, /7,- are, respectively, fractions of percent and 4%-6%. However, gas-phase HP concentrations determined by Eqs. (8.20) and (8.23) differ from those obtained by direct integration of Eq. (8.22) no more than by 0.011 at T = 423 K and 

The fact that coefficients and 7hp can be considerably less than unity (see Figs. 8.3 and 8.4) indicates that aqueous solutions of HP do not obey the ideal 

---

Most real solutions cannot be described in the ideal solution approximation and it is convenient to describe the behaviour of real systems in terms of deviations from the ideal behaviour. Molar excess functions are defined as... 

The ideal solution approximation is well suited for systems where the A and B atoms are of similar size and in general have similar properties. In such systems a given atom has nearly the same interaction with its neighbours, whether in a mixture or in the pure state. If the size and/or chemical nature of the atoms or molecules deviate sufficiently from each other, the deviation from the ideal model may be considerable and other models are needed which allow excess enthalpies and possibly excess entropies of mixing. 

In order to explore the thermodynamic properties, and especially the chemical potential ofthe intercalation compounds, a lattice gas model [10] has been adopted under the assumption that intercalated ions are localized at specific sites in the host lattice, with no more than one ion on any site, and that local and global electroneutrality is observed and there is no strong interaction between the electrons and the intercalated ions. It should be noted that, in solid-state chemistry, this model is often referred to as ideal solution approximation when used to describe the thermodynamics of nonstoichiometric compounds. According to this model, the chemical potential of A in A8MO2 in Equation (5.3) can be divided into two terms as... 

The cell theory plus fluid phase equation of state has been extensively applied by Cottin and Monson [101,108] to all types of solid-fluid phase behavior in hard-sphere mixtures. This approach seems to give the best overall quantitative agreement with the computer simulation results. Cottin and Monson [225] have also used this approach to make an analysis of the relative importance of departures from ideal solution behavior in the solid and fluid phases of hard-sphere mixtures. They showed that for size ratios between 1.0 and 0.7 the solid phase nonideality is much more important and that using the ideal solution approximation in the fluid phase does not change the calculated phase diagrams significantly. 

Figure 8.5 Predicted dependencies of HP mole fractions in the gas phase on the HP mole fraction aqueous solution at T = 523 K J — integration of Duhem s equation 2 — ideal solution approximation and 3 — approximation given by Eq. (8.23).

In most cases the solution is sufficiently dilute to use the ideal-solution approximation for the bulk value,... 

In the case of NaCl particles at AT 1, the value calculated using the approach of Tang (1996) was the lowest, and a higher S value was obtained with the approach of Kreidenweis et al. (2005) and the ideal solution approximation. 

The simplest model is the ideal solution approximation (( )=1). Although it may be too simple (e.g., Young and Warren, 1992), this approximation has been used to calibrate CCNCs in many studies (e.g., Raymond and Pandis, 2002 Roberts and Nenes, 2005). The manufacture of CCNC also relies on this model (Shilling et al., 2007). Thus, it is important to compare this model with more sophisticated models. 

In 1955, Flory applied the ideal-solution approximation (Flory 1954), and assumed that the comonomer B could not enter the crystalline phase of A. He derived that... 

Since the basic Debye s treatment was based on an ideal solution approximation, workers attempted to take into account the various departures from ideal behaviour [66, 67, 68], namely ... 

Figure 5 Nucleation free energy AG at T = 1000 K for different concentration of the solid solution. Square symbols correspond to CVM-TO calculation and the line to the ideal solution approximation (eq. 26). Full and open symbols are respectively for the set of parameters with and without order corrections on triangles and tetrahedrons.

---