The ideal-solution model

The simplest model for describing solution behavior is that for an ideal solution that obeys Raoult s law, 

The insertion of Raoult s law into the relevant thermodynamic functions results in the following mixing expressions for the simple case of a binary solution A-B  

The ideal-solution model is greatly oversimplified, but it provides a good basis for comparing the behaviors of real solutions. 

In many situations, we need to predict the properties of a mixture, given that we already know the properties of the pure species. To do this requires a model that can describe how various components mix. In mathematical terms, this means that we need to relate the Gibbs free energy of a mixture to the Gibbs free energy of the various pure components. One of the simplest models that achieves this is the ideal solution model. In this lecture, we present the ideal solution model. Then we apply this model to describe vapor-liquid equilibria, and as a result, derive Raoult s law. 

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Any convenient model for liquid phase activity coefficients can be used. In the absence of any data, the ideal solution model can permit adequate design. 

The simplest model beyond the ideal solution model is the regular solution model, first introduced by Hildebrant [9]. Here A mix, S m is assumed to be ideal, while A inix m is not. The molar excess Gibbs energy of mixing, which contains only a single free parameter, is then... 

In Section 3.2 the ideal solution model was introduced. The essential assumption of the ideal model is that there is no energy change associated with rearrangements of the atoms A and B. In other words the energies associated with a random distribution of A and B atoms and a severely non-random distribution, in which the A and B atoms are clustered, are equal. 

The entropy of mixing is the same as for the ideal solution model. 

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

The previous summary of activities and their relation to equilibrium constants is not intended to replace lengthier discussions [1,18,25,51], Yet it is important to emphasize some points that unfortunately are often forgotten in the chemical literature. One is that the equilibrium constants, defined by equation 2.63, are dimensionless quantities. The second is that most of the reported equilibrium constants are only approximations of the true quantities because they are calculated by assuming the ideal solution model and are defined in terms of concentrations instead of molalities or mole fractions. Consider, for example, the reaction in solution ... 

It is easily shown that equation 2.65 is a fair approximation only if the ideal solution model is valid and A, B, and C are in very low concentrations. By accepting unity activity coefficients for all the species, equation 2.63 leads one to the equilibrium constant calculated from the molalities (Km) ... 

C-t, which means, of course, that the ideal solution model is adopted, no matter the nature or the concentrations of the solutes and the nature of the solvent. There is no way of assessing the validity of this assumption besides chemical intuition. Even if the activity coefficients could be determined for the reactants, we would still have to estimate the activity coefficient for the activated complex, which is impossible at present. Another, less serious problem is that the appropriate quantity to be related with the activation parameters should be the equilibrium constant defined in terms of the molalities of A, B, and C. As discussed after equation 2.67, A will be affected by this correction more than A f//" (see also the following discussion). 

The calculations are rather easy and have already been performed for models like (1) the ideal solution model where enrichment is always confined to the outmost layer (29), (2) the ideal or regular solution model with one-layer enrichment, taking into account the difference in atomic radii (strain energy) (30-32), (3) the regular solution model with enrichment spread over n (up to 4) layers (35), and (4) intermetallic compounds (37). 

If a < 0, the enrichment is less than computed according to the ideal solution model (45, 50). For temperatures not too high compared with the... 

It should be noted that the use of the KVSj charts implies that both the gas phase and the hydrate phase can be represented as ideal solutions. This means that the Kvsi of a given component is independent of the other components present, with no interaction between molecules. While the ideal solution model is approximately acceptable for hydrocarbons in the hydrate phase (perhaps because of a shielding effect by the host water cages), the ideal solution assumption is not accurate for a dense gas phase. Mann et al. (1989) indicated that gas gravity may be a viable way of including gas nonidealities as a composition variable. 

Thus in the ideal solution model of a two phase field, a knowledge of the enthalpies of fusion of the pure components at their respective melting points allows simultaneous solution of these two equations for the two unknowns, NA(l> and NA(aj at the temperature of interest. 

The basis of the ideal solution model is that the thermodynamic activities of the components are the same as their mole fractions. Implicit in this assumption is the idea that the activity coefficients are equal to unity. This is at best an approximation and has been found to be invalid in most cases. Solutions in which activity coefficients are taken into account are referred to as "real" solutions and are described by equation 3.4. 

Whereas the ideal solution model applies over the entire range of concentrations, but only for very similar components, the ideally dilute solution model applies to any solution, but only over a very limited range of concentrations. From a microscopic point of view, the ideally dilute solution holds as long as solute molecules are almost always completely surrounded by solvent molecules and rarely interact with other solute molecules. 

In all calculations involving the ideal solution model, we assume that we know the molar Gibbs free energy of each of the pure species as a function of temperature and pressure. Mathematically, this means that know the form of the functions (T, p). Physically, this means that we know everything about the thermodynamics of the pure species. 

Now we will use the ideal solution model to develop a mathematical description of vapor-liquid equilibrium in a multicomponent solution. We will make the assumption that we have a system that is separated into a coexisting vapor and liquid phase. The vapor phase will be assumed to behave like an ideal gas, while the liquid phase will be assumed to behave as an ideal solution. 

In Chapter 4, we developed the ideal solution model, which enables the estimation of the properties of mixtures from knowledge of the thermodynamic behavior of the pure species. While the ideal solution model does provide accurate predictions for mixtures of relatively similar substances, many systems do exhibit substantial deviations from the ideal solution model. 

In this chapter, we present methods for mathematically describing the properties of non-ideal solutions — mixtures that deviate from the ideal solution model. 

To model a mixture that phase separates into two coexisting liquid phase, we need to add non-ideal terms (activity coefficients) to the ideal solution model. As an example of this, we examine the stability of the two-suffix Margules model, which has a molar Gibbs free energy of... 

This equation allows the prediction of the freezing curve of a mixture. It requires from knowledge of the freezing temperature and enthalpy of melting of the pure component a, as well as a model of the activity cocl licicnls of the liquid mixture. In the absence of information for the activity cocl ficients. the ideal solution model can be used (i.e., 7a = 1) Eq. (8.7) then reduces to the van t Hoff equation. 

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