Summary Ideal Solutions

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 ... 

The most common model for describing adsorption equilibrium in multi-component systems is the Ideal Adsorbed Solution (IAS) model, which was originally developed by Radke and Prausnitz [94]. This model relies on the assumption that the adsorbed phase forms an ideal solution and hence the name IAS model has been adopted. The following is a summary of the main equations and assumptions of this model (Eqs. 22-29). 

In summary, there are three important characteristics of ideal solutions that one should remember in assessing the properties of any non-ideal system (i) the vapor pressure of each component is proportional to its mole fraction in solution over the whole composition range (Raoult s law) (ii) the enthalpy of mixing is zero (iii) the volume change associated with mixing is zero. The sections which follow deal with non-ideal solutions. 

In summary, then, either the gas or the solid ideal solution model applied to a mixture of exchange cations would convert equation 3.6 to an ion exchange equation of the form... 

Table 4-1 Summary of methods and their applications for separating mixtures which form ideal solutions...

The general principles established for ideal solutions, such as Raoult s law in its various forms, are of course applicable to solutions of any number of components. Similarly, the Gibbs-Duhem equation is applicable to nonideal solutions of any number of components, and as in the case of binary mixtures various relationships can be worked out relating the activity coefficients for ternary mixtures. This problem has now been attacked from several points of view, a most excellent summary of which is presented by Wolil (35). His most important results pertinent to the problem at hand are summarized here. 

In summary, the reference state for species i in the liquid (or solid) phase is no more than a particular state, real or hypothetical, at a given P and Xi (usually that of the system) and at the temperature of the system. We choose the reference state to be that of an ideal solution in which the fugacity is linearly proportional to mole fraction. While the concept of an ideal solution was conjured up in analogy to an ideal gas, there are some interesting differences. A pure gas can be a nonideal gas, while a pure liquid cannot be a nonideal solution because all intermolecular forces in a pure liquid are the same Additionally, an increase in pressure leads to deviations from ideal gas behavior, whereas deviations from ideal solution are caused by changes in composition because nonideal behavior results primarily from the chemical differences of species in a mixture, even at low pressures. 

In summary, we see now how tire change from tire expanded chains in dilute solutions to tire ideal chains in a melt is accomplished. Witli increasing polymer concentration, tire chain overlap increases and tire lengtli scale over... 

Isotherm Models for Adsorption of Mixtures. Of the following models, all but the ideal adsorbed solution theory (lAST) and the related heterogeneous ideal adsorbed solution theory (HIAST) have been shown to contain some thermodynamic inconsistencies. References to the limited available Hterature data on the adsorption of gas mixtures on activated carbons and 2eohtes have been compiled, along with a brief summary of approximate percentage differences between data and theory for the various theoretical models (16). In the following the subscripts i and j refer to different adsorbates. 

In a similar way, anthracene triplet (4>,gj3=0.71, z =6A,700Mr cmr ) and the naphthalene triplet (4>jg = 0.75, e j = 24,500 M" cm" ) in cyclohexane solution have been introduced as transient chemical actinometers for the third-harmonic (355 run) and fourth-harmonic (266 nm) output of Nd YAG lasers, respectively (44). In summary, transient chemical actinometers are ideal for accurately measuring the energy of single laser pulses, provided the quantum yields and extinction coefficients of the transients are well known (45 7). Thus, the well-established benzophenone actinometer (42-44) has been used as a reliable reference to calibrate the azobenzene actinometer (see section "Laser Intensity Measurements with the Azobenzene Actinometer" Doherty S, Hubig SM, unpublished results) and the Aberchrome 540 actinometer (48,49) for intensity measurements with pulsed Nd YAG and/or XeCl excimer lasers. However, such actinometer can only be used when a complete set of laser flash photolysis equipment including a kinetic spectrometer is available. 

In summary, there is an analytical solution of the two-component ideal model in one single, very specific case that of a rectangular pulse of a mixture of two... 

Summary We have derived a deceptively simple equation, AG°= -Rl n K, which enables us to calculate equilibrium constants from AG° data. Having derived and discussed the concept of activity, we can easily take into account the non-ideal behaviour of solutions, and of gases at significant pressures. However, in cases where solutions are dilute, or gas pressures low, calculations for the ideal case are also easily made, by putting all activity coefficients equal to unity. 

In summary, the recent work of Kramers, Wannier, Onsager, and Kaufmann provides an exact solution of two-dimensional nearest neighbor problems of the present idealized type but approximate approaches, for example the quasi-chemical method, must still be used in taking into account second and higher neighbor interactions, heterogeneous surfaces, etc. 

In summary, the major feature of the dynamic model just described is the approximation that solute-solvent and solvent-solvent collisions can be described by hard-sphere interactions. This greatly simplifies the calculations the formal calculations are not difficult to carry out in the more general case, but the algebra is tedious. We want to describe the effects of solute and solvent dynamics on the reactive process as simply as possible, and the model is ideal for this purpose. Specific reactive events among the solute molecules are governed by the interaction potentials that operate among these species. The particular reactive model described here allows us to examine certain features of the coupling between reaction and diffusion dynamics without recourse to heavy calculations. More realistic treatments must of course be handled via the introduction of species operators for the system under consideration. 

In summary, a value of 1.0 for 7 allows us to obtain values for the properties of the standard state, which we mentioned earlier is the essential factor in the choice of standard state. As for rn°, any value could be used, but none has any advantage over 1.0. Therefore the hypothetical ideal one molal standard state is in universal use for dilute solution (molality-based) activities. 

Great significance in the first approximation equation has parameter I, which is the solution ionity or summary ionic strength. It was introduced by Gilbert Newton Eewis and Merle Randall (1888-1950) for the purpose of characterizing the solutions electric field intensity and the extent of its deviation from the ideal state. The value of ionic strength is equal to half of the sum of the product of ion concentrations and their squared charge ... 

Many attempts have been made to find theoretical explanations for the non-ideal behavior of polymer solutions. There exist books and reviews on this topic, e.g., Refs. " " Therefore, only a short summary of some of the most important thermodynamic approaches and models will be given here. The following explanations are restricted to concentrated polymer solutions only because one has to describe mainly concentrated polymer solutions when solvent activities have to be calculated. For dilute polymer solutions, with the second virial coefficient region, Yamakawa s book provides a good survey. 

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