Deviations from Ideal Solutions Difference Measures

For the Lewis-Randall ideal solution, these expressions simplify as follows. 

Item (b) means that, at fixed T and P, work must always be done on a Lewis-Randall ideal solution to separate it into its pure components. Further note that this work does not depend on phase the minimum work to separate a liquid ideal solution at T, P, and x is the same as that to separate an ideal-gas mixture at the same T, P, and x). 

2 DEVIATIONS FROM IDEAL SOLUTIONS DIFFERENCE MEASURES 

Although no real mixture is truly ideal, we can often use the concept of an ideal solution to reduce the labor needed to compute property values for real mixtures. To do so we introduce, for each property f, an excess property/.  

Here/represents an intensive property value for the real mixture, and all three terms in (5.2.1) are at the same temperature T, pressure P, composition x, and phase. The excess properties provide a convenient way for measuring how a real mixture deviates from an ideal solution. In general, an excess property/ may be positive, negative, or zero. An ideal solution will have all excess properties equal to zero. Note that the value for depends on the choice of standard state used to define the ideal solution. Further note that the definition (5.2.1) is not restricted to any phase excess properties may be defined for solids, liquids, and gases, although they are most commonly used for condensed phases. 

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This could also be carried out for the polymer (N2), but as it makes no difference which one is taken (both having started from AG ), Equation 8.31 is more convenient to use. Although this expression is not strictly vahd for the dilute solution regime, it can be converted into a structure that is extremely informative about deviations from ideal solution behavior encoimtered when measuring the molar mass by techniques such as osmotic pressure. If the logarithmic term is expanded using a Taylor series,... 

We start the development in 5.1 by defining ideal solutions and giving expressions for computing their conceptual properties. In 5.2 we introduce the excess properties, which are the differences that measure deviations from ideal-solution behavior, and in 5.3 we show that excess properties can be computed from residual properties. In 5.4 we introduce the activity coefficient, which is the ratio that measures deviations from ideal-solution behavior, and in 5.5 we show that activity coefficients can be computed from fugacity coefficients. This means that deviations from ideal-solution behavior are formally related to deviations from ideal-gas behavior, but in practice, one kind of deviation may be easier to compute than the other. Traditionally, activity coefficients have been correlated by fitting excess-property models to available experimental data simple forms for such models are introduced in 5.6. Those few simple models are enough to allow us to exercise many of the relations presented in this chapter however, more thorough discussions of models for excess properties and activity coefficients must be found elsewhere [1, 2]. 

In addition to the excess properties, which are difference measures for deviations from ideal-solution behavior, we also find it convenient to have ratio measures. In particular, for phase equilibrium calculations, it proves useful to have ratios that measure how the fugacity of a real mixture deviates from that of an ideal solution. Such ratios are called activity coefficients. The activity coefficients can be viewed as special kinds of a more general quantity, called the activity so we first introduce the activity ( 5.4.1) and then discuss the activity coefficient ( 5.4.2). 

This relates a difference measure to a ratio measure for deviations from ideal-solution behavior. 

In 5.3 we showed how excess properties, which are difference measures for deviations from ideal-solution behavior, can be obtained from residual properties, which are difference measures for deviations from ideal-gas behavior. In this section we establish a similar set of equations that relate activity coefficients to fugacity coefficients. As a result, the equations given here, together with those in 5.3, establish a complete connection between the description of mixtures based on models for PvTx equations of state and the description based on models for and y. ... 

Consequently, in the first and third cases the systems behave in the same way, while in the second case the system behaves differently. It is appropriate to note that in the first case IC22 is small in absolute value, in the third case it is positive and large, while in the second case it is large in absolute value but negative. Obviously, the quantity/ 22 is a measure of the nonideality of the system. In the first case, the system behaves almost as an ideal one in the second case it behaves like a nonideal system with a strong positive deviation from ideality and in the third case like a nonideal system with a strong negative deviation from ideality. In the framework of the Kirkwood-Buff theory of solution the... 

There are essentially three significant quantities that can be derived from the inversion of the KB theory. The first is a measure of the extent of deviation from symmetrical ideal (SI) solution behavior, A b. defined below in the next section. It also provides a necessary and sufficient condition for SI solution. The second is a measure of the extent of preferential solvation (PS) around each molecule. In a binary system of A and B, there are only two independent PS quantities these measure the preference of, say, molecule A to be solvated by either A or B molecules. Deviations from SI solution behavior can be expressed in terms of either the sum or difference of these PS quantities. Finally, the Kirkwood-Buff integrals (KBIs) may be obtained from the inversion of the KB theory. These provide information on the affinities between any two species for instance, PaGaa measures the excess of the average number of A particles around A relative to the average number of A particles in the same region chosen at a random location in the mixture. All these quantities can be obtained from the KB integrals. 

At this point it is convenient to discuss very briefly the relative magnitudes of the activity coefficients of the ions and of the uncharged parent electrolyte. In the region of low concentrations where molality and mole firaction are proportional to each other, the extent to which the activity coefficients differ from unity is a measure of the deviations from ideality in the solution, and this in its turn depends on the forces between the various components of the solution. 

Under standard conditions, = 55.54 mol/L, and as a good first approximation, rH20 = 1- For nondissociating chemicals of the type treated here, and in sufficiently dilute solutions, the deviations from ideality may be ignored. One can then determine Km from a measurement of the observed molality of each component in a very dilute solution. Otherwise the different activity coefficients must be determined as prescribed by Section 3.6. This value of Km applies to any other solution generated under the same prescribed conditions of temperature and pressure. Once Km has been fixed in a given experiment, the activities of the three solutes under other conditions are interrelated as shown in Eq. (3.4.25). 

Cyclic voltammetry at different applied potential scan rates is normally the first technique employed to study the properties of polymer-modified electrodes. From current-potential curves recorded with solutions containing supporting electrolyte only and at scan rates sufficiently low for thin layer conditions to prevail both the formal potentials and the amounts of redox species incorporated in the surface coating can easily be measured. In addition, deviations from ideal behaviour provide information on interactions within the polymer phase and, when uncompensated resistance effects can be accounted for, on the reversibility of electron transfer at the electrode-coating interface. Figure 4 shows a typical cyclic voltammogram obtained from a glassy-carbon electrode covered with... 

Another possible variable for the characterization of the goodness of solvents is the interaction parameter x values, expressing the measure of deviations of actual solutions from ideal ones. This value can be determined by several methods, which are, mostly experimentally demanding and time-consuming, x is dependent on both the polymer concentration and molecular weight and information provided about the specific interactions in the solution is of no particular interest [11,30,31], Solvents, obviously different in quality, yield quite close values and thus the resolving capability is low. Comparison of results obtained by various methods and/or experimenters is thus fairly difficult [30,32-34],... 

Therefore we seek ways for computing conceptuals of condensed phases while avoiding the need for volumetric equations of state. One way to proceed is to choose as a basis, not the ideal gas, but some other ideality that is, in some sense, "doser" to condensed phases. By "closer" we mean that changes in composition more strongly affect properties than changes in pressure or density. The basis exploited in this chapter is the ideal solution. We still use difference measures and ratio measures, but they will now refer to deviations from an ideal solution, rather than deviations from an ideal gas. 

In this chapter we developed ways for computing values for conceptuals relative to their values for any well-defined ideal solution. The computational strategy is based on quantities that reveal how a property deviates from its ideal-solution value the excess properties are difference measures, while the activity coefficient is a ratio measure. In other words, the strategy used in this chapter repeats that used in Chapter 4,... 

At this point we have developed two principal ways for relating conceptuals to measurables one based on the ideal gas (Chapter 4) and the other based on the ideal solution (Chapter 5). Both routes use the same strategy—determine deviations from a well-defined ideality—with the deviations computed either as differences or as ratios. Since both routes are based on the same underlying strategy, a certain amoxmt of s)un-metry pertains to the two for example, the forms for the difference measmes— the residual properties and excess properties—are functionally analogous. 

That basic strategy is illustrated in Table 6.1. First we define an ideal mixture whose properties we can readily determine. Then for real mixtures we compute deviations from the ideality as either difference measures or ratio measures. In one route the ideality is the ideal-gas mixture, the difference measures are residual properties, and the ratio measure is the fugacity coefficient. In the other route the ideality is the ideal solution, the difference measures are excess properties, and the ratio measure is the activity coefficient. 

The first term in parenthesis on the right-hand side of eg. fi2.42l reflects entropic effects that arise from the number of possible ways that macromolecules and solvent can be arranged in space this term is also known as the combinatorial contribution. The second term on the right-hand side is the enthalpic contribution and arises from differences between polymer-polymer and polymer-solvent interactions this term is also referred to as the residual contribution (not to be confused with the residual properties introduced earlier, which measure deviations from the ideal-gas state). Even if this term is zero (i.e., x = o), the solution is nonideal due to the size difference between polymer and solvent. 

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