Deviations from Ideal Solutions Ratio Measures

4 DEVIATIONS FROM IDEAL SOLUTIONS RATIO MEASURES 

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). 

Consider the algebraic form (4.3.12) that results from an isothermal integration of the first part of the definition of fugacity. 

For the reference state, lets us choose a pure-component standard state the real (or hypothetical) pure substance at the temperature of the mixture and at some convenient 

The activity is a dimensionless, conceptual, state function. The notation used in the argument list for a, is intended to emphasize that the numerical value for the activity depends, not only on the state of the mixture (T, P, x ), but also on the choice of the standard state. At this point we have not completely identified the standard state we have said it is the pure substance at T but we have not specified the pressure PP or the phase. This leaves some flexibility in using the activity. For example, we might complete the choice of standard state by identifying it as the real pure liquid i at T and at its vapor pressure P (T). This is a common choice. However, as an alternative, we might also choose the (hypothetical) pure ideal gas at T and P of the mixture then the resulting activity would be closely related to the fugadty coefficient. Other choices are also possible, and some are convenient in certain situations. 

---

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]. 

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

By using vapour pressure measurements (14), the activity of the solute, a = yXmiceiie can be determined as the ratio between the partial pressure above the micellar solution and the partial pressure above the pure solute. In this case, the standard state refers to the pure liquid solute or to solute in a supercooled liquid state. The activity coefficient which gives the deviations from ideality is a way of measuring the interaction between solute and surfactant in the aggregates. Once the activity is determined, the standard Gibb s energy of solubilization could be obtained, as follows ... 

The activity coefficient of component i, y(-, is now defined as a measure of the deviation from the ideal solution behaviour as the ratio between the chemical activity and the mole fraction of i in a solution. 

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. 

To have a useful ratio that measures how a real fugacity deviates from that in an ideal solution, we return to the definition of the fugacity (4.3.8), and we integrate that definition from an ideal-solution state to the mixture of interest. For the fugadty of i, the... 

So, when y, and activity coefficient can be interpreted as a ratio measure for how the fugadty coefficient deviates from the standard-state fugacity coefficient ratio measures for deviations from the ideal gas to ratio measures for deviations from an ideal solution. Consequently, it provides a computational means for theories and equation-of-state models based on one kind of ideality (ideal gas) to be used in theories and models based on the other (ideal solution). The activity coefficient (5.5.5) is the one commonly encountered it is simply related to the excess chemical potential. 

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 activity coefficient is the ratio of the actual vapor pressure (Pa) to the vapor pressure calculated brom Raoult s law (xaPa )- It is also the correction factor that converts mole firaction to activity. Finally, it gives a direct measure of the deviation from Raoult s law. If the solution is ideal, Ya 1 When ya > 1> the deviations from Raoult s law are positive when ya < 1> the deviations are negative. 

---