Deviations from Ideal Gases Ratio Measures

But for second-law properties, the relations are not as simple as (4.2.24) and (4.2.25). To have a representative second-law property, consider the entropy. The Maxwell relation (3.3.34) leads to 

With m , and s determined, we can use the defining Legendre transforms to relate the residual Gibbs energy, Helmholtz energy, and chemical potential. The results are 

We caution that (4.2.31) cannot be derived in a simple way by applying the partial molar derivative to the difference in residual Gibbs energies given in (4.2.30). The difficulty is that the partial molar derivative imposes a fixed pressure, but when the Ihs of (4.2.30), g (T, V, x ), is changed at fixed pressure, the mixture and ideal-gas volumes are no longer the same. Consequently, the isobaric derivative of the Ihs of (4.2.30) is not an isometric residual property in particular, it is not the Ihs of (4.2.31). 

In all the equations relating second-law residual properties (4.2.28)-(4.2.31), the compressibility factor Z is to be evaluated at the state (T, P, v, x ) of the real substance of interest. The state dependence of Z is discussed in the next section. 

3 DEVIATIONS FROM IDEAL GASES RATIO MEASURES 

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In 4.1 we introduce ideal gases and their mixtures, and we derive equations for computing their thermodynamic properties. Then, we use the rest of the chapter to develop expressions for computing deviations from ideal-gas values the difference measures in 4.2, the ratio measures in 4.3. 

Besides difference measures, it is frequently convenient to describe deviations from ideality by using ratio measures. In this section we present the ratio measures commonly employed to measure deviations from ideal-gas behavior the compressibility factor and the fugacity coefficient. 

Figure 4.4 compares the two ratio measures, Z and (p, for deviations from ideal-gas behavior for pure ammonia along the subcritical isotherm at 100°C. The figure shows that Z(P) is discontinuous across the vapor-liquid phase transition, while liquid phases have different molar volumes. In contrast, cp(P) appears continuous and smooth, though in fact it is only piecewise continuous. That is, the (p(P) curves for vapor and liquid intersect at the saturation point, but they intersect with different slopes. Near the triple point that difference in slopes is marked, but near the critical point the difference is small the... 

In this chapter we have developed ways for computing conceptual thermodynamic properties relative to well-defined states provided by the ideal gas. We identified two ways for measuring deviations from ideal-gas behavior differences and ratios. Relative to the ideal gas, the difference measures are the isobaric and isometric residual properties, while the ratio measures are the compressibility factor and fugacity coefficient. These differences and ratios all apply to the properties of any single homogeneous phase (liquid or gas) composed of any number of components. 

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]. 

The data in Table 6.2 provide us with clear evidence that real gases are not "ideal." We should comment briefly on the conditions imder which a real gas is ideal or nearly so and what to do when the conditions lead to nonideal behavior. A useful measure of how much a gas deviates from ideal gas behavior is found in its compressibility factor. The compressibility factor of a gas is the ratio PV/nRT. From the ideal gas equation we see that for an ideal gas, PV/nRT = 1. For a real gas, the compressibility factor can have values that are significantly different... 

If measurements of pressure, molar volume, and temperature of a gas do not confirm the relation pV = RT, within the precision of the measurements, the gas is said to deviate from ideality or to exhibit nonideal behavior. To display the deviations clearly, the ratio of the observed molar volume V to the ideal molar volume Vi (=RT/p) is plotted as a function of pressure at constant temperature. This ratio is called the compressibility factor Z. Then,... 

To compute values for the deviation measures, we need volumetric data for the substance of interest such data are usually correlated in terms of a model PvTx equation of state. In 4.4 we develop expressions that enable us to use equations of state to compute difference and ratio measures for deviations from the ideal gas. Finally, in 4.5 we present a few simple models for the volumetric equation of state of real fluids. These few models are enough to introduce some of the problems that arise in attempting to analytically represent the PvTx behavior of real substances, and they allow us to compute values for conceptual, using the expressions from 4.5. However, more thorough expositions on equations of state must be found elsewhere [1-4]. 

The compressibility factor Z serves as a ratio measure for how a real-substance volume deviates from that of an ideal gas at the same T and P,... 

The compressibility factor serves a purpose similar to that of the isobaric residual volume both measure how the volume of a substance deviates from the ideal-gas volume at the same T and P. The distinction is that one is a difference, while the other is a ratio. But the two are related the relation is found by combining (4.2.2) with (4.3.1),... 

In this subsection we introduce a ratio measure that indicates how the fugacity of a real substance deviates from that of an ideal gas. As the reference state, we choose the ideal-gas mixture at the same temperature, pressure, and composition as our real mixture. Then, on integrating the definition of fugacity (4.3.8) from the ideal-gas state to the real state, we obtain an algebraic form analogous to (4.3.12) that is, we find... 

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. 

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. 

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. 

Since a perfect gas behavior was assumed in the derivation of Eq, (6.76), caution is advised in the use of this equation when the pressure of the gas mixture deviates appreciably from this assumption. For example, experimental measurements have shown that the actual water vapor content in air will be over four times that predicted by ideal gas behavior at a temperature of — 227K and a pressure of 20.2 MPa. Familiarity with these deviations is necessary if problems are to be avoided with this method of impurity removal. Data of this type are available as enhancement factors, defined as the ratio of the actual molar concentration to the ideal molar concentration of a specific impurity in a given gas. 

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