Deviations from Ideal Gases Difference Measures

Our results, summarized in Table 4.1, imply that entropy does not necessarily measure the amount of disorder. When ideal gases are mixed (and disorder presumably increases), the entropy may increase, decrease, or remain constant, depending on how the mixing is done and on whether we are mixing different gases or samples of the same gas. Note that none of the results in Table 4.1 violate the second law. 

2 DEVIATIONS FROM IDEAL GASES DIFFERENCE MEASURES 

4 we found that to compute the thermodynamic properties of ideal-gas mixtures, we need only the mixture composition plus the pure ideal-gas properties at the same state condition as the mixture. In other words, tire properties of ideal-gas mixtures are easy to compute. We would like to take advantage of this, even for substances that are not ideal gases. To do so we introduce, for a generic property F, a residual property P, which serves as a difference measure for how our substance deviates from ideal-gas behavior. 

We define two classes of residual properties isobaric ones and isometric ones. The isobaric residual properties ( 4.2.1) are the traditional forms and use P as the independent variable. The isometric ones ( 4.2.2) use v as the independent variable and thereby simplify computations when our equation of state is explicit in the pressure such equations of state are now commonly used to correlate thermodynamic data for dense fluids. Although isometric property calculations may be more complicated than those for isobaric properties, with the help of computers, tiiis is not really an issue. 

These residual properties are defined only for those thermodynamic properties F that can be made extensive  

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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]. 

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. ... 

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... 

PV = nRT at all pressures, so PV/nRT = 1 at all pressures horizontal line). We can test a gas for ideal behavior by measuring P, V, n, and Tfor a sample of the gas at various pressures and then calculating PV/nRT. At pressures up to a few atmospheres, all of these plots show PV/nRT near 1, or nearly ideal behavior. Different gases deviate differently from ideal behavior, and these deviations from ideality become more pronounced at higher pressures. 

Here, both the excess chemical potential, and the activity coefficient measure the deviations from similarity of the two quantities. This is fundamentally different from the deviations from the ideal-gas behavior (i.e., total lack of interactions), discussed in section 6.1. Here the limiting behavior of the activity coefficient is... 

In principle, any device that has one or more physical properties uniquely related to temperature in a reproducible way can be used as a thermometer. Such a device is usually classified as either a primary or secondary thermometer. If the relation between the temperature and the measured physical quantity is described by an exact physical law, the thermometer is referred to as a primary thermometer otherwise, it is called a secondary thermometer. Examples of primary thermometers include special low-pressure gas thermometers that behave according to the ideal gas law and some radiation-sensitive thermometers that are based upon the Planck radiation law. Resistance thermometers, thermocouples, and liquid-in-glass thermometers all belong to the category of secondary thermometers. Ideally, a primary thermometer is capable of measuring the thermodynamic temperature directly, whereas a secondary thermometer requires a calibration prior to use. Furthermore, even with an exact calibration at fixed points, temperatures measured by a secondary thermometer still do not quite match the thermodynamic temperature these readings are calculated from interpolation formulae, so there are differences between these readings and the true thermodynamic temperatures. Of course, the better the thermometer and its calibration, the smaller the deviation would be. 

With few exceptions, thermodynamic property tabulations are calculated from P-V-T meaz.urements and from zero-pressure specific heat values derived from spectroscopic measurements. It should be noted that the zero-pressure (ideal gas) specific heat values, Cp, for the cryogenic fluids are generally known with an uncertainty of less than 3 parts in 10,000 whereas the random deviations of the P-V-T data are of the order of 2 to 5 parts in 1000. The phase boundaries involve a further complication and, consequently, must be defined by additional experimental data. As a minimum requirement, measurements of the vapor pressure are sufficient for the calculation of thermodynamic property differences due to a phase change. This is indicated by the Clapeyron equation, which may be expressed... 

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 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),... 

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. 

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. 

The ideal-gas equation accounts adequately for the properties of most gases under a variety of circumstances. The equation is not exactly correct, however, for any real gas. Thus, the measured volume for given values of P, n, and T might differ from the volume calculated from PV = nRT (A Figure 10.10). Although real gases do not always behave ideally, their behavior differs so little from ideal behavior that we can ignore any deviations for all but the most accurate work. 

To perform these measurements, a known amount of gas is introduced into a chamber containing the porous sample. The system is then allowed to reach equilibrium, after which the deviation of the equilibrium pressure from the ideal pressure that would result if no adsorption was taking place, is measured [38, 39). Under isothermal conditions, which can be achieved by immersing the sample in liquid nitrogen for example, the pressure difference can be related to a volume which, when divided by the mass of the sample, represents the specific volume of gas adsorbed (in cm g ). The results of these measurements are known as isotherm curves (see Figure 11.8). Depending on the experimental apparatus used, the pressure can either be adjusted incrementally or, in some of the latest adaptations, adjusted continuously at a rate that guarantees the attainment of quasi-equihb-rium conditions [40]. The pressure (as represented on the x-axis) is the relative pressure p/po where p is the supplied pressure and po is the saturation pressure... 

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