Corrections for non-ideality

K = Specific heat ratio, at inlet conditions given for some substances in Table 1. Note Published values of K at 15 °C and one atmosphere may be used. If K is unknown, a conservative value of K = 1.001 may be used, in which case the factor C = 315. Note that a correction for non-ideal gases may be necessary. 

Later, we will make equilibrium calculations that involve activities, and we will see why it is convenient to choose the ideal gas as a part of the standard state condition, even though it is a hypothetical state/ With this choice of standard state, equations (6.94) and (6.95) allow us to use pressures, corrected for non-ideality, for activities as we make equilibrium calculations for real gases.s... 

However, it is usually easy, if a little laborious, to correct for non-ideality. Measurement of Mw,app over a range of loading concentrations is necessary followed by an extrapolation to zero concentration using an equation of the form [76]... 

Work Wei is then identified with the correction for non-ideal behaviour AGB and the activity coefficient is obtained from the equation... 

Hence, the expression used to calculate Molecular weight corrected for non-ideal behaviour as also for asymmetric scattering will be... 

The object of this work was to extend the field of application of the equation-of-state method. The method was applied to aqueous systems in conjunction with a model that treats water as a mixture of a limited number of polymers, an approach similar to that previously adopted for the carboxylic acids (2). Association is calculated by the law of mass action corrections for non-ideal behaviour are made by means the equation of state. A major problem of the method is the large number of parameters needed to describe the properties and concentrations of the polymers together with their interaction with molecules of other substances. The Mecke-Kemptner model (15) (also known as the Kretschmer-Wiebe model (16) and experimental values for hydrogen-bond energies were usecT for guidance in fixing these parameters. 

Analytical solutions for x and y as functions of the bed-length, z, and time, t, are available [45,52], The expressions are a useful extension of two-phase model applied to plug-flow. These two models are appropriate in describing the extraction of crushed or broken seeds to recover the seed oil, either in shallow beds or in plug flow. As shown by Sovova [52], applying the plug-flow model requires corrections for non-ideal residence-time distribution (non-plug flow) of the fluid in contact with the solid. 

There is often considerable confusion about the use of equilibrium constants, K (Frames 39, 41, 42, 43, 45, 46, 47, 49) (and whether they have units or not) and indeed about the use of formulae (like that for chemical potential, p., Frames 5,27, 28, 29, 35, 37, 38 and 39) which contain logarithmic (Frames 6 and 36) terms. The importance of logarithmic terms as corrections for non-ideal behaviour has been referred to earlier (see section 39.3, Frame 39). 

Ft is the molar attraction function, Fp its polar component (both as discussed earlier) V is the molar volume of the solvent molecule or the structural unit of the polymer. AT is the Lydersen correction for non-ideality, used in the auxiliary equations. The values for low-molecular liquids were derived by Lydersen (1955) the corresponding values for polymers, which are slightly different, have been derived by Hoy (AT(P)). 

Table 3.2. Correction for non-ideality for a few gases commonly used in adsorption experiments.

This method of correcting for non-ideal gas behavior can be extended to gas mixtures if the concepts of pseudo-critical pressure and pseudo-critical temperature are introduced. These quantities are defined... 

All of the equilibrium relations quoted above are ideal expressions, where quotients of concentrations are constant. But in fact all the solutions will be non-ideal and the observed equilibrium constant will have to be corrected for non-ideality using activity coefficients for each species appearing in the equilibrium expression. In practice, extrapolation procedures... 

The three topics of ion pairing, complex formation and solubilities are typical aspects of equilibrium in electrolyte solutions, and are handled in precisely the same manner as acid-base equilibria. As in the calculations on acid-base equilibria, only the ideal case is considered. Discussion of corrections for non-ideality are deferred until Sections 8.22 to 8.28. In this chapter pay special attention to ... 

The quotients or products which appear in the expressions for AG, AG and K are of activities, and are therefore applicable to all conditions, ideal or non-ideal. So far in this book, such quantities have been given in terms of concentrations, and, as such, are non-ideal quantities unless they are for infinite dilution where all concentrations —> 0. However, many experimental determinations are based on concentrations and, in principle, should be corrected for non-ideality. This will be dealt with in Sections 8.21, 8.24 to 8.27. 

The solubility of Hg2Cl2(s) is so low that the saturated solution will be so dilute as to be ideal. Hence corrections for non-ideality are not needed. 

Experimental determinations made in terms of concentrations give concentration quotients which are non-ideal constants. Corrections for non-ideality are made in terms of the calculated ionic strength and the various Debye-Huckel expressions. However, emf experiments, including pH measurements, can sometimes furnish equilibrium constants directly in terms of activities, and as such these will be ideal equilibrium constants. 

From this it is clear that the equilibrium constant expressed in concentrations, i.e. A non-ideai> is not a tme constant but depends on the particular ionic strength at which the measurements are made. This applies, in particular, to any equilibrium which involves charged species or charge-separated species, since these are more likely to cause non-ideality than molecular species. All concentration equilibrium constants should thus be corrected for non-ideality. 

For most equilibrium constant measurements the amount of a reactant or product present is found as a concentration, e.g. spectrophotometric or conductance analyses. However, pH and some emf methods determine the activity of a species directly rather than a concentration, and so corrections for non-ideality for these species will not be necessary. But, there are also some situations where, although the basic experimental measurement is an activity, subsequent calculations involve stoichiometric relations given in concentrations. Unless care is taken, the final equilibrium constant could end up involving terms in activities and concentrations, i.e. is mixed. Here corrections for non-ideality will still have to be made. Specific cases will make this clearer. 

Correcting for non-ideality gives a much smaller correction here, where pH measurements are the basis, than was found for the same solution when spectrophotometry was the basis. 

Comparison of non-graphical and graphical methods of correcting for non-ideality... 

Be very careful here The total potential at any distance, r, from the origin can be identified with the potential at any distance, r from the central z e ion due to that ion itself, only if nonideality is ignored. This means that corrections for non-ideality must be superimposed onto the Bjerrum theory after the association constant has been derived (see Section 10.12.4). 

For higher concentrations, Aiassoc will have to be corrected for non-ideality by including activity coefficients calculated from the Debye-Htickel equation. 

In solutions of weak electrolytes, the concentration of ions is often sufficiently low for the solution to approximate to ideality. However, with moderately weak electrolytes, it becomes possible to have higher concentrations of ions present and deviations from the behaviour described above become apparent. For such electrolytes corrections for non-ideality must be made (see Chapter 12). 

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