Nonideal behavior

Deviations from ideal behavior are related to the atomic and molecular nature of a gas and to the existence of intermolecular forces. There are fundamentally two assumptions undertaken with the ideal gas equation of state. It is assumed that the gas particles have no volume and that there are no interactions between the gas particles. For instance, at T = 0, the ideal gas law implies PV = 0, which for any fixed pressure means = 0. Atoms and molecules are not hypothetical points in space that can collapse into a zero-volume existence their volume does not approach zero as the temperature approaches absolute zero. Thus, a proper description of a real gas requires a correction to the ideal gas law. As one example of such a correction, we could introduce a constant, Vq, that is the gas volume at T = 0. This means P(y - Vq) = nRT. Another difference is that ideal gas particles exhibit no attraction, whereas real atoms and molecules do. So, in many ways, the PVT behavior of a real gas tends to differ from that of an ideal gas, and this is manifested in different forms for P-V isotherms. 

The most complete knowledge about the behavior of some real gas offers the same predictive capability as for an ideal gas, that is, a mathematical relationship among P, V, and T. Finding such a formula for a particular real gas might be accomplished through measurement of numerous P, V, and T values for a given sample of the gas. Such a PVT data set might be used 

If the gas were ideal, all X, values would be zero, and so X represents the nonideality of the real gas. We can regard each X, as the value of some function X P,V,T) at the point (Pi, Vi, T, and then the objective in fitting is to find that function. There is a mathematical basis for claiming that this function in general can be written as an infinite power series of the three variables, though there may be simpler expressions that are more suitable. 

Appendix A includes a sechon on curve fitting that outlines a means to find the cs for a truncated power series expansion. We have, then, a general approach for finding an expression for X in ferms of P, V, and T. When that is obtained, we return to Equation 2.28 and write X = PV - nRT, now using the fitting expression in P, V, and T in place of X. Such a result is a real gas law based on the PVT laboratory data that are available. 

The general power series expansion often proves to be more elaborate than is necessary. For some gases, the deviation from ideality depends mostly on the pressiue, and then it is appropriate to limit the general power series expansion to a simpler form  

A tail at the beginning of a chromatographic peak, usually due to injecting too much sample. 

Now that we have defined capacity factor, selectivity, and column efficiency we consider their relationship to chromatographic resolution. Since we are only interested in the resolution between solutes eluting with similar retention times, it is safe to assume that the peak widths for the two solutes are approximately the same. Equation 12.1, therefore, is written as 

Solving equation 12.17 for Wb and substituting into equation 12.19 gives 

The retention times for solutes A and B are replaced with their respective capacity factors by rearranging equation 12.10 

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A component in a vapor mixture exhibits nonideal behavior as a result of molecular interactions only when these interactions are very wea)c or very infrequent is ideal behavior approached. The fugacity coefficient (fi is a measure of nonideality and a departure of < ) from unity is a measure of the extent to which a molecule i interacts with its neighbors. The fugacity coefficient depends on pressure, temperature, and vapor composition this dependence, in the moderate pressure region covered by the truncated virial equation, is usually as follows ... 

Condensed phases of systems of category 1 may exhibit essentially ideal solution behavior, very nonideal behavior, or nearly complete immiscibility. An illustration of some of the complexities of behavior is given in Fig. IV-20, as described in the legend. 

Green, D. B. Rechtsteiner, G. Honodel, A. Determination of the Thermodynamic Solubility Product, Xsp, of Pbl2 Assuming Nonideal Behavior, /. Chem. Educ. 1996, 73, 789-792. 

Physical Equilibria and Solvent Selection. In order for two separate Hquid phases to exist in equiHbrium, there must be a considerable degree of thermodynamically nonideal behavior. If the Gibbs free energy, G, of a mixture of two solutions exceeds the energies of the initial solutions, mixing does not occur and the system remains in two phases. Eor the binary system containing only components A and B, the condition (22) for the formation of two phases is... 

Thixotropy and Other Time Effects. In addition to the nonideal behavior described, many fluids exhibit time-dependent effects. Some fluids increase in viscosity (rheopexy) or decrease in viscosity (thixotropy) with time when sheared at a constant shear rate. These effects can occur in fluids with or without yield values. Rheopexy is a rare phenomenon, but thixotropic fluids are common. Examples of thixotropic materials are starch pastes, gelatin, mayoimaise, drilling muds, and latex paints. The thixotropic effect is shown in Figure 5, where the curves are for a specimen exposed first to increasing and then to decreasing shear rates. Because of the decrease in viscosity with time as weU as shear rate, the up-and-down flow curves do not superimpose. Instead, they form a hysteresis loop, often called a thixotropic loop. Because flow curves for thixotropic or rheopectic Hquids depend on the shear history of the sample, different curves for the same material can be obtained, depending on the experimental procedure. 

Since activity coefficients have a strong dependence on composition, the effect of the solvent on the activity coefficients is generally more pronounced. However, the magnitude and direc tion of change is highly dependent on the solvent concentration, as well as the liquid-phase interactions between the key components and the solvent. The solvent acts to lessen the nonideahties of the key component whose liquid-phase behavior is similar to the solvent, while enhancing the nonideal behavior of the dissimilar key. 

Chemical and physical nonlinearities are caused by interactions among the components of a system. They include such effects as peak shifting and broadening as a function of the concentration of one or more components in the sample. Instrumental nonlinearities are caused by imperfections and/or nonideal behavior in the instrument. For example, some detectors show a... 

The chloroform/polystyrene solution exhibits highly nonideal behavior. As shown by curve C in Figure 4, the x parameter for this solution rises from a low value to a high value as solvent concentration increases. However, as shown in Figure 5, the partial pressure of chloroform above a mixture of... 

DMSO and water form a solution with nonideal behavior, meaning that the properties of the solution are not predicted from the properties of the individual components adjusted for the molar ratios of the components. The strong H-bonding interaction between water and DMSO is nonideal and is the primary driver for the very hygroscopic behavior of DMSO. Even short exposure of DMSO to humid air results in significant water uptake. Water and DMSO nonideal behavior results in an increase in viscosity on mixing due to the extensive H-bond network. 

It can be seen in Table 6.9 that as pressure increases, nonideal behavior changes the equilibrium conversion. Note that like Ka, K also depends on the specification of the number of moles in the stoichiometric equation. For example, if the stoichiometry is written as ... 

Note that the standard state fugacity now carries a subscript, because for A it is defined by the nonideal behavior of A. Substituting Equation 15 into Equation 14 yields44 5... 

A common way of following the progress of a gas phase reaction with a change in the number of mols is to monitor the time variation of the total pressure, it. From this information and the stoichiometry, the partial pressures of the participants can be deduced, and a rate equation developed in those terms. Usually it is adequate to assume ideal gas behavior, but nonideal behavior can be taken into account with extra effort. Problem P3.03.06 is an example of nonideality. 

The furfural-cyclohexane phase diagram (Fig. 146) shows that you can have mixtures that exhibit nonideal behavior, without having to form an azeotrope. In sum, without the phase diagram in front of you, you shouldn t take the distillation behavior of any liquid mixture for granted. 

Thixotropy is one of the reversible time-dependent effects that constitute nonideal behavior (Fig. 54). 

Because of the highly nonideal behavior of polymer-solvent solutions, polymer-vapor equilibrium relations account for the third major difference found in stripping operations with polymeric solutions. The appro-... 

Should the macromolecules interact with each other, then d In yildcj does not vanish. In actual experience, its value is almost always positive, largely because of excluded volume effects. Then, c,[0 In Jt/dc,] will then increase in magnitude as Cj increases and r decreases. Thus, the downward curvature shown in curve B of Figure 21.3 is typical of nonideal behavior. 

Seawater has high concentrations of solutes and, hence, does not exhibit ideal solution behavior. Most of this nonideal behavior is a consequence of the major and minor ions in seawater exerting forces on each other, on water, and on the reactants and products in the chemical reaction of interest. Since most of the nonideal behavior is caused by electrostatic interactions, it is largely a function of the total charge concentration, or ionic strength of the solution. Thus, the effect of nonideal behavior can be accoimted for in the equilibrium model by adding terms that reflect the ionic strength of seawater as described later. 

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