Dilute Behavior

A dilute system is one in which the number of particles in any level is much less than the number of particles that can be in the level. Returning to Equation 11.10, we see that with s, the effect of 6 becomes insignificant. Consider one level with n = 2, first with s = 4 and then with s = 100. From Equation 11.10 we obtain 

With s = 4, A for 8 = 1 differs by 25% of the value of A for 8 = 0, whereas with s = 100, the difference is only 1%. Under dilute conditions, the number of arrangements approaches the same value for all three types of particles. For sufficiently dilute systems the statistics of indistinguishable boltzons is appropriate for all particles. 

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For gas-liquid solutions which are only moderately dilute, the equation of Krichevsky and Ilinskaya provides a significant improvement over the equation of Krichevsky and Kasarnovsky. It has been used for the reduction of high-pressure equilibrium data by various investigators, notably by Orentlicher (03), and in slightly modified form by Conolly (C6). For any binary system, its three parameters depend only on temperature. The parameter H (Henry s constant) is by far the most important, and in data reduction, care must be taken to obtain H as accurately as possible, even at the expense of lower accuracy for the remaining parameters. While H must be positive, A and vf may be positive or negative A is called the self-interaction parameter because it takes into account the deviations from infinite-dilution behavior that are caused by the interaction between solute molecules in the solvent matrix. 

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

Comparison with the standard form for the chemical potential, p = p° + RT In a [Eq. 47 of Chapter 6], shows that in the ideally dilute solution activities are equal to mole fractions for both solvent and solute. In order to find the standard state of the solvent in the ideally dilute solution, we note that at xA = 1 (infinite dilution, within the range of applicability of the model), we have p = p. The standard state of the solvent in the ideally dilute solution is pure solvent, just like the standard states of all components in an ideal solution. The solvent in the ideally dilute solution behaves just like a component of the ideal solution. Although it is also true that p° becomes p at x, = 1, this is clearly outside the realm of applicability of Eq. (43). In order to avoid this difficulty, in determining p° we make measurements at very low values ofx, and extrapolate to x, = 1 using p = p, — RT In x as if the high dilution behavior held to x, = 1. In other words, our standard state for a solute in the ideally dilute solution is the hypothetical state of pure solute with the behavior of the solute in the infinitely dilute solution. 

As shown in Example 1, the chemical potential of a solute can be of a more complicated form than given by Eqs. (43)-(45), even though the solution shows ideally dilute behavior. This can result from a transformation of the substance when it dissolves in the solution. 

The standard state for solutes in the (HL) reference is therefore the hypothetical state of pure solute (x, = 1), but with solute molecules interacting only with solvent molecules (y, = 1). Practically, chemical potentials in the standard state are obtained by making measurements at very low concentrations and extrapolating them to X,- = 1, assuming that Henry s law continues to hold to this concentration. At nonzero concentration of solutes, activity coefficients in the (HL) reference measure deviations of the solution from ideally dilute behavior. 

The Henry s law reference activity coefficients are plotted in Fig. 4. Note that this system shows positive deviation with respect to the ideally dilute behavior of Henry s law. 

Equations (47)-(50) indicate how thermodynamic quantities can be obtained from cell potentials measured under standard conditions. However, standard states are hypothetical states (e.g., infinitely dilute behavior at 1.0 m concentration), which cannot be prepared in the cell. As a result, an extrapolation procedure is used to find 8° from measured cell voltages as a function of concentration. From Eq. (47), we write the dependence of 8 on the concentration of the electrolyte in the form... 

The dielectric constant of water, 78.4, is very large. Deviations from ideally dilute behavior are much greater in less polar solvents than they are in water. 

Two Independent methods have been utilized to examine the nature of the sorption Isotherm. An analysis of the experimental Isotherm compared to an extrapolation of the Infinite dilution behavior allows calculation of an enhancement number for any of the polymers at any given partial pressure. Calculation of a cluster number based on an Independent method shows very close concordance with the enhancement ntimber, providing strong support for the postulate that associated groups of water molecules sorb In the polymer, and account for the anomolous sorption. 

Thus, the Henry s law constant is the hypothetical fugacity of a solute species as a pure liquid extrapolated from its infinite-dilution behavior we will denote this by f (T, P) (see Fig. 9.7-3a). Thus... 

The standard state for a pure liquid or solid is taken to be the substance in that state of aggregation at a pressure of 1 bar. This same standard state is also used for liquid mixtures of those components that exist as a liquid at the conditions of the mixture. Such substances are sometimes referred to as liquids that may act as a solvent. For substances that exist only as a solid or a gas in the pure component state at the temperature of the mixture, sometimes referred to as substances that can act only as a solute, the situation is more complicated, and standard states based on Henry s law may be used. In this case the pressure is again fixed at 1 bar, and thermal properties such as the standard-state enthalpy and heat capacity are based on the properties of the substance in the solvent at infinite dilution, but the standard-state Gibbs energy and entropy are based on a hypothetical state.of unit concentration (either unit molality or unit mole fraction, depending on the form of Henry s law used), with the standard-state fugacity at these conditions extrapolated from infinite-dilution behavior in the solvent, as shown in Fig. 9.1-3a and b. Therefore just as for a gas where the ideal gas state at 1 bar is a hypothetical state, the standard state of a substance that can only behave as a solute is a hypothetical state. However, one important characteristic of the solute standard state is that the properties depend strongly upon the solvent. used. Therefore, the standard-state properties are a function of the temperature, the solute, and the solvent. This can lead to difficulties when a mixed solvent is used. 

In general, within the sometimes stringent limitation of temperature dependence of Cy one can map the infinite dilution fugacity of any polar compound into hydrocarbon systems. Further, if the infinite dilution behavior follows known patterns with hydrocarbon type, this can be made the basis for a correlation of Ci . This ability to incorporate polar compounds over narrow ranges of concentration is extremely useful in refining and hydrocarbon processing applications. 

To determine an overlap concentration, we examined the viscometric behavior of solutions of a-PS in decahydronaphthalene a function of and concentration. Figure 27 shows results for a 4 x 10 sample. Curve b shows a typical dilute behavior (as in Figure 5) where the extension of isolated molecules determines the higher compared with the solvent (curve a). As the concentration is increased, the dilatant non-Newtonian behavior is progressively developed and shifted to lower strain rates. Simple visual inspection shows a sudden marked change in dilatancy, both in character and degree for concentrations greater than 0.25% (curve c). 

Figure 15.2 Cerium oxide slurry dilution behavior.

From Eqs. 9.2.46 and 9.2.50, the solute chemical potential is given by /t-b = C/b + pV — TS. In the dilute solution, we assume Ub and Ir are linear functions of xr as explained above. We also assume the dependence of 5b on xr is approximately the same as in an ideal mixture this is a prediction from statistical mechanics for a mixture in which aU molecules have similar sizes and shapes. Thus we expect the deviation of the chemical potential from ideal-dilute behavior, /tr = + RT InxB, can be... 

The deviation of from unity is a measure of the deviation of jXA from infinite-dilution behavior, as we can see by comparing the preceding equation with a rearrangement of Eq. 9.6.11 ... 

Equation 10.2.10 predicts that the activity of HCl in aqueous solutions is proportional, in the limit of infinite dilution, to the square of the HCl molality. In contrast, the activity of a nonelectrolyte solute is proportional to the first power of the molality in this limit. This predicted behavior of aqueous HCl is consistent with the data plotted in Fig. 10.1 on page 285, and is confirmed by the data for dilute HCl solurions shown in Fig. 10.2(a). The dashed line in Fig. 10.2(a) is the extrapolation of the ideal-dilute behavior given by o-m,B = (mB/m°). The extension of this line to niB = m° establishes the hypothetical solute reference state based on molality, indicated by a filled circle in Fig. 10.2(b). (Since the data are for solutions at the standard pressure of 1 bar, the solute reference state shown in the figure is also the solute standard state.)... 

Chapter 6 is the extension of Chapter 5 to include mixtures of two or more liquids. The most important concepts here are ideal behavior and small deviations from it. Most of the treatment is based on the Kirkwood-Buff theory of solutions. The derivation and a sample application of this powerful theory are presented in detail. We also present the elements of the McMillan-Mayer theory, which is more limited in application. Its main result is the expansion of the osmotic pressure in power series in the solute density. The most useful part of this expansion is the first-order deviation from ideal dilute behavior, a result that may also be obtained from the Kirkwood-Buff theory. 

No actual solutions are ideal, and many solutions deviate from ideal-dilute behavior as soon as the concentration of solute rises above a small value. In thermodynamics we try to preserve the form of equations developed for ideal systems so that it becomes easy to step between the two types of system. This is the thought behind the introduction of the activity, flj, of a substance, which is a kind of effective concentration. The activity is defined so that the expression... 

The best approach is, therefore, to compare the results of simulations made at different length scales with real measurements to determine the validity of the approach since the causes for changes in e.g. viscosity can be caused by changes in local interaction energy or with the PE structure or both [103]. In this section, the results of computer simulations with relation to the PE structure, complex formation and dilution behavior are summarized. The focus lies on molecular dynamics simulations since Monte Carlo simulations [102, 113] are discussed in detail in chapter Thermodynamic and Rheological Properties of Polyelectrolyte Systems . 

The power of the polymer volume fraction V2 in the above expression has also been verified experimentally [4] with some indication that it holds well into the concentrated regime. The transition from dilute to semi-dilute behavior occurs when the concentration of polymers is sueh that the coils begin to overlap. This concentration has been denoted in the literature as the overlap concentration, c or Vj. For high-molecular-weight polymers, this eoncentration occurs at fairly low values, V2 of the order of 0.01. Solutions are generally considered concentrated when V2 > 0.1. For good solvents, c scales with M [3]. Other scaling laws for polymer systans can be found in Reference 3. 

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