Nonideal dilute solutions

How would we expect the activity coefficient of a nonelectrolyte solute to behave in a dilute solution as the solute mole fraction increases beyond the range of ideal-dilute solution behavior  

The following argument is based on molecular properties at constant T and p. 

We focus our attention on a single solute molecule. This molecule has interactions with nearby solute molecules. Each interaction depends on the intennolecular distance and causes a change in the internal energy compared to the interaction of the solute molecule with solvent at the same distance. The number of solute molecules in a volume element at a given distance from the solute molecule we are focusing on is proportional to the local solute concentration. If the solution is dilute and the 

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For nonideal dilute solutions we can study the osmotic pressure using virial expansion. For simplicity, let us specifically assume that the solute is one component. The virial expansion of the osmotic pressure. 

Nonideality requires that especially dilute solutions be used. 

For ideal solutions, the activity coefficient will be unity, but for real solutions, 7r i will differ from unity, and, in fact, can be used as a measure of the nonideality of the solution. But we have seen earlier that real solutions approach ideal solution behavior in dilute solution. That is, the behavior of the solvent in a solution approaches Raoult s law as. vi — 1, and we can write for the solvent... 

These expressions comprise the nonideal terms in the previous equations for the chemical potential, Eqs. (30) and (31 ). They may therefore be regarded as the excess relative partial molar free energy, or chemical potential, frequently used in the treatment of solutions of nonelectrolytesi.e, the chemical potential in excess (algebraically) of the ideal contribution, which is —RTV2/M in dilute solutions. 

A theory close to modem concepts was developed by a Swede, Svante Arrhenins. The hrst version of the theory was outlined in his doctoral dissertation of 1883, the hnal version in a classical paper published at the end of 1887. This theory took up van t Hoff s suggeshons, published some years earlier, that ideal gas laws could be used for the osmotic pressure in soluhons. It had been fonnd that anomalously high values of osmotic pressure which cannot be ascribed to nonideality sometimes occur even in highly dilute solutions. To explain the anomaly, van t Hoff had introduced an empirical correchon factor i larger than nnity, called the isotonic coefficient or van t Hoff factor,... 

The constant of proportionality Yi is the species activity coefficient, which accounts for the nonideality of the aqueous solution. The species activity coefficients approach unity in very dilute solutions,... 

We will proceed in our discussion of solutions from ideal to nonideal solutions, limiting ourselves at first to nonelectrolytes. For dilute solutions of nonelectrolyte, several limiting laws have been found to describe the behavior of these systems with increasing precision as infinite dilution is approached. If we take any one of them as an empirical mle, we can derive the others from it on the basis of thermodynamic principles. 

In what follows we shall always write A, = fC. We assume that the ligand is provided from either an ideal gas phase or an ideal dilute solution. Hence, A, is related to the standard chemical potential and is independent of the concentration C. On the other hand, for the nonideal phase, A will in general depend on concentration C. A first-order dependence on C is discussed in Appendix D. Note also that A, is a dimensionless quantity. Therefore, any units used for concentration C must be the same as for (Aq) . 

Therefore, the distribution ratio of B remains constant only if the ratio of the activity coefficients is independent of the total concentration of B in the system, which holds approximately in dilute solutions. Thus, although solutions of metal chelates in water or nonpolar organic solvents may be quite nonideal, Nernst s law may still be practically obeyed for them if their concentrations are very low (JCchehte< 10" ). Deviations from Nernst s law (constant D ) will in general take place in moderately concentrated solutions, which are of particular importance for industrial solvent extraction (see Chapter 12). 

In solution thermodynamics, the concentration (C) of ions is replaced by their activity, a, where a = Cy and y is the activity coefficient that takes into account nonideal behavior due to ion-solvent and ion-ion interactions. The Debye-Hiickel limiting law predicts the relationship between the ionic strength of a solution and y for an ion of charge Z in dilute solutions ... 

The analysis of mixed associations by light scattering and sedimentation equilibrium experiments has been restricted so far to ideal, dilute solutions. Also it has been necessary to assume that the refractive index increments as well as the partial specific volumes of the associating species are equal. These two restrictions are removed in this study. Using some simple assumptions, methods are reported for the analysis of ideal or nonideal mixed associations by either experimental technique. The advantages and disadvantages of these two techniques for studying mixed associations are discussed. The application of these methods to various types of mixed associations is presented. 

For concentrations sufficiently high that the infinite dilution approximation breaks down, Eq. 2.17 must be modified by the incorporation of an activity coefficient y, which compensates for additions to the chemical potential from such nonidealities as solute-solute interactions. We have... 

The effect of solvent can be represented without too much additional difficulty for the case of dilute solutions which are nonideal." For such solutions, Henryks law is obeyed with reasonable accuracy, i.e., the solubility of the vapor is proportional to its concentration in solution. For purposes of comparison with our preceding work we can express the vapor pressure of the solute in terms of the deviations of the system from Raoult s law ... 

Only extremely dilute solutions (in which there are no interactions between particles or particles and solvent) obey the relationship ttV =nRT. Yet, relatively concentrated solutions must be used to obtain good measurements of TT. Usually, the observed value of tt must be corrected for nonideality. This is accomplished by plotting ir/C (the reduced or specific osmotic strength) against C, and extrapolating to zero C. The MW is then calculated from ... 

In contrast to previous papers (Ruckenstein and Shulgin, 2003a-d), the solubility of the drug in a binary solvent is considered to be finite, and the infinite dilution approximation is replaced by a more realistic one, the dilute solution approximation. An expression for the activity coefficient of a solute at low concentrations in a binary solvent was derived by combining the fluctuation theory of solutions (Kirkwood and Buff, 1951) with the dilute approximation. This procedure allowed one to relate the activity coefficient of a solute forming a dilute solution in a binary solvent to the solvent properties and some parameters characterizing the nonidealities of the various pairs of the ternary mixture. 

In thermodynamically nonideal solutions, the effect of concentration on diffusion coefficient ( >A)conc can be estimated in terms of the activity coefficient of the solute 7a> the binary diffusivities andDa in very dilute solutions, and the mole fractions Za and ATb of A and B (V8) ... 

FIGURE 11.15 Vapor pressures above a mixture of two volatile liquids. Both ideai (biue lines) and non-ideai behaviors (red curves) are shown. Positive deviations from ideal solution behavior are illustrated, although negative deviations are observed for other nonideal solutions. Raoult s and Henry s laws are shown as dilute solution limits for the nonideal mixture the markers explicitly identify regions where Raoult s law and Henry s law represent actual behavior. 

There are ideal solutions, ideally dilute solutions, and nonideal solutions. Ideal solutions are solutions made from compounds that have similar properties. In other words, the compounds can be interchanged within the solution without changing the spatial arrangement of the molecules or the intermolecular attractions. Benzene in toluene is an example of a nearly ideal solution because both compounds have similar bonding properties and similar size. In an ideally dilute solution, the solute molecules are completely separated by solvent molecules so that they have no interaction with each other. Nonideal solutions violate both of these conditions. On the MCAT, you can assume that you are dealing with an ideally dilute solution unless otherwise indicated however, you should not automatically assume that an MCAT solution is ideal. 

The derivation of Equations (435) and (436) from dilute solutions is only approximate. It is also possible to thermodynamically derive more fundamental types of these equations by using the activity concept, but initially we need to define non-localized, ideal and nonideal monolayers. If all the solute molecules are mobile in a monolayer, this is called a non-localized monolayer. We may consider three types of molecules in non-localized mono-layers ideal point molecules having no mass and volume where no lateral interactions are present between these point molecules non-ideal molecules having their mass and volume but no lateral interactions taking place between them, as above and non-ideal molecules having their mass and volume, and in addition appreciable lateral interactions taking place between them. 

A substance in solution has a chemical potential, which is the partial molar free energy of the substance, which determines its reactivity. At constant pressure and temperature, reactivity is given by the thermodynamic activity of the substance for a so-called ideal system, this equals the mole fraction. Most food systems are nonideal, and then activity equals mole fraction times an activity coefficient, which may markedly deviate from unity. In many dilute solutions, the solute behaves as if the system were ideal. For such ideally dilute systems, simple relations exist for the solubility of substances, partitioning over phases, and the so-called colligative properties (lowering of vapor pressure, boiling point elevation, freezing point depression, osmotic pressure). 

Nonideality of solutions is discussed in Section 2.2.5. It can be expressed as the deviation of the colligative properties from that of an ideal, i.e., very dilute, solution. Here we will consider the virial expansion of osmotic pressure. Equation (2.18) can conveniently be written for a neutral and flexible polymer as... 

Note that unlike the case for binary gas mixtures the diffusion coefficient for a dilute solution of. 4 in 5 is not the same as for a dilute solution of B in A, since fi, Mb, and will be different when the solute and solvent are exchanged. For intermediate concentrations, an approximate value of is sometimes obtained by interpolation between the dilute solution values, but this method can lead to large errors for nonideal solutions. 

Solutions of macromolecules are often sufficiently dilute that Eq. (13.5.21) applies. Moreover for large molecules can be computed from hydrodynamics. For a sphere with stick boundary conditions Cs = 6jirjas. Thus in dilute solutions D° and thereby as, the particle radius, can be determined (see Chapters 5 and 8). Since D° depends on the temperature and the solvent, it is important to report the data in a standardized manner. Usually the measurements are performed at room temperature and are extrapolated to inifinite dilution. Thus for example the notation D%0,a denotes the diffusion coefficient of the solute at 20°C in the solvent H20 extrapolated to infinite dilution. For nonideal solutions... 

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