Ideal-dilute solution solvent behavior

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. 

The practical system of activities and activity coefficients is useful for solutions in which only the solvent has a mole fraction near unity all of the solutes are present in relatively small amounts. For such a system we use the rational system for the solvent and the practical system for the solutes. As the concentration of solutes becomes very small, the behavior of any real solution approaches that of the ideal dilute solution. Using a subscript j to identify the solutes, then in the ideal dilute solution (Section 14.11)... 

We now use the Gibbs-Duhem equation to investigate the behavior of the solvent in an ideal-dilute solution of one or more nonelectrolyte solutes. The Gibbs-Duhem equation applied to chemical potentials at constant T and p can be written xt dp,i = 0 (Eq. 9.2.43). We use subscript A for the solvent, rewrite the equation as xa d/iA + = 0,... 

The solvent and the key component that show most similar liquid-phase behavior tend to exhibit little molecular interactions. These components form an ideal or nearly ideal liquid solution. The ac tivity coefficient of this key approaches unity, or may even show negative deviations from Raoult s law if solvating or complexing interactions occur. On the other hand, the dissimilar key and the solvent demonstrate unfavorable molecular interactions, and the activity coefficient of this key increases. The positive deviations from Raoult s law are further enhanced by the diluting effect of the high-solvent concentration, and the value of the activity coefficient of this key may approach the infinite dilution value, often aveiy large number. 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

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... 

In the preceding chapters we considered Raoult s law and Henry s law, which are laws that describe the thermodynamic behavior of dilute solutions of nonelectrolytes these laws are strictly valid only in the limit of infinite dilution. They led to a simple linear dependence of the chemical potential on the logarithm of the mole fraction of solvent and solute, as in Equations (14.6) (Raoult s law) and (15.5) (Heiuy s law) or on the logarithm of the molality of the solute, as in Equation (15.11) (Hemy s law). These equations are of the same form as the equation derived for the dependence of the chemical potential of an ideal gas on the pressure [Equation (10.15)]. 

DEBYE-HOCKEL LIMITING LAW. The departure from ideal behavior in a given solvent is governed by the ionic strength of the medium and the valences of the ions of the electrolyte, but is independent of their chemical nature. For dilute solutions, the logarithm of the mean activity is proportional to the product of the cation valence, anion valence, and square root of ionic strength giving the equation... 

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. 

Consider now the case of a dilute solution for which one can sensibly expect near—ideal behavior. Let the subscript 1 designate the solvent. Then Eq. (2.10.7) reduces to (for... 

Any solution that obeys Raoult s law is called an ideal solution. One might say that Raoult s law is to solutions what the ideal gas law is to gases. As with gases, ideal behavior for solutions is never perfectly achieved, but is sometimes closely approached. Nearly ideal behavior is often observed when the solute-solute, solvent-solvent, and solute-solvent interactions are very similar. That is, in solutions where the solute and solvent are very much alike, the solute simply acts to dilute the solvent. However, if the solvent has a special affinity for the solute, such as if hydrogen bonding occurs, the tendency of the solvent molecules to escape will be lowered more than expected. In such cases the observed vapor pressure will be lower than the value predicted by Raoult s law there is a negative deviation from Raoult s law. 

Forms of the Activity Coefficient.—The equations given above are satisfactory for representing the behavior of liquid solutes, but for solid solutes, especially electrolytes, a modified form is more convenient. In a very dilute solution the mole fraction of solute is proportional both to its concentration (c), i.e., moles per liter of solution, and to its molality (m), i.e., moles per 1000 g, of solvent hence for such solutions, which are known to approach ideal behavior, it is possible to write either... 

According to (36.1), the behavior of the solvent tends toward that required by Raoult s law, as stated above. Further, since d is constant, equation (36.2), for the solute, is equivalent to / = Hftf k, where h is equal to e and hence is also constant. It is obvious that in ute solution the solute cannot obey Raoult s law unless k is unity, that is, unless /3 is very small or zero. In other words, although the solvent in a dilute solution satisfies Raoult s law, nevertheless, the solute does not do so unless the system as a whole, i.e., over the whole range of composition, exhibits little or no departure from ideal behavior. This conclusion is in harmony with the results depicted in Fig. 22, A and B. However, although the solute in the dilute solution does not necessarily obey Raoult s law, it does conform to the simple expression / = Nsf k, which may be written as... 

Raoult s law /rah-oolz/ A relationship between the pressure exerted by the vapor of a solution and the presence of a solute. It states that the partial vapor pressure of a solvent above a solution (p) is proportional to the mole fraction of the solvent in the solution (X) and that the proportionaUty constant is the vapor pressure of pure solvent, (po), at the given temperature i.e. p = PqX. Solutions that obey Raoult s law are said to be ideal. There are some binary solutions for which Raoult s law holds over all values of X for either component. Such solutions are said to be perfect and this behavior occurs when the intermolecular attraction between molecules within one component is almost identical to the attraction of molecules of one component for molecules of the other (e.g. chlorobenzene and bromobenzene). Because of solvation forces this behavior is rare and in general Raoult s law holds only for dilute solutions. 

Solvents dissolved in water may volatilize into flie atmosphere or soil gases. A Henry s Law constant (Kg) can be used to classify flie behavior of dissolved solvents. Henry s Law describes the ratio of the partial pressure of the vapor phase of an ideal gas (Pj) to its mole fraction (Xj) in a dilute solution, viz.. 

---