The ideal-dilute solution

An ideal-dilute solution is a real solution that is dilute enough for each solute to obey Henry s law. On the microscopic level, the requirement is that solute molecules be sufficiently separated to make solute-solute interactions negligible. 

Note that an ideal-dilute solution is not necessarily an ideal mixture. Few Uquid mixtures behave as ideal mixtures, but a solution of any nonelectrolyte solute becomes an ideal-dilute solution when sufficiently dilute. 

Within the composition range that a solution effectively behaves as an ideal-dilute solution, then, the fugacity of solute B in a gas phase equilibrated with the solution is proportional to its mole fraction xb in the solution. The chemical potential of B in the gas phase, which is equal to that of B in the liqvtid, is related to the fugacity by //.b = + RTlnifs/p°) (Eq. 9.3.12). Substituting /b = A h,bxb (Henry s law) into this equation, we obtain 

The expression in brackets in Eq. 9.4.23 is a function of T and p, but not of xb, and represents the chemical potential of B in a hypothetical solute reference state. This chemical potential will be denoted by B where the x in the subscript reminds us that the reference state is based on mole fraction. The equation then becomes 

Here the notation emphasizes the fact that /zb and g are functions of T and p. 

The rigid requirement of the ideal solution that every component obey Raoult s law over the entire range of composition is relaxed in the definition of the ideal dilute solution. To arrive at the laws governing dilute solutions, we must examine the experimental behavior of these solutions. The vapor-pressure curves for three systems are described below. 

Note that if the solution were ideal, then K would equal p° and both Henry s law and Raoult s law would convey the same information. 

In the acetone-chloroform system shown in Fig. 14.13, the vapor pressure curves fall below the Raoult s law predictions. This system exhibits negative deviations from Raoult s law. The total vapor pressure has a minimum value that lies below the vapor pressure of either of the pure components. The Henry s law lines, the fine dashed lines in the figure, also lie below the Raoult s law lines for this system. 

Algebraically, we can express the properties of the ideal dilute solution by the following equations  

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In analogy to the gas, the reference state is for the ideally dilute solution at c, although at the real solution may be far from ideal. (Teclmically, since this has now been extended to non-volatile solutes, it is defined at... 

However, a detailed model for the defect structure is probably considerably more complex than that predicted by the ideal, dilute solution model. For higher-defect concentration (e.g., more than 1%) the defect structure would involve association of defects with formation of defect complexes and clusters and formation of shear structures or microdomains with ordered defect. The thermodynamics, defect structure, and charge transfer in doped LaCo03 have been reviewed recently [84],... 

Whereas the ideal solution model applies over the entire range of concentrations, but only for very similar components, the ideally dilute solution model applies to any solution, but only over a very limited range of concentrations. From a microscopic point of view, the ideally dilute solution holds as long as solute molecules are almost always completely surrounded by solvent molecules and rarely interact with other solute molecules. 

Calling the solvent component A and a solute component i, we define the ideally dilute solution as one that has, in a range around infinite dilution,... 

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. 

It is more common to use molality or molarity as the concentration unit for the solute in the ideally dilute solution. The relevant equations are then... 

In order to find the vapor pressures above the ideally dilute solution, we equate the chemical potentials of the components in the solution with those in the vapor. Because the solvent in the ideally dilute solution behaves just like a component of an ideal solution, its vapor pressure follows Raoult s law. For the solute in an ideally dilute solution, we obtain... 

Because the ideally dilute solution approximation only holds in the limit of very dilute solutions (i.e., where mi <5iC 1 /MA), this becomes... 

Particularly simple forms of the equations for the freezing-point depression, boiling-point elevation, and osmotic pressure are obtained when the solution is ideal or when it is sufficiently dilute, so that the ideally dilute solution approximation is appropriate. In both of these cases, the activity of the solvent is equal to its mole fraction, so that... 

The x - extends over all the solute species and is designated as xB, which we have previously used to designate the mole fraction of the single solute in a binary solution. At a concentration small enough to make the ideally dilute solution approximation, it is usually sufficient to use only the first term in the Taylor series approximation of ln(l — xB),4... 

When doing calculations with colligative properties, it is usually advantageous to make measurements at a number of concentrations and extrapolate the results to zero concentration, where the ideally dilute solution theory is clearly applicable. 

In both the ideal and the ideally dilute solution models, the activity can be set equal to the mole fraction, a = x, for all components. In order to maintain the... 

When the ideally dilute solution is used as the reference for real solutions, thermodynamic properties are designated by (HL) for Henry s law reference. This reference is always used when some components are not liquids at the temperatures employed and may also be used if they are all liquids, but only very dilute solutions are being considered. For this reference, we treat the solvent in the same manner as for the (RL) reference ... 

The ionization of electrolytes is clearly manifest in the thermodynamic properties of their solutions. For example, in the ideally dilute solution limit, a solution of a strong electrolyte behaves as ions, rather than molecules, interacting with solvent molecules. A NaCl solution of molality m behaves, in the limit of infinite dilution, as an ideally dilute solution of concentration 2m, as 2 mol of ions are produced from each mole of NaCl dissolved in solution. A general strong electrolyte, dissociating by the equation... 

In contrast to a perfect solution, a solution is called an ideal solution, if Eq. 8.1 is valid for solute substances in the range of dilute concentrations only. Moreover, the unitary chemical potential p2(T,p) of solute substance 2 is not the same as the chemical potential p2( T,p) of solute 2 in the pure substance p2(T,p) p2(T,p) Henry s law. For the main constituent solvent, on the other hand, the unitary chemical potential p[( T,p) is normally set to be equal to f l p) in the ideal dilute solution p"(T,p) = p°(l p). The free enthalpy per mole of an ideal binary solution of solvent 1 and solute 2 is thus given by Eq. 8.10 ... 

In Frame 36, section 36.3 we studied the ideal dilute solution, consisting of solvent, B and solute, A, for which we found ... 

The reference condition of the ideal diluted solution and the ion strength Ix is approximated with the mass content... 

Debye-Hiickel developed a theory for the activity coefficients of an ionic solution at a molecular level. A selected ion in the ideally diluted solution is statistically well distributed and there are no interactions between ions present in the solution. In contrast, the ion in the concentrated solution is surrounded by the excess of counter ions in the vicinity of the ion, as the counter ions are attracted by Coulombic forces, while ions of the same charge are repelled. Thus, ion atmosphere is created. As a result, there is a difference in reversible work between the concentrated wrev and dilute solutions wrev ideal ... 

Corresponding to each chemical potential there is an activity coefficient defined in terms of equation (20.4). By convention, the activity coefficients of electrolytes are always expressed in terms of the ideal dilute solution as standard reference state, cf. chap. XXI, 3. Thus in the case of an aqueous NaCl solution we may write... 

Relationships analogous to those given above may be derived in an exactb similar manner for the activities referred to mole fractions or molarities. As seen in 37c, the activities for the various standard states, based on the ideal dilute solution, can be related to one another by equation (37.7). The result is, however, applicable to a single molecular species the corresponding relationships between the mean ionic activity coefficients of a strong electrolyte, assumed to be completely ionized, are found to be... 

Single ion activities should be used for components where the activities are expressed in terms of the ideal dilute solution as the standard state (Henrian activities). 

It should be noted that gives the standard chemical potential for the ideally dilute solute in a hypothetical system in which the mole fraction of B is unity. This is obviously a fictitious state which is impossible in reality but whose properties are obtained by extrapolating the Henry s law line to Xb = 1 (see fig. 1.12). When Henry s law is not obeyed, an activity coefficient 73 introduced so that the product Yb h b is equal to the vapor pressure Pb- The activity of the dilute component Ub is defined to be 73 3- Thus, the general expression for the concentration dependence of pb becomes... 

The concept of the ideal dilute solution i s extended to include nonvolatile solutes by requiring that the chemical potential of such solutes also have the form given by Eq. (14.18). 

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)... 

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