Ideal solutions dilute

An additional option allows the user to fit data for binary mixtures where one of the components is noncondensable. The mixture is treated as an ideal dilute solution. The solute... 

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... 

The standard state of an electrolyte is the hypothetical ideally dilute solution (Henry s law) at a molarity of 1 mol kg (Actually, as will be seen, electrolyte data are conventionally reported as for the fonnation of mdividual ions.) Standard states for non-electrolytes in dilute solution are rarely invoked. 

For example, the measurements of solution osmotic pressure made with membranes by Traube and Pfeffer were used by van t Hoff in 1887 to develop his limit law, which explains the behavior of ideal dilute solutions. This work led direcdy to the van t Hoff equation. At about the same time, the concept of a perfectly selective semipermeable membrane was used by MaxweU and others in developing the kinetic theory of gases. 

Equation 13 has an important implication a clathrate behaves as an ideally dilute solution insofar as the chemical potential of the solvent is independent of the nature of the solutes and is uniquely determined by the total solute concentrations 2K yK1.. . 2x yKn in the different types of cavities. For a clathrate with one type of cavity the reverse is also true for a given value of fjiq (e.g. given concentration of Q in a liquid solution from which the clathrate is being crystallized) the fraction of cavities occupied 2kVk s uniquely determined by Eq. 13. When there are several types of cavities, however, this is no longer so since the individual occupation numbers 2k2/ki . ..,2k yKn, and hence the total solute concentration... 

When a small amount of a strong molten electrolyte is dissolved in another strong molten electrolyte, the laws of ideal dilute solutions are obeyed until relatively high concentrations are attained, assuming occurrence of a virtually complete dissociation. 

The ideality of the solvent in aqueous electrolyte solutions is commonly tabulated in terms of the osmotic coefficient 0 (e.g., Pitzer and Brewer, 1961, p. 321 Denbigh, 1971, p. 288), which assumes a value of unity in an ideal dilute solution under standard conditions. By analogy to a solution of a single salt, the water activity can be determined from the osmotic coefficient and the stoichiometric ionic strength Is according to,... 

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],... 

If the activity coefficients jj for all ions and electrons are constant, i.e. Henry s or Raoult s law or the law of ideally diluted solutions holds, Eqn (8.27) reads... 

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) . 

It should be noted that estimating Hemy s law constant assumes the gas obeys the ideal gas law and the aqueous solution behaves as an ideally dilute solution. The solubility and vapor pressure data inputted into the equations are valid only for the pure compound and must be in the same standard state at the same temperature. 

The solute and the solvent are not distinguished normally in such ideal mixtures, which are sometimes called symmetric ideal mixtures. There are, however, situations where such a distinction between the solute and the solvent is reasonable, as when one component, say, B, is a gas, a liqnid, or a solid of limited solubility in the liquid component A, or if only mixtures very dilute in B are considered (xb 0.5). Such cases represent ideal dilute solutions. 

Although Pb tends to - °o as Xb tends to 0 (and In Xb also tends to - °o), the difference on the right-hand side of Eq. (2.18) tends to the finite quantity pi, the standard chemical potential of B. At infinite dilution (practically, at high dilution) of B in the solvent A, particles (molecules, ions) of B have in their surroundings only molecules of A, but not other particles of B, with which to interact. Their surroundings are thus a constant environment of A, independent of the actual concentration of B or of the eventual presence of other solutes, C, D, all at high dilution. The standard chemical potential of the solute in an ideal dilute solution thus describes the solute-solvent interactions exclusively. 

Fig. 2.4 The vapor pressure diagram of a dilute solution of the solute B in the solvent A. The region of ideal dilute solutions, where Raoult s and Henry s laws are obeyed by the solvent and solute, respectively, is indicated. Deviations from the ideal at higher concentrations of the solute are shown. (From Ref. 3.)...

For ideal (dilute) solutions the activity is replaced by the mole fraction x]s, which, for a two-component surface solution, equals (1 — x ). With the customary expansion of the logarithm as a power series (see Appendix A), these substitutions yield... 

Equation (1) may be applied to the equilibrium between vapor and liquid of a pure substance (X = vapor pressure) or to the equilibrium between an ideal dilute solution and the pure phase of a solute X = solubility) or to the equilibrium of a chemical reaction (X = equilibrium constant). 

It can be seen from Eqn. (2.65) and equivalent relations that phenomenological point defect thermodynamics does not give us absolute values of defect concentrations. Rather, within the limits of the approximations (e.g., ideally dilute solutions of irregular SE s in the solvent crystal), we obtain relative changes in defect concentrations as a function of changes in the intensive thermodynamic variables (P, T, pk). Yet we also know that the crystal is stoichiometric (i.e., S = 0) at the inflection... 

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. 

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... 

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