Dilute ideal

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 situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

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. 

First approaches at modeling the viscoelasticity of polymer solutions on the basis of a molecular theory can be traced back to Rouse [33], who derived the so-called bead-spring model for flexible coiled polymers. It is assumed that the macromolecules can be treated as threads consisting of N beads freely jointed by (N-l) springs. Furthermore, it is considered that the solution is ideally dilute, so that intermolecular interactions can be neglected. 

In most cases polymer solutions are not ideally dilute. In fact they exhibit pronounced intermolecular interactions. First approaches dealing with this phenomenon date back to Bueche [35]. Proceeding from the fundamental work of Debye [36] he was able to show that below a critical molar mass Mw the zero-shear viscosity is directly proportional to Mw whereas above this critical value r 0 is found to be proportional to (Mw3,4) [37,38]. This enhanced drag has been attributed to intermolecular couplings. Ferry and co-workers [39] reported that the dynamic behaviour of polymeric liquids is strongly influenced by coupling points. 

These classical molecular theories may be used to illustrate good agreement with the experimental findings when describing the two extremes of concentration ideally dilute and concentrated polymer solutions (or polymer melts). However, when they are used in the semi-dilute range, they lead to unsatisfactory results. 

Relaxation Time Behaviour in Ideally Dilute and Concentrated Solutions... 

First approaches to approximating the relaxation time on the basis of molecular parameters can be traced back to Rouse [33]. The model is based on a number of boundary assumptions (1) the solution is ideally dilute, i.e. intermolecular interactions are negligible (2) hydrodynamic interactions due to disturbance of the medium velocity by segments of the same chain are negligible and (3) the connector tension F(r) obeys an ideal Hookean force law. 

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],... 

The second boundary condition arises from the continuity of chemical potential [44], which implies - under ideally dilute conditions - a fixed ratio, the so-called (Nernst) distribution or partition coefficient, A n, between the concentrations at the adjacent positions of both media ... 

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.)...

This further incubation should be sufficiently long enough to observe measurable metabolite formation. There is some debate as to the most ideal dilution scheme that should be followed, but the general consensus is that a higher dilution (e.g., >10-fold) reduces the influence of competitive inhibition. Also the concentration of probe substrate should ideally be at least 5 Km, the purpose being that the high probe concentration together with the dilution step minimizes competitive inhibition of... 

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). 

We now proceed to more realistic and complicated systems by considering crystals in which the point defects interact. If the interaction is due to forces between nearest neighbors only, then one may calculate the point defect concentrations by assuming that, in addition to single point defects, e.g. it and i2, pairs (or still higher clusters) of point defects form and that they are in internal equilibrium. These clusters are taken to be ideally diluted in the crystal matrix, in analogy to the isolated single defects. All the defect interactions are thus contained in the cluster formation reaction... 

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... 

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