In ideally diluted solution

Figure 33.3 Henry s Law behaviour for solutes in ideal dilute solutions. Reproduction of Figure 33.2 but with tangents (C K, DL, HJ and Gl) appropriately drawn to the partial vapour pressure curves at the ends where they are acting as solute. ...

In real (non-ideal) liquid mixtures quite often equation (39.1) is simply not valid. Nor is the one (i.e. equation (36.8), Frame 36) holding for the solute in ideal dilute solution, which took the form ... 

In ideally dilute solutions, the mole fraction of the solvent is approximately equal to one. 

The intrinsic viscosity [q] has the same unit, [ml g" ], as the reduced viscosity rjred a better understanding, the intrinsic viscosity can be considered as a measure for the volume demand of the single polymer coil in ideally diluted solution. The intrinsic viscosity is proportional to the reciprocal density of the polymer coil in solution according to Eqs. (4.6) and (4.7). 

If the range of homogeneity of the reaction product is sufficiently narrow, then the average diffusion coefficient as defined in eq. (8-9) can be calculated by means of defect thermodynamics, if it is assumed that the defects behave as the solute in ideally dilute solutions. In section 4.2 it was shown how the concentrations of the defect centers depend upon the component activities for a given type of disorder in binary ionic crystals. As an example, let us consider the formation of copper (I) oxide on copper sheet at 1000 °C in an oxidizing atmosphere whenis about 1 torr. The following defect equilibrium can be written ... 

There are a number of mathematically simple problems involving single polymer chains which are basically exactly soluble by the standard methods. By the term exactly soluble we imply that a closed form analytical solution is available which may be evaluated, possibly with the use of computers, to any desired numerical accuracy. This category of problems includes the configurational statistics of polymers in ideal dilute solutions - (i.e., at the 0-point). (See Section II for a brief discussion of the 0-point.)... 

Diffusion coefficients, both translational and rotational, can be calculated from the equations D = kT/f and =kT/f ((3.38) and (3.39)) if the frictional coefficient / or f is known. For particles of simple shape, in ideal dilute solution in a continuous fluid, / can be expressed in terms of the size and shape and the solvent viscosity, by the methods of classical hydrodynamics [7]. 

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. 

Turning now to the non-ideal solution, we may answer question (1) by saying that the value of (163) will vary with concentration only insofar as the solution differs from an ideal solution and we can proceed to ask a third question how would the value of (163) vary with concentration for an ionic solution in the extremely dilute range We must answer that in a series of extremely dilute solutions the value of (163) would be constant within the experimental error it is, in fact, a unitary quantity, characteristic of the solute dissolving in the given solvent. As in See. 55, this constant value adopted by (163) in extremely dilute solutions may conveniently be written as the limiting value as x tends to zero thus... 

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... 

The value obtained by Hamm et alm directly by the immersion method is strikingly different and much more positive than others reported. It is in the right direction with respect to a polycrystalline surface, even though it is an extrapolated value that does not correspond to an existing surface state. In other words, the pzc corresponds to the state of a bare surface in the double-layer region, whereas in reality at that potential the actual surface is oxidized. Thus, such a pzc realizes to some extent the concept of ideal reference state, as in the case of ions in infinitely dilute solution. 

The changeover to thermodynamic activities is equivalent to a change of variables in mathematical equations. The relation between parameters and a. is unambiguous only when a definite value has been selected for the constant p. For solutes this constant is selected so that in highly dilute solutions where the system p approaches an ideal state, the activity will coincide with the concenttation (lim... 

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,... 

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. 

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

In infinitely dilute solutions (in the standard state) ions do not interact, their electric field corresponds to that of point charges located at very large distances and the solution behaves ideally. As the solution becomes more concentrated, the ions approach one another, whence their fields become deformed. This process is connected with electrical work depending on the interactions of the ions. Differentiation of this quantity with respect to rc, permits calculation of the activity coefficient this differentiation is identical with the differentiation 3GE/5/iI and thus with the term RT In y,. 

In principle, any physical property that varies during the course of the reaction can be used to follow the course of the reaction. In practice one chooses methods that use physical properties that are simple exact functions of the system composition. The most useful relationship is that the property is an additive function of the contributions of the different species and that each of these contributions is a linear function of the concentration of the species involved. This physical situation implies that there will be a linear dependence of the property on the extent of reaction. As examples of physical properties that obey this relationship, one may cite electrical conductivity of dilute solutions, optical density, the total pressure of gaseous systems under nearly ideal conditions, and rotation of polarized light. In sufficiently dilute solutions, other physical properties behave in this manner to a fairly good degree of approximation. More complex relationships than the linear one can be utilized but, in such cases, it is all the more imperative that the experimentalist prepare care-... 

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],... 

This expression has been written in terms of concentration if activity coefficients sue known or estimated, then a thermodynamically ideal solubility product may be obtained from the Emalogous product of ionic activities. As the concentration of ions in solutions of lanthanide fluorides is low, the concentration and activity solubility products will not differ markedly, although activity coefficients for these salts of 3 + cations are significantly less than unity even in such dilute solutions (4a). 

K is equilibrium constant. If we consider the reference standard state to be gaseous, K = K because the activity and fugacity coefficients are unity in very dilute solution and ideal gas, respectively. Then ( g)0 and (ks)o will be the same. 

The volumetric systems of Bradshaw and Schleicher (106) and Kell and Whalley (26,32) are the most precise methods of directly measuring the absolute densities or volumes at high pressures. These methods, however, are not ideally suitable for making systematic density studies of aqueous electrolyte solutions as a function of P and T because of the arduous nature of the experimental work and the loss of precision in very dilute solutions. 

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.)...

The activity coefScient enables corrections for the non-ideality of a system to be made. It decreases as ionic strength increases. In inSnitely dilute solutions, a = C ,... 

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