Activity coefficient postulate

We have simply restated the requirement of Hammett s activity coefficient postulate by this analysis, but the simplification of eq. (14) to eq. (16) is clearly seen to stand or fall on the cancellation, and we may expect trouble whenever this is impossible. We note from eq. (16) that a base which does obey the postulate should give a linear plot of unit slope for the log indicator ratio vs. Ho. Furthermore, the Ho at which the indicator ratio becomes equal to unity (equal concentrations of base and conjugate acid) will correspond to the pK of the conjugate acid. It is customary to use the pKj of the conjugate acid of the base as the measure of basicity and we shall use this convention henceforth. 

For veral classes of nitrogen bases, the position of protonation is a controversial matter and it appears that the pjnroles, indoles, and possibly amides undergo protonation on all possible basic sites depending on the circumstances. These same compounds are some of the womt offenders for non-adherence to the activity coefficient postulate, and furthermore have in common a resonance-stabilized system in which two of the basic sites are in a position 1, 3 to each other. Nitrobenzene and benzoic acid which exhibit abnormal activity be-... 

In the linear or first-order approximation, it is postulated that these activity coefficient terms are directly proportional, as in Eq. (8-92) ... 

Independently of T, reaction 5.159 thus appears to proceed spontaneously toward the right, in agreement with the principle of Fe-F incompatibility in silicates, postulated long ago by Ramberg (1952). This fact also implies the nonideality of the system. According to Munoz (1984), the activity coefficients of OH-phlogopite and F-phlogopite in mixture are defined by... 

From Eq. (1.14) it is clear that in dilute aqueous solution, as the activity coefficients tend to unity, the Hammett acidity function becomes identical with pH. On the other hand, by making the fundamental assumption that the ratio/b//bh+ is the same for different bases in a given solution, Hammett postulated that the //0 function was unique... 

Modified Gouy-Chapman theory has been applied to soil particles for many years (Sposito, 1984, Chapter 5). It postulates only one adsorption mechanism -the diffuse-ion swarm - and effectively prescribes surface species activity coefficients through the surface charge-inner potential relationship contained implicitly in the Poisson-Boltzmann equation (Carnie and Torrie, 1984). Closed-form... 

An understanding of the concentration dependence of activity coefficients required the postulation of the concepts of ion-pair formation and complex formation. Certain structural questions, however, could not be answered unequivocally by these considerations alone. For instance, it was not possible to decide whether pure Cou-lombic or chemical forces were involved in the process of ion association, i.e., whether the associated entities were ion pairs or complexes. The approach has been to postulate one of these types of association, then to work out the effect of such an association on the value of the activity coefficient, and finally to compare the observed and calculated values. Proceeding on this basis, it is inevitable that the postulate will always stand in need of confirmation because the path from postulate to fact is indirect. 

The TLM (Davis and Leckie, 1978) is the most complex model described in Figure 4. It is an example of an SCM. These models describe sorption within a framework similar to that used to describe reactions between metals and ligands in solutions (Kentef fll., 1988 Davis and Kent, 1990 Stumm, 1992). Reactions involving surface sites and solution species are postulated based on experimental data and theoretical principles. Mass balance, charge balance, and mass action laws are used to predict sorption as a function of solution chemistry. Different SCMs incorporate different assumptions about the nature of the solid - solution interface. These include the number of distinct surface planes where cations and anions can attach (double layer versus triple layer) and the relations between surface charge, electrical capacitance, and activity coefficients of surface species. 

In the experimental determination of activity coefficients of strong electrolytes, by the methods described below, the molalities, etc., of the ions are taken as the stoichiometric values, that is, the total possible molality, etc., disregarding incomplete dissociation, For example, in the last problem, the molalities of the sodium and sulfate ions in the 0.5 molal solution of sodium sulfate were taken as exactly 1.0 and 0.5, respectively, without allowing for the possibility that the salt may be only partially dissociated at the specified concentration. The activity coefficients obtained in this manner are called stoichiometric activity coefficients they allow for all variations from the postulated ideal behavior, including that due to incomplete dissociation. If the treatment is based on the actiuil ionic molalities, etc., in the given solution, as in the Debye-Httckel theory (Chapter XVII), there is obtained the true (or actual) activity coefficient. TTie ratio... 

The Nernst-Planck model is based on limiting laws for ideal systems. It accounts only for diffusion and electric transference of ions, not for electroosmotic solvent transfer in the ion-exchanger phase, swelling or shrinking of the ion-exchange material, variations of activity coefficients and diffu-sivities, and possible slow structural relaxation of the exchanger matrix. It also postulates the existence of individual diffusion coefficients for ions. 

It is furthermore postulated that the saturated solution in eqniUbrinm with the sparingly solnble solid is sufficiently dilute to be considered ideal, and we may thns set the activity coefficients... 

In the analytical solution of groups (ASOG) method of Derr and Deal [14], the four postulates of Wilson and Deal are implemented. The activity coefficient due to molecular size differences is given by the Flory-Huggins equation, Equation (4.414), and the group activity coefficient is given by the Wilson equation. 

The UNIFAC (universal functional activity coefficient) method [16] is similar to the ASOG method and is based on the four postulates of Wilson and Deal [13] regarding solution of groups. In UNIFAC, the activity coefficient is made of two parts. 

In smy event, and to the extent that the postulate can be tested with the data at hand, superposition upon the solute partition coefficients of the solvent-component deviations from Raoult s law, equation 22, sppears to go far in accounting for deviations from the diachoric solutions relation. That is, the non-interactive probe-solute partition coefficients reflect interactions that are extant between the solvents, negative deviations from equation 12 being indicative (as here) of solvent-solvent activity coefficients of less than unity. Moreover, even if this were in fact to prove to be only approximately so, the way would nonetheless be open to a very powerful method of assessment (to the same degree of approximation) of (finite-concentration) solvent/solvent activity coefficients one need only me ure (infinite-dilution) partition coefficient then deduce via equation 22,... 

We now take a look at how numerical values are assigned to the thermodynamic properties of single ions. There is more than one method, of course. Most physical chemistry courses for example will mention the Macinnes convention, which postulates that because the K+ and Cr ions have similar properties, their activity coefficients should be identical. This means that... 

From the azeotropic data an activity coefficient can bo worked out for each component at the azeotropic composition (equations (7 69) and (7 70)). When these are substituted in the given equations, two almost equal values of 6 are obtained. The data are thus not inconsistent with the postulates (but, of course, are quite insufficient to prove them). 

We define the linear adsorption regime such that the activity coefficient/ = 1. In this case, one can solve Eq. (131) under quasistationary conditions by postulating that the characteristic surface-transport relaxation time is very small, as estimated earlier. An integration of Eq. (131) leads to the following expression for the particle flux [112,115] ... 

This model better reflects the physical reality of particle adsorption processes by considering a three-dimensional motion of the wandering particle within the adsorption layer (see Fig. 36a). Thus, the ASF (blocking function) depends not only on particle coverage and structure but also on the distance from the interface h [112]. One may, therefore, postulate that the ASF is connected with the activity coefficient occurring in the expression for the chemical potential, Eq. (130), by the simple relationship... 

The thermodynamic parameters, measured at various temperatures for the isotropic, nematic, or smectic phase of a liquid crystalline stationary phase, are related to changes in the ability of a solute to interact with the solvent. Several theoretical treatments, including the refined infinite dilution solution model [453, 480, 481] and the lattice model [455,456], have been postulated for the general interpretation of the thermodynamic parameters in terms of the retention and selectivity parameters in gas chromatographic separations. However, attempts to correlate the thermodynamic quantities with specific solute-solvent interactions remain qualitative. Nevertheless, a few general observations can be noted. Upon cooling from an isotropic phase into a liquid crystalline phase, the values of decrease and those of 7 increase. Plots of Vg or 7 versus temperature show discontinuities at phase transition temperatures. The activity coefficient of any solute in an anisotropic environment (Ya) is found to be more positive (and typically, 7r>l) than in its isotropic phase (7i°°). This trend in is a reflection of vari-... 

One way to correct this situation is to consider the heterogeneous nature of the surface and asymmetries in gas/solid systems. Myers (2004) postulated that reliable methods for the predictions of the activity coefficients needed in RAST may be discovered in the future . We do not think that this has been accomplished as yet. We are not aware of successful and most importantly predictive versions of the RAST approach. 

Failure to obey Hammett s postulate may be the result of inadequate treatment of the experimental data (see III-A) to correct for spectral solvent shifts, or it may result from the base in question actually having different activity coefficient behavior from a true Hammett base. Clearly recognized examples of the latter condition... 

In 1868 two Scottish scientists, Crum Brown and Fraser [4] recognized that a relation exists between the physiological action of a substance and its chemical composition and constitution. That recognition was in effect the birth of the science that has come to be known as quantitative structure-activity relationship (QSAR) studies a QSAR is a mathematical equation that relates a biological or other property to structural and/or physicochemical properties of a series of (usually) related compounds. Shortly afterwards, Richardson [5] showed that the narcotic effect of primary aliphatic alcohols varied with their molecular weight, and in 1893 Richet [6] observed that the toxicities of a variety of simple polar chemicals such as alcohols, ethers, and ketones were inversely correlated with their aqueous solubilities. Probably the best known of the very early work in the field was that of Overton [7] and Meyer [8], who found that the narcotic effect of simple chemicals increased with their oil-water partition coefficient and postulated that this reflected the partitioning of a chemical between the aqueous exobiophase and a lipophilic receptor. This, as it turned out, was most prescient, for about 70% of published QSARs contain a term relating to partition coefficient [9]. 

Despite the work of Overton and Meyer, it was to be many years before structure-activity relationships were explored further. In 1939 Ferguson [10] postulated that the toxic dose of a chemical is a constant fraction of its aqueous solubility hence toxicity should increase as aqueous solubility decreases. Because aqueous solubility and oil-water partition coefficient are inversely related, it follows that toxicity should increase with partition coefficient. Although this has been found to be true up to a point, it does not continue ad infinitum. Toxicity (and indeed, any biological response) generally increases initially with partition coefficient, but then tends to fall again. This can be explained simply as a reluctance of very hydrophobic chemicals to leave a lipid phase and enter the next aqueous biophase [11]. An example of this is shown by a QSAR that models toxicity of barbiturates to the mouse [12] ... 

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