Four-Parameter Equations

Later sections describe several empirical relationships with the form of Eq. (7-19). Some free energy relationships require expression as four-parameter equations of the form... 

Drago and co-workers have correlated a large body of enthalpies of adduct formation in Lewis acid-base systems, including some solvents as reactants, with this four-parameter equation ... 

In 1955 Swain, et al. proposed a four-parameter equation, Eq. (8-74), to describe the solvent dependence of solvolytic reactions. 

Swain et al. ° analyzed solvent effects on 1080 pieces of rate and equilibrium data, showing that more than 98% of the effects could be correlated by the four-parameter equation... 

Trebble, M.A. Bishnoi, P.R. Development of a New Four-Parameter Equation of State, Fluid Phase Equilibria, 35, 1-18 (1987). 

To establish the operational pH scale [168-170], the pH electrode can be calibrated with a single aqueous pH 7 phosphate buffer, with the ideal Nemst slope assumed. Because the % calculation requires the free hydrogen ion concentration (as described in the preceding section) and because the concentration scale is employed for the ionization constants, an additional electrode standardization step is necessary. That is where the operational scale is converted to the concentration scale pcH (= log [H+]) using the four-parameter equation [116,119,171,172]... 

Hydrogen bonding is a special type of acid-base interaction (see Chapter 9). Probably the most important equation relating hydrogen bond strengths is the equation known as the Drago four-parameter equation,... 

In the case of the four-parameter equation, the enthalpies of interaction of a large number of acids and bases were determined calorimetrically in an inert solvent. With these values being known, a value of 1.00 was assigned for EA and CA for the Lewis acid iodine. The experimental enthalpies for the interaction of iodine with several molecular Lewis bases were fitted to the data to determine EB and CB values for the bases. Values were thus established for the four parameters for many acids and bases so that they can be used in Eq. (9.112) to calculate the enthalpies of the interactions. The agreement of the experimental and calculated enthalpies is excellent in most cases. However, the four-parameter approach is used primarily in conjunction with interactions between molecular species, although extensions of the approach to include interactions between charged species have been made. Table 9.7 gives the parameters for several acids and bases. 

Table 9.7 Acid and Base Parameters for Use in the Drago Four-Parameter Equation.

Recently the data concerning to interaction of propanthiole with chlorine dioxide in 8 solvents have been published [1], In this work it was shown, that the dependence of process rate from solvents properties is satisfactory described for seven solvents, after the exclusion of data for ethyl acetate, by the Koppel-Palm four parameters equation (coefficient of multiple correlation R 0,96) at determining role of medium polarity (coefficient of pair correlation between lg(k) and (s - l)/(2e + 1) - r 0.90). Chemical mechanism of the reaction including the formation of ion-radical RS H and radical RS has been proposed by authors [ ] ... 

Taft has also introduced the steric substituent constant E, which is used separately in an expression of the type shown in Eq. (3) or together with polar (T constants in a four-parameter equation. 

We do not discount the possibihty that a two term, four parameter equation could be found using trigonometric or other functions which can produce an energy term that becomes large for soft-soft or hard-hard combinations, but not for others. Clearly, none has been reported to date. 

These authors have shown that this four-parameter equation for W fits the early data of Treloar29 on natural rubber. 

Retention Modeling of Individual Solutes. In order to calculate the total response surface, the retention of each individual solute in the sample must be accurately modeled. For the results that we report here, a four parameter equation based on the relationships in equations 2-5 was selected to describe the retention of each solute. The model employed for the fit was ... 

Eaborn and his associates (Eaborn and Waters, 1961 Eaborn and Pande, 1961b, 1961c) have shown that the correlation of the relative rates in degermylation, destannylation, and deplumbylation are considerably improved by the application of the four-parameter equation. 

For electrophilic side-chain reactions, it is pointed out that such an expression is not appropriate because of the serious differences in the mode of delocalization of charge. Just as it is difficult to assess the necessity of applying the four-parameter equation for aromatic substitution so it is equally difficult to ascertain the usefulness of a five-parameter equation. 

A four-parameter equation for predicting the calorimetric enthalpies of acid-base interactions, —Ai/ab, in neutral solvents, proposed by Drago et al. [146], with spectroscopically determined shifts of the OH stretching frequency of phenol when interacting with a variety of bases of dilute solutions in carbon tetrachloride and tetrachloroethylene. 

Also, Fowkes [26,148] actually developed a technique for the determination of the work of adhesion, WA, by applying a four-parameter equation for acid-base interaction in the system of solid-liquid using Eqs. (29) and (32)... 

Doan and Drago [153,154] have reported an extended work that Eq. (61) include the spectral shifts obtained from the LSERs approach. They showed that a four-parameter equation for a general set of electron donors (acceptors), in which the specific interaction or spectral shift with an electron acceptor (donors) can be fit by LSERs. 

These are four-parameter equations, and Lola Deming spent weeks at her desk calculating machine to fit the curves that Lambert and Clark obtained for the adsorption of benzene on ferric oxide gel (21). These isotherms are shown in Figure 5. I do not blame anyone for not using these equations, and actually very few have used them to date. Joyner and Emmett (20) were among the hardy souls who did. Clampitt and German published another paper very recently (II), which is well worth reading for all those who deal with Type IV isotherms. 

The aforementioned macroscopic physical constants of solvents have usually been determined experimentally. However, various attempts have been made to calculate bulk properties of Hquids from pure theory. By means of quantum chemical methods, it is possible to calculate some thermodynamic properties e.g. molar heat capacities and viscosities) of simple molecular Hquids without specific solvent/solvent interactions [207]. A quantitative structure-property relationship treatment of normal boiling points, using the so-called CODESS A technique i.e. comprehensive descriptors for structural and statistical analysis), leads to a four-parameter equation with physically significant molecular descriptors, allowing rather accurate predictions of the normal boiling points of structurally diverse organic liquids [208]. Based solely on the molecular structure of solvent molecules, a non-empirical solvent polarity index, called the first-order valence molecular connectivity index, has been proposed [137]. These purely calculated solvent polarity parameters correlate fairly well with some corresponding physical properties of the solvents [137]. 

Another statistical treatment of a set of 32 solvent parameter scales for 45 solvents using the program SMIRC ( election of a set of minimally mterrelated columns) has been carried out by Palm et al. [246], who, incidentally, introduced the first multi(four)-parameter equation for the correlation analysis of solvent effects in 1971 [cf. Eq. (7-50) in Chapter 7]. The minimum sufficient set of residual descriptors for the multilinear description of solvent effects consists of nine solvent parameter scales. This set of nine (purified) descriptors has been successfully applied to an extended set of 359 different solvent-dependent processes for more details, see reference [246]. 

To account for nucleophilically solvent-assisted processes, Grunwald, Winstein et al. [42] later provided a four-parameter equation of the type shown in Eq. (7-15) b... 

A more rigorous approach has been suggested by Koppel and Palm [6, 112], who argue that a complete description of all solute/solvent interactions must include both non-specific and specific effects. They proposed the general four-parameter equation (7-50), which relates the variation of a given property A to two non-specific (T and P) and two specific characteristics of the solvent E and B). 

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