Two-Parameter Equations

Similarly, we can derive the weight average degree of polymerization  

The second term is always less than 0.5. This indicates a very narrow molecular weight distribution, much narrower than that of a condensation polymer. 

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There were two schools of thought concerning attempts to extend Hammett s treatment of substituent effects to electrophilic substitutions. It was felt by some that the effects of substituents in electrophilic aromatic substitutions were particularly susceptible to the specific demands of the reagent, and that the variability of the polarizibility effects, or direct resonance interactions, would render impossible any attempted correlation using a two-parameter equation. - o This view was not universally accepted, for Pearson, Baxter and Martin suggested that, by choosing a different model reaction, in which the direct resonance effects of substituents participated, an equation, formally similar to Hammett s equation, might be devised to correlate the rates of electrophilic aromatic and electrophilic side chain reactions. We shall now consider attempts which have been made to do this. 

The more extensive problem of correlating substituent effects in electrophilic substitution by a two-parameter equation has been examined by Brown and his co-workers. In order to define a new set of substituent constants. Brown chose as a model reaction the solvolysis of substituted dimethylphenylcarbinyl chlorides in 90% aq. acetone. In the case ofp-substituted compounds, the transition state, represented by the following resonance structures, is stabilized by direct resonance interaction between the substituent and the site of reaction. 

The applicability of the two-parameter equation and the constants devised by Brown to electrophilic aromatic substitutions was tested by plotting values of the partial rate factors for a reaction against the appropriate substituent constants. It was maintained that such comparisons yielded satisfactory linear correlations for the results of many electrophilic substitutions, the slopes of the correlations giving the values of the reaction constants. If the existence of linear free energy relationships in electrophilic aromatic substitutions were not in dispute, the above procedure would suffice, and the precision of the correlation would measure the usefulness of the p+cr+ equation. However, a point at issue was whether the effect of a substituent could be represented by a constant, or whether its nature depended on the specific reaction. To investigate the effect of a particular substituent in different reactions, the values for the various reactions of the logarithms of the partial rate factors for the substituent were plotted against the p+ values of the reactions. This procedure should show more readily whether the effect of a substituent depends on the reaction, in which case deviations from a hnear relationship would occur. It was concluded that any variation in substituent effects was random, and not a function of electron demand by the electrophile. ... 

There are many simple two-parameter equations for Hquid mixture constituents, including the Wilson (25), Margules (2,3,18), van Laar (3,26), nonrandom two-Hquid (NRTI.v) (27), and universal quasichemical (UNIQUAC) (28) equations. In the case of the NRTL model, one of the three adjustable parameters has been found to be relatively constant within some homologous series, so NRTL is essentially a two-parameter equation. The third parameter is usually treated as a constant which is set according to the type of chemical system (27). A third parameter for Wilson s equation has also been suggested for use with partially miscible systems (29,30,31). These equations all require experimental data to fit the adjustable constants. Simple equations of this type have the additional attraction of being useful for hand calculations. 

Whereas this two-parameter equation states the same conclusion as the van der Waals equation, this derivation extends the theory beyond just PVT behavior. Because the partition function, can also be used to derive aH the thermodynamic functions, the functional form, E, can be changed to describe this data as weH. Corresponding states equations are typicaHy written with respect to temperature and pressure because of the ambiguities of measuring volume at the critical point. 

This equation is useful for gases above the critical point. Only reduced pressure, P, and reduced temperature, T, are needed. In the form represented by equation 53, iteration quickly gives accurate values for the compressibiUty factor, Z. However, this two-parameter equation only gives accurate values for simple and nonpolar fluids. Unless the Redhch-Kwong equation (eq. 53) is expHcifly solved for pressure in nonreduced variables, it does not give accurate hquid volumes. 

Inspection of a plot of against [B] should suggest a trial equation that can be tested graphically or numerically. Five of the more common two-parameter equations for kj, and one three-parameter expression are these ... 

Rate law flooding. The second-order rate constant for the reaction between the hydrated ions of vanadium(3+) and chromium(2+) depends on [H+ ]. From the data given, which refer to T = 25.0 °C and a constant ionic strength of 0.500 M, formulate a two-parameter equation that describes the functional dependence. Evaluate the two constants. Compare your result to the one derived in to Problem 1 -2. 

Edwards two-parameter equation 549 Electrochemical oxidation of sulphides 76, 252, 253 of sulphoxides 968, 987, 1043 Electrochemical reduction of sulphones 962, 963, 1002-1041 of sulphoxides 933, 1041, 1042 Electronegativity, of the sulphur atom 584 Electronic effects 390, 484-535 Electron scavengers 892, 896 Electron spin resonance spectroscopy 874, 890-895, 1050-1055, 1082, 1083, 1090-1093... 

A better description of the interaction energy is provided by the following two-parameter equation ... 

Upon calibrating the columns with narrow MWD polystyrene standards, it was noted that the polydichlorophosphazenes eluted over a relatively straight region of the calibration plot. Therefore a two parameter equation. 

Several assumptions were made in using the broad MWD standard approach for calibration. With some justification a two parameter equation was used for calibration however the method did not correct or necessarily account for peak speading and viscosity effects. Also, a uniform chain structure was assumed whereas in reality the polymer may be a mixture of branched and linear chains. To accurately evaluate the MWD the polymer chain structure should be defined and hydrolysis effects must be totally eliminated. Work is currently underway in our laboratory to fractionate a low conversion polydlchlorophosphazene to obtain linear polymer standards. The standards will be used in polymer solution and structure studies and for SEC calibration. Finally, the universal calibration theory will be tested and then applied to estimate the extent of branching in other polydlchlorophosphazenes. 

Figure 5.7 Comparison of four-parameter fy-maxi mum, v-minimum. IC50, and h) and two-parameter (IC50 and h) fits of non-ideal concentration-response data. In panels A and B the data indicate a nonzero plateau at low inhibitor concentration that might reflect a low-amplitude, high-affinity second binding interaction. In panels C and D the data indicate a plateau at high inhibitor concentration that does not achieve full inhibition of the enzyme. There could be multiple causes of behavior such as that seen in panels C and D. One common cause is low compound solubility at the higher concentrations used to construct the concentration-response plot. Note that the discordance between the experimental data and the expected behavior is most immediately apparent in the plots that are fitted by the two-parameter equation.

There are several systematic nuclear magnetic resonance studies of the interaction between the substituents and the protons and ring atoms of five-membered heterocycles. In some 2-substituted furans, thiophenes, selenophenes, and tellurophenes there is a linear correlation between the electronegativity of the chalcogen and several of the NMR parameters.28 As there also is a good correlation between the shifts of the corresponding protons and carbons in the four heterocycles, the shifts of unknown selenophene and tellurophene derivatives can be predicted when those of thiophene are known. This is of special interest for the tellurophene derivatives, since they are difficult to synthesize. In the selenophene series, where a representative set of substituents can be introduced in the 2- as well as in the 3-position, the correlation between the H and 13C shifts and the reactivity parameters according to Swain and Lupton s two-parameter equation... 

Redlich-Kwong Equation. The Redlich-Kwong equation, which was proposed in 1949 [5], has been found to reproduce experimental P-V-T data for gases just as well as several equations that use more than two empirical constants and better than other two-parameter equations [6]. It has the form... 

Each two-parameter equation for real gases yields a slightly different equation. That such an equation exists for all two-parameter equations was shown by Meslin in 1893 [10]. 

If a gas follows any two-parameter equation of state, such as the van der Waals or the Redlich-Kwong, it has been shown in Section 5.2 that Z, the compressibility factor, is a universal function of the reduced pressure P = P/Pc and the reduced temperature Pj = P/ Pe- Then if Z is plotted as a function of Pj, at a given reduced temperature P, all gases fit a single curve. At another reduced temperature Tj, a new curve is obtained for Z versus Pj, but it too fits all gases. Gases at equal reduced pressures and reduced temperatures are said to be in corresponding states. 

The rate constants for methanolysis of alkyl /j-toluenesulfonates conform to a two-parameter equation of the Hammett-Taft type, the governing structural factor being steric hindrance. " However, the alcoholysis rates of alkenyl and alkynyl jo-toluenesulfonates do not obey this relation, probably because the multiple bonds stabilize the reaction intermediate. 

The small value of the coefficient a and the similar values of px and py in the one- and two-parameter equations indicate that the substituents X and Y exert independent effects on the ring-chain equilibrium. 

One may question the need for a four parameter enthalpy equation, i.e., whether describing an acid or base by two parameters each is redundant. The following simple matrix algebra shows the conditions whereby a four parameter model reverts to a less redundant two parameter equation. Letting A be the transformation matrix, E and C represent the parameters for the four parameter model, and a represent the acid parameters for the two parameter model, the following equation results ... 

Pearson (2) and Drago and Weyland 38) proposed a two-parameter equation for the equilibrium constant K for the reaction between A and B... 

Addition of sodium dodecyl benzene sulfonate to dilute alkaline electrolyte depresses the passivation of zinc surface [275]. Owing to the dodecyl benzene sulfonate adsorption, the passive layer on zinc has a loose and porous structure. Zinc electrodissolution was inhibited by the presence of sodium metasdicate [276] and some acridines [277]. The protection effect was described by a two-parameter equation. 

One of the surprising features of Ritchie s studies was a constant selectivity. While the absolute rate constants differed considerably over the series of cations, the selectivities toward pairs of nucleophiles were constant and independent of the reactivities of R" ". Rate constants were correlated by a simple two parameter equation... 

The two-parameter equation suggested by Jencks and co-workers (Eq. 22) separates substituent effects into a polar or inductive contribution described by p and a resonance contribution described by p. ... 

Sorption Analysis. Specific surface areas and porosity can be calculated from the adsorption isotherm of nitrogen at — 196 °C. The method of Brunauer, Emmett, and Teller [4.29] is generally accepted for the evaluation of specific surface areas (BET surface area in square meters per gram). The two-parameter equation is applicable to carbon black. The BET surface area comprises the outer surface area as well as the surface area of the pores. 

A linear correlation between 13C chemical shifts and local n electron densities has been reported for monocyclic (4n + 2) n electron systems such as benzene and nonbenzenoid aromatic ions [76] (Section 3.1.3, Fig. 3.2). In contrast to theoretical predictions (86.7 ppm per n electron [75]), the experimental slope is 160 ppm per it electron (Fig. 3.2), so that additional parameters such as o electron density and bond order have to be taken into account [381]. Another semiempirical approach based on perturbational MO theory predicts alkyl-induced 13C chemical shifts in aromatic hydrocarbons by means of a two-parameter equation parameters are the atom-atom polarizability nijt obtained from HMO calculations, and an empirically determined substituent constant [382]. 

Many semiempirical equations of slate with varying degree of theoretical foundations are in use. The Van der Waals equation, a two-parameter equation that gives a qualitatively correct picture of the P-V-T relations of a gas and of the gas liquid transition, is an example. 

Complexes of Th4+ with cupferron (as well as oxime) were investigated by determining the distribution of tracer amount of 234Th between the organic (chloroform or isobutyl methyl ketone) and aqueous phases.118 The distribution curves show the presence of complexes other than the tetra ones, but formation constants for the neutral tetrakis products were calculated on the basis of the two-parameter equation method of Dryssen and Sillen. 

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