The Principle of Congruence

The Principle of Congruence.—In 1946, Brensted and Koefoed demonstrated that there is a simple empirical relation between the properties of the normal alkanes and their mixtures. The relation, termed the principle of congruence, is formulated in terms of h, the average chain length of the mixture, which is a function of the mole fractions and the number nj of carbon atoms of each of the components 

To a very good approximation the properties of a mixture are the same as the properties of the pure alkane of chain length h. 

Barker and Linton were the first to apply the principle of congruence to virial coefficients. They treated data on three hydrocarbon mixtures and a number of binary mixtures of alkanes with hydrogen. (The chain length = 0 was assigned to H2.) Within experimental error the virial coefficients of the mixtures were equal to the virial coefficient of the hydrocarbon with chain length h. Newer measurements made it possible to test the principle of congruence more thoroughly for mixtures of n-alkanes and also for n-perfluoro-alkanes. The principle holds reasonably well for the hydrocarbons and somewhat more poorly for the perfluorocarbons. Stryjek applied the principle of 

Halm and L. I. Stiel, Amer. Inst. Chem. Engineers J., 1971,17, 259. 

---

Recently Loci, Mayhew, and Williamson have examined the extension of the principle of congruence to ternary alkane mixtures. As shown in Figure 10, the enthalpy of mixing of mixtures of alkanes of end-point indices 6 and 16 is very nearly independent of whether or not the end-point index (w = 16) is that of a pure substance (n-hexadecane) or of a mixture (n= 13 + = 19, X = 0.5). 

The data of the multicomponent mixtures were thus in agreement with the principle of congruence introduced by Brondsted and Kofoed (1946). 

The congruence between theory, model experiment and historical record enables us to regard the principle of life as one of nature s regularities. 

Correlations of this kind which assert that properties of mixing depend only on the ratio of numbers of groups are in effect restatements of Bronsted and Koefoed s simple principle of congruence. This principle assumes that the properties of a mixture of homologues are determined by the index of the mixture and that mixtures with the same index v (congruent mixtures) will have the same properties. The index of a mixture is given by... 

Cycloaddition reactions are close to the heart of many chemists - these reactions have fascinated the chemical community for generations. In a series of communications in the sixties. Woodward and Hoffmann [2] laid down the fundamental basis for the theoretical treatment of all concerted reactions. The basic principle enunciated was that reactions occur readily when there is congruence between the orbital symmetry characteristics of reactants and products, and only with difficulty when that congruence is absent - or to put it more succinctly, orbital symmetry is conserved in concerted reactions [3]. 

Reaction characterisation by calorimetry generally involves construction of a model complete with kinetic and thermodynamic parameters (e.g. rate constants and reaction enthalpies) for the steps which together comprise the overall process. Experimental calorimetric measurements are then compared with those simulated on the basis of the reaction model and particular values for the various parameters. The measurements could be of heat evolution measured as a function of time for the reaction carried out isothermally under specified conditions. Congruence between the experimental measurements and simulated values is taken as the support for the model and the reliability of the parameters, which may then be used for the design of a manufacturing process, for example. A reaction modelin this sense should not be confused with a mechanism in the sense used by most organic chemists-they are different but equally valid descriptions of the reaction. The model is empirical and comprises a set of chemical equations and associated kinetic and thermodynamic parameters. The mechanism comprises a description of how at the molecular level reactants become products. Whilst there is no necessary connection between a useful model and the mechanism (known or otherwise), the application of sound mechanistic principles is likely to provide the most effective route to a good model. 

RESOLUTION WITH THE CONGRUENCE PRINCIPLE OF VISCOSITY IN MIXTURES OF N-HEXADECANE AND 1-N-CHLOROALKANES AT 25 DEGREES. 

A rule of logic is that first principles cannot be tested by deductions that themselves rely on the same first principles. An important issue is, therefore, the question of whether the probability that a character is a homology can be estimated using a cladistic analysis, that is, if it is possible to identify homologies a posteriori (Patterson s test of congruence). The answer is complex (Wagele, 2000). 

Abstract. Onsager s model to describe the behavior of molecules in liquids is put in the appropriate historical context of the evolution of chemistry. Some key aspects of the model that justify its success in the past decades are discussed, with emphasis on general features shared with many other good models we have in theoretical chemistry and that should be kept in mind for the development of further models congruence with physical principles, simplicity and robustness. The present and future evolution of this model is briefly considered, with the aim of learning better from this example how to exploit our studies for the advancement of theoretical chemistry. 

---