Independent constituents

If these structural features are not well represented by a mild redistribution of random independent constituents from an initially given prior prejudice, and arise instead from some degree of correlation between the scatterers, they cannot be expected to be satisfactorily dealt with by the method. For these reasons, substructures which scatter well beyond the experimental resolution should be left out of the subset of scatterers distributed at random. The data sets commonly collected for charge density studies do not as a rule extend beyond 0.4 A resolution, but scattering from the atomic core does extend well beyond this limit.2... 

THE. 14.1. Prigogine and R. Defay, On the number of independent constituents and the phase rule (On the number of independent constituents and the phase rule), J. Chem. Phys. 15, 614 (1947). 

The stoichiometric description of a chemical reaction requires answers to the following questions. How many equations are to be written How can we verify that they are independent How can we verify that they correctly describe the experimental results . Answers to these questions are provided by the criteria of Brinkley and Jouguet and by the theory of invariants. But, before this, the concepts of independent constituents and stoichiometries need to be defined. 

Let us consider a complex reaction involving c components Cls C2,. .., Cc, none of which is chemically inert under the experimental conditions and which may appear in the formulation of s independent stoichiometric equations. Stoichiometric equations are independent if none of them may be obtained by a linear combination of the others. On the other hand, one particular component is not independent when it may be obtained from the other constituents. Therefore, the number, c, of independent constituents is given by the relationship... 

Chemically inert constituents are obviously independent constituents this is the case in systems where only physical transformations take place. For a given chemical system, the nature of the independent constituents is not determined in an unequivocal way indeed, when several constituents are linked by a chemical reaction, any one of them can be generated from the others. However, the number of independent constituents is rigorously determined and is characteristic of the system under consideration. Therefore, the number of independent stoichiometries is also perfectly fixed, but their choice is arbitrary, because all linear combination of stoichiometries is also a stoichiometry. This brings out the fact that stoichiometries do not, in general, represent the true reaction mechanism. 

This is a homogeneous system of m linear equations in the c unknowns vn > vn > vic Let us denote by c the rank of the matrix of element indices akJ in the constituent formulae. The number, m, of elements is less than or equal to the number, c, of the constituents i.e. m < c. Furthermore, since all constituents may be formed from the m elements, the number, c, of independent constituents is, at the most, equal to m i.e. c less than m if some constituents may be obtained by chemical reaction between other constituents. Thus, c < m < c. 

The statement of Brinkley s criterion is, therefore the number, c, of independent constituents is equal to the rank of the matrix of element indices in the constituent formulae. The number, s, of independent stoichiometries to be written is s = c — c. ... 

Let us point out that the minimum number of invariants is equal to c = c— s, i.e. to the number of independent constituents the corresponding invariants are classically obtained by doing the mass balances for each element. But it has been seen that it is possible, in fact, to have a number, s, of stoichiometric equations less than s = c —c if this happens, the number of invariants is greater than the number, c, of independent constituents or of elements involved in the system. Consequently, the invariant theory provides an optimal exploitation of the stoichiometric results. 

Chemical reactions involve finite changes in E, H, S, A, and G these are called AE, AH, AS, A A, and AG. It would be wonderful if one could refer everything back to 0 K, and to separate atoms, so that the formation of a molecule would involve an enthalpy of formation AHf which would be negative, since all molecules should be more stable than their independent constituent atoms. However, this sensible ideal is defeated by the fact that heats of atomization (needed to split a molecule, say ethanol C2H5OH, into 2C atoms, 1 O atom, and 6 H atoms) are hard to measure, and OK is impossible to reach. 

IV.l SCHEME OF CLASSIFICATION The methods available for the detection of anions are not as systematic as those which have been described in the previous chapter for cations. No really satisfactory scheme has yet been proposed which permits of the separation of the common anions into major groups, and the subsequent unequivocal separation of each group into its independent constituents. It must, however, be mentioned that it is possible to separate the anions into major groups dependent upon the solubilities of their silver salts, of their calcium or barium salts, and of their zinc salts these however, can only be regarded as useful in giving an indication of the limitations of the method and for the confirmation of the results obtained by the simpler procedures to be described below. 

A chemical molecule, by contrast consists of many particles. In the most general case N independent constituent electrons and nuclei generate a molecular Hamiltonian as the sum over N kinetic energy operators. The common wave function encodes all information pertaining to the system. In order to constitute a molecule in any but a formal sense it is necessary for the set of particles to stay confined to a common region of space-time. The effect is the same as on the single confined particle. Their behaviour becomes more structured and interactions between individual particles occur. Each interaction generates a Coulombic term in the molecular Hamiltonian. The effect of these terms are the same as of potential barriers and wells that modify the boundary conditions. The wave function stays the same, only some specific solutions become disallowed by the boundary conditions imposed by the environment. 

Cyclohexane is rly easily dehydrogenated into benzene, and even at very low extents of reaction, stoichiometry reaction (6) can be replaced by the secondary stoichiometry reaction (12). For cyclohexane, the constituents are (apart from cyclohexane) hydrogen, methane, ethane, ethylene, acetylene, propene, 1-butene, 1,3-butadiene, cydohexene, and benzene [3]. However, one can check that the equations written are independent, using the Jouguet [IS] criterion, (t.e., if/ = n-o)). In this criterion, the number of the independent constituents, 1//, for a chemical system is equal to the required constituents. n, (i e., H3, Cl, C3, CsHe, Q, C4, Cj. c-Q, CaJ, subtracting the number of independent stoichiometric equations. q>. 

Moreover, by combining the Brinkley s [19] criterion, m rt-if/, one can calculate the number of independent stoichiometric equations. In this criterion the number y/ of independent constituents of a chemical system is equal to the rank of the matrix of the indexes of the elements in the formula of the constituents, hence cd = n-y/ = 9-2 = 7. Kinetic considerations have led to write 11 stoichiometric equations. It must be checked by Jouguet s criterion [18] that these 11 stoichiometric equations are independent. In the case of cycohe.xane the stoichiometric equations are 2, 3, 4, 8, 9, 11, and 12. Thus these seven stoichiometric equations are independent and describe the decomposition of cyclohexane. 

The methods available for the detection of anions are not as systematic as those which have been described in the previous chapter for cations. No really satisfactory scheme has yet been proposed which permits of the separation of the common anions into major groups, and the subsequent unequivocal separation of each group into its independent constituents. 

The simplicity of the conservation equation (5.269) is due to the fact that the ideal system that it represents is composed of independent constituent parts. When this independence is compromised by interactions, characterized by a coupling parameter g, the system becomes nonideal. Then, we propose in place of (5.269) the following generalization... 

Component a chemically independent constituent of the system. It is best understood in relation to the phrase number of components which is the minimum number of independent species necessary to define the composition of all the phases present in the system. 

Starting by determining the number of independent constituents, the formula matrix can be written as follows ... 

He weighed some gas in a bound state, referring to it as bound or fixed air. The gas was liberated during fermentation processes or combustion of charcoal but it did not sustain respiration or combustion. Black believed this gas to be an independent constituent of atmospheric air. 

The solid dark areas in Fig. 4 illustrate the anticipated additive effects for each combination based on the sums of the effects of each of their independent constituents. The striped area in each bar represents the... 

An intimate relationship of nucleolus and chromosomes is so well established in the minds of some that White [28] concluded The bodies known as nucleoli are attached to certain chromosomes (always the same ones, for a particular organism) and are hence to be looked upon as part of those chromosomes, rather than as independent constituents of the nucleus. This concept of nucleoli originating from chromosomes is not without experimental support. Already in 1928, Heitz [29] demonstrated that the nucleolus is formed from a specific part of the chromosome, a chromosomal structure later called the organizing element by McClintock [30]. It was soon observed that the nucleolus is not formed by one or a few chromosomes, but that all chromosomes contribute material that is later incorporated into the nucleolus. 

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