Orthonormal chemical reactions and chemical relaxation. Part 1.—Reactions in dilute solution

Hayman, H. J. G.

doi:10.1039/tf9696502918

Cited by 10 publications

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“…and can therefore be used equally well for chemical reactions which are not normal modes of chemical relaxation. This enables us to introduce the concept of a set of orthonormal chemical reactions [2][3][4] defined as a set of reactions, not necessarily the normal modes of chemical relaxation, which satisfy the orthonormality conditions (70). Such a set of reactions have all the thermodynamic properties, though not of course the kinetic properties, of the normal modes of relaxation.…”

Section: Comparison Of Methodsmentioning

confidence: 99%

“…We now consider an alternative general procedure for discussing the normal modes of relaxation of a chemical system; this method, originally devised by Jost [I 2] , by Hearon [13] and by Wei and Prater [14] for first-order reactions has been generalized recently by Schwarz [15] and by Hayman [2][3][4]16]. Equation (21) As we shall see, the transformation matrix R can be chosen so that the matrix RTR-1 \}1'L\}1~R is diagonal; the diagonal elements of this matrix, i.e.…”

Section: Normal Coordinates As Linear Combinations Of Deviations Frommentioning

confidence: 99%

“…We shall not repeat here the details of our earlier discussions [2][3][4] regarding the significance of the normal modes corresponding to the c infinite relaxation times obtained in the present method but shall merely point out that these normal modes do not represent balanced chemical reactions but rather "improper" chemical reactions corresponding to the removal or addition of the various solutes in amounts proportional to the corresponding "stoichiometric coefficients"; moreover, if our system is initially in chemical equilibrium, small changes in composition corresponding to these improper reactions will not disturb this equilibrium. The normal coordinates corresponding to these improper normal modes are constant in time and will, in general, give the deviations of the final equilibrium state from the reference equilibrium state; our choice of the final equilibrium state as our reference state thus means that these normal coordinates are all zero.…”

Section: Normal Coordinates As Linear Combinations Of Deviations Frommentioning

confidence: 99%

“…where Pi is the chemical potential of the ith substance while <Il x and <l»y are defined by (2) Since it is unnecessary to repeat the full details of our earlier discussions [2][3][4] regarding the physical meaning of the "improper" normal modes of relaxation, we simplify our previous treatment by choosing the reference equilibrium state of our system equal to its final equilibrium state; we distinguish the properties of our system in this state by the addition of a bar, e.g. T. V and mi" We denote the deviations of the molal ratios of our system at time t from those in the above reference state by 0i defined by (i= 1,2, ... ,r).…”

Section: Introductionmentioning

confidence: 99%

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Matrix Methods and Normalization Procedures in Chemical Relaxation

Hayman

1973

Israel Journal of Chemistry

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Two very general methods for dealing with chemical relaxation in solution near equilibrium are developed in detail along parallel lines; in the first of these the normal modes of relaxation are obtained as linear combinations of a set of basic reactions whereas in the second the normal coordinates are obtained as linear combinations of the deviations of the solute concentrations from those at equilibrium. The general procedures described in the literature, differing both in their basic assumptions (some being based on the theory of irreversible thermodynamics, others on kinetic considerations) and also in the concentration units adopted and the normalization procedures used, form special cases of the above two general methods and are considered from this point of view. The two general methods are compared critically and shown to be complementary since each has its own special advantages and disadvantages. In particular, the significance of the normal modes of relaxation corresponding to infinite relaxation times are shown to be entirely different in the two cases; in the first method these modes are “reactionless” and have little significance whereas the “improper” normal modes obtained by the second method show how the concentrations of the various solutes can be changed without disturbing the chemical equilibrium of the system.

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Section: Comparison Of Methodsmentioning

confidence: 99%

Section: Normal Coordinates As Linear Combinations Of Deviations Frommentioning

confidence: 99%

Section: Normal Coordinates As Linear Combinations Of Deviations Frommentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Matrix Methods and Normalization Procedures in Chemical Relaxation

Hayman

1973

Israel Journal of Chemistry

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Orthonormal Chemical Reactions and Chemical Relaxation Part 4. Reactions in Solution with Concomitant Entropy and Volume Changes

Hayman

1971

Israel Journal of Chemistry

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The method described previously [2,3 J for obtaining the relaxation times, normal coordinates and normal modes of relaxation of a chemical system is extended to the case in which heat transfer and volume changes occur at rates too slow to maintain isothermal and isobaric conditions. The four types of chemical relaxation treated previously (isothermal and adiabatic relaxation at constant pressure and at constant volume) form limiting cases of this problem whose solution thus provides a generalization and unification of the concept of orthonormal chemical reactions.In previous papers of this series [1, 2, 3J we confined our attention to chemical relaxation under four simple thermodynamic conditions (isothermal and adiabatic relaxation at constant pressure and at constant volume). These problems form limiting cases of the somewhat; more general relaxation problem treated here in which heat transfer and volume changes occur, though at rates too slow to maintain isothermal and isobaric conditions. We treat this problem by an extension of our previous method in which we look upon the entropy and volume of our system as if they were two additional chemical substances and assume that the entropy and volume changes which occur can be ascribed to two additional elementary "chemical" reactions.As in part 3, we consider a closed system which, at time t, has a uniform terrperature T and pressure P and contains Ni molecules of the ith reacting substance (i=1,2, ... ,r) and N s molecules of the solvent s. We define the molal ratio mi of the ith substance by (i=1,2, ... ,r) ( 1) and denote the corresponding chemical potential by ].Ii. In the present paper we suppose, however, that our system is in thermal contact with a heat bath at a constant terrperature~and that it is fitted with a piston working against a constant pressure pO • As before, we choose a suitable arbitrary reference equilibrium state corresponding to the terrperature~and pressure pO and to molal ratios m~, m~, ••• ,m'j. and denote the small deviations of our system fran this reference state

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Theory and Simulation of Chemical Relaxation Spectra

Ilgenfritz¹

1977

Molecular Biology Biochemistry and Biophysics

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Orthonormal chemical reactions and chemical relaxation. Part 1.—Reactions in dilute solution

Cited by 10 publications

References 0 publications

Matrix Methods and Normalization Procedures in Chemical Relaxation

Matrix Methods and Normalization Procedures in Chemical Relaxation

Orthonormal Chemical Reactions and Chemical Relaxation Part 4. Reactions in Solution with Concomitant Entropy and Volume Changes

Theory and Simulation of Chemical Relaxation Spectra

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