Energy equation, general form

I Liming now to the numerator in the energy expression (Equation (2.95)), this can be broken do, n into a series of one-electron and two-electron integrals, as for the hydrogen molecule, l ach of these individual integrals has the general form ... 

The dihedral angle or torsional energy interaction in MM-t is of the general form of equation (12) on page 175 but explicitly includes n=l, 2, and 3 with a phase angle (j) =0 ... 

If Equation 2,36 is rewritten in a general form using specific energy notation,... 

Monthly evaporation estimates can be made using the Kohler-Nordenson-Fox equation with a pan coefficient of 0.7. The Kohler-Nordenson-Fox equation describes evaporation as the combination of water loss due to radiation heat energy and the aerodynamic removal of water vapor from a saturated surface. The general form for the combination equation is ... 

Potential energy surfaces calculated by means of the London equation (5-15) cannot be highly accurate, but the results have been very useful in disclosing the general shape of the surface and the reaction coordinate. The London equation also forms the basis of some semiempirical methods. 

In Chapter 2, the general energy equation for the flow of a fluid through a pipe has been expressed in the form ... 

In this derivation a, the charge transfer cross-section is assumed to remain constant within the energy range applicable. Equation 3 describes the energy distribution of the ions at the cathode, and its general form has been substantiated experimentally (2). 

Equation 17 can be viewed as the general form of a sum rule for an arbitrary one-electron operator O expressed in terms of the square of the transition moment of the operator and its excitation energies. 

The first step in the solution of equation (10.28b) is to hold the two nuclei fixed in space, so that the operator drops out. Equation (10.28b) then takes the form of (10.6). Since the diatomic molecule has axial symmetry, the eigenfunctions and eigenvalues of He in equation (10.6) depend only on the fixed value R of the intemuclear distance, so that we may write them as tpKiy, K) and Sk(R). If equation (10.6) is solved repeatedly to obtain the ground-state energy eo(K) for many values of the parameter R, then a curve of the general form... 

However, the energy balance equation appropriate for use in this illustration differs from that employed in the previous case because thermal losses through the reactor walls must be accounted for. It will be of the same general form as equation 12.7.48, but with the wall heat transfer coefficient replaced by an overall heat... 

This is a general form of the energy equation for a control volume useful in combustion problems. The terms can be literally expressed as... 

The desired enthalpic information for the phenomenon under investigation is obtained through a simple energy balance, whose terms obviously depend on the processes that occur in each specific system. The general form of that balance is given by equation 13.6,... 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

It has been mentioned above that with nonuniform catalyst surfaces, as they mostly occur in practice, the above equations give merely an upper limit of the velocity although in such catalysts the true surface would be expected to be greater than the geometrical one. In nearly all the cases where a reaction has been measured on different catalysts, it has been found that fco is by no means a universal constant as would have to be expected from the simple theory. There is a regular connection between fco and the activation energy, of the general form ... 

Hohenberg and Kohn have proved generally that the total ground state energy E of a collection of electrons in the presence of an externally applied potential (e.g. the valence electrons in the presence of the periodic potential due to the cores in a lattice), when no net magnetic moment is present, depends only on the average density of electrons n(R). By this proof, n(R) becomes the fundamental variable of the system (as it is in the Thomas-Fermi theory ). Variational minimization of the most general form of E, with respect to n lends to the Hartree-Fock equations formalism. 

Panned (P4) and Jurges (J8) laid the basis for such predictions and more recently analogue computers have been applied to this problem (J6, S6). In principle, such predictions involved the solution of the general energy equation which may be written in the following form for two-... 

As we have seen, the Hammett equation can, in principle, be applied to both equilibria and rate data. This implies that in certain cases, it is feasible to relate rate constants to equilibrium constants when both reflect the effects of a given structural moiety. In a general form a rate-equilibrium relationship can be written in terms of the corresponding changes in free energies of activation and of equilibration ... 

In the general vector form, the thermal-energy equation is De ... 

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