Following equation

Daubert has recently published a new method (Hydrocarbon Processing, September 1994, page 75) to convert D 86 data to TBP results using the following equations ... 

The compressibility factors Zy are obtained after solving the following equations in for i = 1 and i = 2 ... 

The simplest way is to weight the Cp/ of each component according to the following equation ... 

The general problem is to determine at given conditions of temperature and pressure, the quantities and compositions of the two phases in equilibrium starting from an initial quantity of material of known composition and to resolve the system of the following equations ... 

The solubility of a solid in the liquid phase of a mixture depends on the properties of the two phases for the components that crystallize, the equilibrium is governed by the following equation [ XI... 

In a normal pressure regime the pressure in a hydrocarbon accumulation is determined by the pressure gradient of the overlying water (dP / dD), which ranges from 0.435 psi/ ft (10 kPa/m) for fresh water to around 0.5 psi/ft (11.5 kPa/m) for salt saturated brine. At any depth (D), the water pressure (PJ can be determined from the following equation, assuming that the pressure at the surface datum is 14.7 psia (1 bara) ... 

Eddy currents and the magnetic flux that is associated to them are proportional to the radial distance of the coil center. The magnetic flux is proportional to the probe induction and consequently to the passing current. The theoretic calculation of this induction is given by the following equation ... 

Very shortly, the first one is based on the stress measurement performed using a rosetta strain gauge located in an area of sufficiently uniform stress distribution. In this case, the calibration factor Cr can be easily obtained by the following equation ... 

If the entropy and the enthalpy for the separate mixing in each of the half-mole superlattices are calculated and then combined, the following equation is obtained ... 

The second-order rate law for bimolecular reactions is empirically well confinned. Figure A3.4.3 shows the example of methyl radical recombination (equation (A3.4.36)) in a graphical representation following equation (A3.4.38) [22, 23 and 24]. For this example the bimolecular rate constant is... 

We note that, by energy conservation, the following equation must hold ... 

Figure A3.13.9. Probability density of a microcanonical distribution of the CH cliromophore in CHF within the multiplet with cliromophore quantum nmnber V= 6 (A. g = V+ 1 = 7). Representations in configuration space of stretching and bending (Q coordinates (see text following (equation (A3.13.62)1 and figure A3.13.10). Left-hand side typical member of the microcanonical ensemble of the multiplet with V= 6...

The component of the DC quadnipolar potential in the z-axis direction is described by the following equation. 

These values are related to the initial wavenumber intervals by the following equations ... 

Here, the first factor (r, R) in the sum is one of the solutions of the electronic BO equation and its partner in the sum, Xt(R) is the solution of the following equation for the nuclear motion, with total eigenvalue... 

We reinstate the velocity of light c in this and in Section VI.F in order to appreciate the order of magnitude of the various terras. When contributions from C) are neglected, the expression in Eq, (162) equated to zero gives the following equations, in which the large (i = 1,2) and small (i = 3,4) components are separated. 

In the same manner, we obtain the following equations from Eq. (160)... 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

Substitution of the ansatz (31) into the Schrddinger equation (1) for the full system, together with the above approximations, yields the following equations for the coefficients Co(t),djaj/0 t) of the Cl expansion (31) ... 

There are various approaches to the problem of coupling quantum degrees of freedom to classical degrees of freedom. The QCMD model is given by the following equations of motion ... 

Using Eqs. (29) and (30) we obtain that the coherent potential v i with elastic scattering correction is determined by the following equations... 

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