Molecular equation defined

MWR = is used, where Af is a melt-viscosity-average molecular weight defined by the equation Af = antilog, (0.493 log Tf + 3.5576, where Tj is... 

The rapidly changing trigonometric denominator gives rise to divergences that correspond to the molecular resonances [16,27,28]. We can see that the DOS contains the same divergent term as the transmission, therefore, at least close to the molecular resonances, the transmission seems to be proportional to the scaled DOS, as suggested by the equation defining t (equation (32)). The spectral function (see equation (37)) contains the term Ti + U2, which refers to the contacts... 

The differences between the cluster skeletons of the three molecules of 2 are very small with the mean values of the Ru-Ru distances being similar and the mean Ru-C(carbide) distances being identical. The most notable differences between the structures arise from the orientation of the tricarbonyl units attached to the apical Ru atoms above and below the molecular equator of the octahedral cluster (the molecular equator is defined as the Ru4 plane in which the bridging carbonyl ligand is present). The two tricarbonyl units are almost exactly staggered in the crystal obtained from benzene, whereas they approach an eclipsed conformation in the other polymorph. Although the 13C-NMR spectrum of 2 has not been recorded in solution (or in the solid state), it is not unreasonable to anticipate that... 

MARK.-HOUW1NK EQUATION. Defines the relationship between the intrinsic viscosity and molecular weight for homogeneous linear polymers. 

These two types of molecular weight averages are representative of the type called absolute methods, in that well-established thermodynamic equations can be used to convert the experimental data directly into a value of the molecular weight. However, some other methods require calibration. The most important of these indirect methods involves a measurement of the intrinsic viscosity. This quantity is a measure of the extent to which a polymer molecule increases the viscosity of the solvent in which it is dissolved. The viscosity method can be calibrated to yield a viscosity-average molecular weight, defined by... 

Consider some molecular property defined by the matrix A with the elements Ay = (( /-(l/Ur. R)l P,)) of a Hermitian operator A. One then easily derives that this matrix A obeys the following equation of motion [37],... 

As described in the previous contributions by Cances and by Tomasi, in such a family of methods the solvent effects on the molecular solutes are evaluated by introducing a set of apparent charges representing the polarization of the dielectric medium. These charges are obtained by solving integral equations defined on the domain of the boundary of the cavity which hosts the molecular solute. The solution of such equations can be divided in two main steps. 

As several workers have shown (for example, Ref. 6), it is possible to avoid any explicit rate expression for the overall decomposition of a paraffin. One defines all the larger number of free radical and molecular equations of significance, develops an appropriate computer program and adjusts, within credible limits, Arrhenius parameters of the various rate expressions to fit an available body of rate and yield data. 

Peclet Number, Pe dimensionless number appearing in enthalpy or species mass conservation equations (defined for heat transfer and mass transfer, respectively). It is interpreted again as the ratio of the convective transport to the molecular transport and is defined as... 

Figure 4. Nuclear trajectories generated by the simulated annealing molecular dynamics defined by the Lagrangian in Equation(4) using masses for all parameters equal to either 0.1 or 0.01 times the proton mass. For comparison, the exact nuclear trajectory on the Born-Oppenheimer surface is shown.

Molecular size may also be roughly measured by the dimensions of a molecular box defined by the extreme points of the molecule along the inertial axes (Figure 12.3). Here, D, Dy and can be interpreted as the minimum, intermediate and maximum dimensions of the molecule. The three moments of inertia, Mx>My>M, can be calculated in magnitude and direction as described in Appendix 12-1, Equations (10-11). 

Equations defining an average molecular area demand for all surfactant components of a mixture taking into account different , have been proposed by Rusanov [48], Lucassen-... 

As mentioned earlier, the equations defining the NLO properties are usually complicated by the fact that the electric fields of interest are time-dependent (i.e., from lasers). This means that different combinations of types or frequencies of fields give rise to multiple molecular and bulk properties at each order, as illustrated in Eq. [3]. To differentiate these effects, the molecular quantities need to be written to reflect the nature of the experimental setup. This is done as a(—o) a)), p(—and y( —a) a)j,a)2,(U3), where the quantities to the right of the semicolon are the input radiation(s) and the quantity to the left is the resulting (outgoing) radiation. 0) is simply the sum of the frequencies on the right side of the semicolon (conservation of energy). Definitions of the most commonly computed quantities are shown in Table 1. 

The FN-DQMC calculation in the table uses the diffusion quantum Monte Carlo method. quantum Monte Carlo (QMC) method uses a random process to solve the Schrftdinger equation. Many QMC methods exist, but the diffusion QMC (DQMC) method is most commonly used for molecular calculations. Defining the imaginary time variable t s itjh. 

The electronic potentials in the last row of this equation have relatively small contributions to Kdft-pb at equilibrium due to the fact that they essentially are confined inside the solute molecular domain. Note that Eq. 12.27 has the same structure as the potential-driven geometric flow equation defined in the models presented in earlier in this chapter. As t oo, the initial profile of S evolves into a steady-state solution, which offers an optimal surface function S. 

Equations defining an average molecular area demand for all siufactant components of a mixture, taking into account different cOj have been proposed by Lucassen-Reynders (38, 39) and Joos and coworkers (33,41). An example in which the contribution of each component to coj is determined by its adsorption relative to the other adsorptions (31, 38) is... 

For flexible molecules, the correlation functions are also functions of the molecular conformation. We consider small substances those for which conformation effects can be ignored. The molecular Omstein-Zemicke (OZ) equation defines the molecular DCF, c, (ri2, 0)i, otj) (Gray and Gubbins 1984),... 

This equation defines a sedimentation coefficient, s, which may be thought of as the intrinsic speed of the particle and which depends on the molar mass and on the frictional coefficient. The sedimentation coefficient is expressed in Svedberg units, which equal 10 s. Since it varies with temperature and solvent viscosity, it is often expressed as S20,wj that is, the equivalent in 20°C water. The quantity ni — vp) is called the buoyant molecular weight, and corresponds to the mass of the particle less than that of the displaced medium. 

If the conditions of consistent ionization volume, a fixed electron beam, and consistent alignment of the cell orifice with the molecular beam-defining apertures are met, the integration over the electron beam and the molecular beam shape in Equation 48.29 will be constant from cell to cell. Provided that the cells are isothermal, the comparison between different molecular beams the ion ratio is given by ... 

The Mark-Houwink equation defines the viscosity average molecular mass as ... 

In order to define the resolving power of a column in tenns of molecular weight, there must be an equation that relates A and This can be calculated by an equation defining the linear portion of the SEC calibration curve ... 

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