Analog quantities

The quantities that are compared in Fig. 5.2 are mean square displacements, gi(t), of inner monomers in the laboratory frame and analogous quantities, g2 (t), in the center of the mass frame of each chain, the center of mass mean square displacement, g3(t), and mean square displacement of monomers at chain ends, (g4(t), gs(t). The precise definitions of these mean square displacements are as follows [12,20] ... 

The charge transfer reaction (5.2.39) is characterized by the formal electrode potential the conditional rate constant of the electrode reaction kf and the charge transfer coefficient aly while the reaction (5.2.40) is characterized by the analogous quantities E2y kf and a2. If the rate constants of the electrode reactions, which are functions of the potential, are denoted as in Eqs (5.2.39) and (5.2.40) and the concentrations of substances Au A2 and A3 are cly c2 and c3, respectively, then... 

Since the volumetric flow rate is a function of X, T, and P, the residence time V/v depends on these variables. Instead of using the reactor residence time T to describe performance, an analogous quantity called the space time ST, defined as... 

Analogous quantities to die electric moments can be defined when the external perturbation takes the form of a magnetic field. In this instance die first derivative defines the permanent magnetic moment (always zero for non-degenerate electronic states), the second derivative the magnetizability or magnetic susceptibility, etc. 

The Kelvin equation may also be applied to the equilibrium solubility of a solid in a liquid. In this case the ratio p/p0 in Equation (40) is replaced by the ratio a/a0, where a0 is the activity of dissolved solute in equilibrium with a flat surface, and a is the analogous quantity for a spherical surface. For an ionic compound having the general formula MmXn, the activity of a dilute solution is related to the molar solubility S as follows ... 

The constant kd2 is assumed to be of the same form as that given by Equation (47) for the first layer, with one important modification. While desorption from the first layer involves detaching a molecule from the adsorbent as a substrate, desorption from the second layer involves detachment from another adsorbed molecule of the same kind. The adsorption energy e was used in the Boltzmann factor in the first case the energy of vaporization ev is a more appropriate value to use for the analogous quantity in the second case. We shall assume the frequency factor to be unchanged and write... 

Table VI summarizes half-wave potentials of cathodic waves. These values are linearly dependent on the energies of lowest free 7r-molecular orbitals. The correlation is just about as close (Table IV) as those for analogous quantities with benzenoid and other hydrocarbons and derivatives of benzenoid hydrocarbons.

An analogous quantity, the generalized oscillator strength, is found to be useful in electron-scattering theory. It is a function of the momentum K transferred from the incident electron to the molecule and has the form5... 

It follows from (31.2) that for large Z this energy separation (wavelength of the appropriate electronic transition) is proportional to Z2. The analogous quantity for configurations with the same n (i.e. An = 0) is... 

Equation (88) was used to demonstrate [44] the differences in distributions of molecular sizes in pre-gel stages of an RA3 polymerization with substitution effects as calculated according to a statistical and a kinetics model. The moments of distribution and the gel points were calculated by the numerical solution of a few ordinary differential equations that were derived from Eq. (88) and compared with analogous quantities calculated from a statistical model. 

The analogous quantities for the dissociation laser, Ed(z), ed(o>), and ed(a>) are defined similarly, with the parameters td and cod replacing tx and mx, and so forth. %The pump pulse Ex(t) induces a transition to a linear combination of the eigenstates p of the excited electronic state. The pump pulse may be chosen to encompass any. timber of states. Here we choose the pump pulse sufficiently narrow in frequency to incite only two of these states, E2) and IE3). The dump pulse Erf(r) dissociates the Sfjnolecuie by further exciting it to the continuous part of the spectrum. Both fields are chosen sufficiently weak for perturbation theory to be valid. 

Then die solution of a mass ditfusion problem can be obtained directly from the analytical or graphical solution of the corresponding heat conduction problem given in Chapter 4. The analogous quantities between heat and mass transfer are summarized in Table 14-11 for easy reference. Tor the case of a seiiii-Lnfinite medium with constant surface concentration, for example, the solution can be expressed in an analogous manner to Eq. 1-45 as... 

An analogous quantity, called the bond vertex degree 6f, is calculated from the - atom connectivity matrix C as ... 

The reduced Markovian phase space is now given by the Euler angles specifying the position of the solute rotator 11 j and the three components of the corresponding angular momentum vector L, plus the analogous quantities and L2 for the solvent structure plus the fast field X and its conjugate linear momentum P. The conditional probability for the system... 

For the osmium EXAFS, the first term in Eq. 4.10 represents the contribution of osmium backscattering atoms. In this term, the quantity /V, represents the number of nearest neighbor osmium atoms about an osmium absorber atom and R, represents the distance between the osmium atoms. The phase shift function 28, (/0 is that for an OsOs atomic pair. The quantity f,(/0 exp(—2/C 2tr,2) differs from the analogous quantity for pure metallic osmium by a factor exp(—2X 2Ao-,2), where Ao-,2 is the difference between the value of o-,2 for the OsOs pair in the osmium-copper catalyst and the value for the same pair in the pure metallic osmium. Note that the quantity F,(K) exp( —2/C2cr,2) for the pure metallic osmium is known from the analysis of EXAFS data on it, as indicated earlier. 

AF and that of the third, dissociative, step is fV . The average of Xi and Xj is X. The free energy of activation for proton transfer within the properly oriented, structured, and solvated symmetrical donor-acceptor complex, AH-A, is Xi/4. The analogous quantity for the other partner in the original proton-transfer reaction is X2/4. 

The total volume per unit mass, V, the total entropy per unit mass, S, and so on, are related to the analogous quantities in each of the phases by the equations... 

It is useful to identify the physical significance of the quantities used here and to relate them to the analogous quantities for simple mixtures. In a simple liquid mixture, the properties of the pure components dominate the partial molar properties, and we have... 

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