Basic expression for

We begin, as we did before with the basic expression for the transmittance of a sample since this is a repeat of previous equations we use the same numbers instead of starting with new numbering for the same equations ... 

This is the basic expression for the response of the microscope to a uniform isotropic specimen at a defocus z. It should be permanently imprinted on the mind A V(z) curve for glass calculated using this expression is shown in Fig. 7.5. The primary difficulty in achieving exact agreement with measured curves lies in knowing the exact pupil function of the lens in the experiment. 

The basic expressions for the mass fluxes and the equations of continuity for multi-component mixtures are given in Sec. II,B. For a -component mixture of ideal gases in a system in which there is no pressure diffusion, forced diffusion, or thermal diffusion, the fluxes are given by... 

Such expressions can be extended to permit the evaluation of the distribution of concentration throughout laminar flows. Variations in concentration at constant temperature often result in significant variation in viscosity as a function of position in the stream. Thus it is necessary to solve the basic expressions for viscous flow (LI) and to determine the velocity as a function of the spatial coordinates of the system. In the case of small variation in concentration throughout the system it is often convenient and satisfactory to neglect the effect of material transport upon the molecular properties of the phase. Under these circumstances the analysis of boundary layer as reviewed by Schlichting (S4) can be used to evaluate the velocity as a function of position in nonuniform boundary flows. Such analyses permit the determination of material transport from spheres, cylinders, and other objects where the local flow is nonuniform. In such situations it is not practical at the present state of knowledge to take into account the influence of variation in the level of turbulence in the main stream. 

The basic expressions for the rate constant within a fully classical version of conventional transition-state theory were derived in Chapter 5. According to Eq. (5.53), we may write... 

The basic expression for molar flux density, Eq. 3.23, can be written as... 

The experimental details of dispersion polymerization with various polymeric dispersants and macromonomers are fairly well established. A basic expression for particle size control has also been derived for the formation of clear-cut core-shell particles based on highly incompatible core-shells such as polystyrene-PVP and polystyrene-PEO. However, results deviate considerably from theory in compatible polymers such as PMMA with PEO macromonomer. The detailed structures of the hairy shells need to be discovered in order to better understand the exact mechanism of their formation and stabilizing function. 

The Gibbs--Duhem Relation as a Basic Expression for Gas Adsorption... 

By applying equation 49 to standard-state partial molar enthalpies of species in a reaction, we obtain the basic expressions for describing the effect of change in temperature on the equilibrium constant of the reaction. 

Substituting for [ROO ] in equation 5, for the rate-limiting reaction of peroxyl radicals, gives the basic expression for suppressed oxygen uptake in the presence of the antioxidant (equation 14). 

Note it is important to remember that in the basic expression for the emf the logarithmic terms are to base e. When Debye-Hiickel equations are being used to relate the activity coefficients to the ionic strength, logarithms to base 10 are involved. Conversion between the two bases is essential. Hence the 2.303 in the above equation loge = 2.3031ogior. 

The basic expression for the quantization of the electromagnetic field is the expansion Eq(54). In the quantized theory the numbers Ck,, C x become operators of the creation C x and the annihilation Ck,x of photons. These operators are acting on the state vector < ) that is defined in the Fock space (occupation number space). The C xt Ck, operators satisfy the commutation relations ... 

The basic expression for the rate of oxidation is the same as for isobutane... 

The basic expressions for the matrix elements of the spin-orbit coupling operator have been derived by Griffith [48]. A double tensor operator is Xf[y where, for fixed y and varying M, X fy is an irreducible tensor operator with respect to spin variables. Similarly, for fixed S and varying y, Xf[y is an irreducible tensor operator with respect to space variables. A matrix element of a double tensor operator is now reduced as follows... 

The basic expression for scattering from a collection of scattering objects (equation 4.18) is... 

The basic expressions for ion-pair extraction are given by eqns. (1) and (2). The distribution ratio expression (3) derived therefrom is valid if further processes can be disregarded. 

When Noy is evaluated at a point on the tray, the basic expression for poim eflficiency is obtained ... 

It may be observed that the previously presented basic expression for Fick s law does not include pressure in either the differential or... 

The first term in the decomposition of the supermolecular interaction energy E s has the same formal expression as the corresponding term in the perturbation theory (Eq. 5). The basic expression for the use of the electrostatic potential as an index of chemical reactivity does not depend upon the theory adopted in describing molecular interactions. 

Troe has described how one can estimate the value of the partition function internal states at the dissociation limit, which treats the vibrations in RadiRad ) as harmonic. Multiplicative factors are then estimated to allow, in turn for (i) the anhar-monicity of the vibrations (ii) the energy dependence of the density of vibrational states (iii) an overall rotation factor, which allows for the existence of centrifugal barriers and (iv) an internal rotation factor allowing for the barriers associated with internal rotors. 

To properly account for such effects a more sophisticated model is necessary. The most promising approach, developed by Krishna and his associates, is based on the generalized Maxwell-Stefan (GMS) model [22-30]. The basic expression for the flux in a multicomponent system is ... 

In order to present some basic expressions for the Auger lifetime let us denote Auger generation (impact ionization) rate as Ga and Auger recombination rate as Ra- We consider first the Auger 1 process. Since two electrons and one hole take part in it, the recombination rate will be proportional to n p... 

The basic expressions for material balances and their graphical interpretation were presented for any mass-transfer operation in Chap. 5. Here they are adapted to the problems of gas absorption and stripping. 

On the basis of our formalism it is possible to derive a cluster expansion formula for the restoring force of a chain. This quantity is basic to rubber-like elasticity. In cases when many chains form a network as in the case of a slightly vulcanized rubber one ould take the connections of chains into consideration. However, since such a consideration has been given before, we shall provide here only the basic expression for the restoring force. 

System (2.4) is the basic expression for determining the charges of the individual reactors and for establishing the material balance of complex combined processes. 

The maximum displacement of the isolation system, D/, is computed from the basic expression for a spring force, Vb = kbDb- Noting that kb = (dIj and the relationship between spectral acceleration and displacement is given by A (d ,C) =, the maximum isolator... 

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