The General Expression

For reasons that will become clear in the following paragraphs, the chemical potential cannot be expressed as a simple integral involving the pair correlation function. 

Consider, for example, the pressure equation that we have derived in section 5.8.3, which we denote by 

By this notation, we simply mean that we have expressed the pressure as a function of p and r, and also in terms of g(i ), which is itself a function of p and T. 

We see that in order to express a in terms of g R), we must know the explicit dependence of g R) on the density. Thus, if we used the pressure equation in the integrand of (5.9.4), we need a second integration, over the density, to get the Helmholtz energy per particle. The chemical potential follows from the relation 

A second method of computing the chemical potential is to use the energy equation derived in section 5.8.2, which we express as 

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The magnetic field diffracted by the edge of the crack in the material has the general expression... 

For a circularly polarized light experiment, one can measure the cross sections for either right (r) or left (1) polarized scattered light. Suppose that right polarized light is made incident on a Raman active sample. The general expressions for the Raman cross sections are [176]... 

X-ray scattering arises from fluctuations in electron density. The general expression of the absolute scattered intensity (simplified as I(q) from now on) from the tliree-dimensional objects iimnersed in a different... 

The general expression for particle scattering can best be described by the correlation fimction y(r). Using the definition in ( Bl.9.48). we have... 

Thus the overall interaction energy 0(z) of a molecule at distance z from the surface may be represented hy the general expression - ... 

The effect of an uncertainty in potential on the accuracy of a potentiometric method of analysis is evaluated using a propagation of uncertainty. For a membrane ion-selective electrode the general expression for potential is given as... 

The concept of a 1,5-dipolar cyclization gives rise to a general method for the synthesis of an appreciable number of heterocyclic systems. 1,5-Dipoles are derived from 1,3-dipoles by conjugation with different double bond systems, and it is possible to derive 98 theoretically possible 1,5-dipolar systems. The general expression for a 1,5-dipole and some possible combinations of double bond systems are shown in Scheme 14. 

This type of ring interconversion is represented by the general expression shown in Scheme 15. Analogous rearrangements occur in benzo-fused systems. The known conversions are limited to D = O in the azole system, i.e. cleavage of the weak N—O bond occurs readily. Under the reaction conditions, Z needs to be a good nucleophile in its own right or by experimental enhancement (base catalysis, solvent, etc.) and Z is usually O, S, N or C. 

Additional thermodynamic properties are related to these and arise by arbitrary definition. Multiplication ofEq. (4-11) hy n and differentiation yields the general expression ... 

The general expression given by Eq. (14-8) is more complex than normally is required, but it must be used when the mass-transfer coefficient varies from point to point, as may be the case when the gas is not dilute or when the gas velocity varies as the gas dissolves. The values of yi to be used in Eq. (14-8) depend on the local hquid composition Xi and on the temperature. This dependency is best represented by using the operating and equilibrium lines as discussed later. 

Rapid Adsorption-Desorption Cycles For rapid cycles with particle diffusion controlling, when the cycle time is much smaller than the time constant for intraparticle transport, the LDF approximation becomes inaccurate. The generalized expression... 

Equation (18-36) is the general expression for impeller-induced nucleation. In a fixed-geometry system in which only the speed of the circulating pump is changed and in which the flow is roughly proportional to the pump speed, Eq. (18-36) may be satisfactorily replaced with... 

The resistances, when incorporated into equations descriptive of cross-flow filtration, yield the general expression for the permeate flux for particulate suspensions in cross-flow-electrofiltration systems. 

In nonpolar solvents, the observed rate of bromination is frequently found to be described as a sum of the first two terms in the general expression. The second-order term... 

Derive the general expression for the observed rate constant for hydrolysis of A as a function of pH. Assume, as is the case experimentally, that intramolecular general acid catalysis completely outweighs intermolecular catalysis by hydronium ion in the pH range of interest. Does the form of your expression agree with the pH rate profile given for this reaction in Fig. 8.6 (p. 489) ... 

Verify that the general expression for the nxxfulus of a dispersion-stiffened composite material reduces to... 

Start with the general expression for the force per unit width, N, In terms of the middle-surface strains and curvatures to derive the specific expression for for a two-layered, equal-thickness [0 /90°l laminate. Your final expression must be in terms of Qy and t, the laminate thickness. What Is such a laminate called What deformation characteristics does this laminate exhibit when subjected to N., i.e., what does this laminate do ... 

The general expression for the diffusion thermoelectric power for a given group of charge carriers is given by the Mott formula ... 

The general expression for involves both rj and Its final term performs an interpolation for mixed spin cases. 

By iteration, the general expression for the Newton Raphson method may be written (if f can be evaluated and is continuous near the root) ... 

This is the general expression for film growth under an electric field. The same basic relationship can be derived if the forward and reverse rate constants, k, are regarded as different, and the forward and reverse activation energies, AG are correspondingly different these parameters are equilibrium parameters, and are both incorporated into the constant A. The parameters A and B are constants for a particular oxide A has units of current density (Am" ) and B has units of reciprocal electric field (mV ). Equation 1.114 has two limiting approximations. 

In Fig. 7A is given the steady state scheme for two sites which defines each of the elemental rate constants and in Fig. 7B are the steady state equations for the rate of change with time of the probability, %, of each of the occupancy states of the channel oo, xo, ox and xx. C and Cx are the concentrations of the x ion on the left-and right-hand sides, respectively. The general expression for the current, ix, due to the ionic species, x, passing through a single channel is... 

To obtain a quantitative relation between cell voltage and concentration, it is convenient to start with the general expression for the free energy change discussed in Chapter 17 ... 

The equilibrium constants pi,p2,...,p are called the overall stability constants and are related to the stepwise stability constants by the general expression... 

For the desorption processes that follow the first-order kinetics, we thus obtain [neglecting again ([Pg.360]

If the desorption process follows the second-order kinetics, it is necessary to solve a quadratic equation for 0 resulting from Eq. (8) and the general expression (17) becomes rather complex. 

Of special interest is the hyperbolic heating schedule, z = 0, since the general expressions (35) reduce to simple forms ... 

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