Alternative Expression for

Alternative expression for cfU Equation (4.161b) can also be written as 

Other higher-order expressions are obtained in the same manner. The algebra is tedious, but the method is straightforward. Another compact form of Eq. (4.182) is 

In this equation, if and yjj approach zero, appi oaches 

The above simple relationship provides the criteria of stability for multicomponent systems. When 

In the derivation of the criteria of criticality, we will first use a simple approach that relies on geometrical representation. Then an alternative approach will be presented which is more suitable for multicomponent systems with a large number of components. 

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Appendix 3.1 Alternative Expression for a Wavefunction Satisfying Bloch s Function... 

An alternative expression for the detection limit, which minimizes both type 1 and type 2 errors, is the limit of identification, (Sa)loi> which is defined as... 

Since G is a state variable and forms exact differentials, an alternative expression for dG is... 

Tbe mass-transfer coefficients k c and /cf by definition are equal to tbe ratios of tbe molal mass flux Na to tbe concentration driving forces p — Pi) and (Ci — c) respectively. An alternative expression for tbe rate of transfer in dilute systems is given by... 

An alternative expression for resolution was put foreword by Karger, Snyder and Horvath (1), viz. 

The alternative expression for resolution given in equation (7) demonstrates that the plate resolution, as in other forms of chromatography, depends on the number of theoretical plates, the selectivity and the capacity ratio of the solute for the particular plate concerned. In practice, however, the expression given in equation (7) appears to be the more practically useful for TLC. separations. 

We denote the ratio of cake volume to filtrate volume as Xq. Hence, the cake volume is XoV. An alternative expression for the cake volume is hcA where he is tire cake height in meters. Consequently ... 

Equations (4-5) and (4-7) are alternative expressions for the estimation of the diffusion-limited rate constant, but these equations are not equivalent, because Eq. (4-7) includes the assumption that the Stokes-Einstein equation is applicable. Olea and Thomas" measured the kinetics of quenching of pyrene fluorescence in several solvents and also measured diffusion coefficients. The diffusion coefficients did not vary as t) [as predicted by Eq. (4-6)], but roughly as Tf. Thus Eq. (4-7) is not valid, in this system, whereas Eq. (4-5), used with the experimentally measured diffusion coefficients, gave reasonable agreement with measured rate constants. 

It is conventional to write the multipole expansion as a series in 1/r, and so we need to find alternative expressions for terms higher than the first. From elementary vector calculus we have... 

Equations (44), (45), and (45 ) are alternative expressions for the theoretical equation of state for an ideal rubber. Equations of state for swollen networks, derived in Appendix B, have the same form. Only the magnitude predicted for r at a given elongation is affected by swelling. 

Salters et al. (2002) have recently proposed an alternative expression for calculating D j and Dm in clinopyroxene as a function of crystal and melt composition. The expressions are calibrated on over 40 experimental determinations of Ehj and Dm- Salters et al. (2002) do not give values for the average absolute deviation. The full expressions (with 1 s.d. uncertainties in brackets) are ... 

With these alternative expressions for (57) it is now possible to write from (54)... 

The (m—1) factor will be dealt with below. An alternative expression for s2 is obtained by developing the squared parentheses as... 

Appropriate manipulation of Eqs. 6-60 and 6-62 gives alternate expressions for the two monomer reactivity ratios... 

A useful class of alternative expressions for may be derived by inserting an arbitrary projection tensor into the divergence of on the RHS of (2.183). We note that... 

The main result of this section is the derivation of a class of alternative expressions for the Cartesian drift velocity that is based on an expansion of the divergence of the diffusivity of as a sum... 

Note that, in Equation 12.3, is a negative quantity since it is opposed to the direction x (see Fignre 12.5) and this explains the negative sign in the second member of Equation 12.3 where the absolnte qnantity It/ l is introduced. Equation 12.4 relates (4 to P- Useful alternative expressions for F are obtained by combining Equation 12.4 with Einstein s equation... 

This formula generalizes the conclusion reached in Tables 1.2 and 1.3. It shows clearly that for a fixed amount of material, the surface area is inversely proportional to the radius for uniform, spherical particles. At the same time, the formula reminds us that some lower limit for Rs must be imposed since the relationship is undefined for Rs = 0. If SI units were used consistently, Asp would be expressed in m2 kg-1 however, m2 g 1 are the most commonly used units for this quantity. In the event of nonuniform or nonspherical particles, alternate expressions for Equation (2) have to be used. The following example considers the case of cylindrical particles. 

The coefficients are then matched with their counterparts in an alternative expression for dG dG = Vdp - SdT. When surface effects are included, these two expressions become... 

Because [ES] in Equation 6-11 is not easily measured experimentally, we must begin by finding an alternative expression for this term. First, we introduce the term [Et], representing the total enzyme concentration (the sum of free and substrate-bound enzyme). Free or unbound enzyme can then be represented by [Et] — [ES]. Also, because [S] is ordinarily far greater than [Et], the amount of substrate bound by the enzyme at any given time is negligible compared with the total [S]. With these conditions in mind, the following steps lead us to an expression for V0 in terms of easily measurable parameters. 

The decrease of Aa Tg must lead to a considerable decrease in free-volume with decreasing temperature. According to52, it was necessary to revise the postulate of constant free-volume at Tg. Taking into account the temperature dependence of Og and ai, Simha and Weil gave alternative expressions for the free-volume fraction/in terms of volumes and expansivities, respectively ... 

Let us consider the non-relativistic limit of the relativistic operators describing radiation. Expressing the small components of the four-component wave functions (bispinors) in terms of the large ones and expanding the spherical Bessel functions in a power series in cor/c, we obtain, in the non-relativistic limit, the following two alternative expressions for the probability of electric multipole radiation ... 

FIGURE 5.5 Alternative expression for corrected model allowing for distortion of hydrate (v 5 vH). Equation 5.29 is the analogous expression for the processes shown. 

Equation (11.116) requires that the thermodynamic equilibrium constant cannot vary with pressure. There are, however, alternate expressions for the equilibrium constant that do change with pressure. For a gas-phase reaction such as the one that we are considering, the concentrations of species are often given in terms of their partial pressures, pt. Then,... 

Consideration of the oxygen balance at the phase boundary between layers i and i + 1 thus leads to an alternate expression for the formation rate of layer i at the interface xt - Lt... 

An alternative expression for the maximum radius of influence is that given by Morrison and Szecsody (1985), which was derived from Eq. (10.2) but with k (unsaturated hydraulic conductivity) expressed as a function of the hydraulic head (see also Hart el al., 1994 and Hart and Lowery, 1997) ... 

Alternative expressions for the global rates of Reactions 14 and 16 were tried while developing the model. For the CO + CO2 conversion (Reaction 16) the overall correlation derived by Howard et al. (2) was initially used, but with this the calculated values of SL were considerably lower than the measured ones. For the methane disappearance rate (Reaction 14) the correlations proposed by Westbrook and Dryer (7) were tried, and these gave results negligibly different from those obtained by Equation 18. 

Alternative expressions for yield and conversion can be obtained using masses of P, and B, respectively. 

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