General expressions

5 FREE ENERGY OF A SOFT SURFACE 5.5.1 General Expression 

FIGURE 5.2 A soft particle composed of the particle core of surface area A coated by an ion-penetrable surface layer in an electrol3fe solution of volume V. Pdir) is the volume charge density at position r resulting from electrol3fe ions and is the volume charge 

We derive an alternative expression for Fei, which corresponds to Eq. (5.70) for a hard particle, by considering a discharging process in which all the charges (both the volume charge density pfi (r) and the charge of electrolyte ions) in the system are decreased at the same rate. From Eq. (5.72), we obtain 

Note that j/(r, v, X) is related to the volume charge density p iir, v, X) resulting from electrolyte ions at position r at stage X in the discharging process starting from the stage V by the following Poisson-Boltzmann equation  

Here n is the outward normal from the particle core to the electrolyte solution and the integral taken on the external surface of V may be neglected. Sine we have assumed that the particle core is uncharged, that is, diJ/Zdn = 0, the quantity given by Eq. (5.81) is zero. Also from the form of pe (r, v, 1), namely. 

As the oscillators of the OPP model vibrate independently of each other, the frequencies are dispersionless, that is, independent of a wavevector q. For the internal modes of a molecular crystal, this tends to be a very good approximation. For the external modes, the dispersion can be pronounced, as shown in Figs. 2.1 and 2.2. In order to obtain the mean-square vibrational amplitudes for the latter, a summation over all phonon branches in the Brillouin zone must be performed. 

The displacement of an atom j, in unit cell /, at time t, is obtained by the summation over all 3nN normal modes, combined into 3n phonon branches, 

The tensor U)j( describing the mean-square displacements of atom j, is the time average u(y )u( )r , where u is the 3 x 1 column matrix of the displacements of atom j along the Cartesian axes, and T indicates the transpose. Since the normal modes are independent of each other, cross terms between modes disappear in the averaging. The result is 

Application of Eq. (2.58) to calculate the temperature factors requires knowledge of the full frequency spectrum of the crystal throughout the Brillouin zone. Such information is only available for relatively simple crystal structures such as Al, Ni, KC1, and NaCl (Willis and Pryor 1975, p. 13ff.). Agreement between theory and experiment for such solids is often quite reasonable. 

A considerable simplification is achieved when molecules can be treated as rigid bodies, as was done for naphthalene and anthracene (Fig. 2.2), the frequency spectra of which were derived using atom-atom potential functions. The mean-square displacements due to the internal modes can be calculated from the experimental infrared and Raman force constants, and added to the values obtained with Eq. (2.58). The rigid-body model for thermal vibrations is further discussed in section 2.3.3. 

The correlation function (q) of the segment density within a given chain in the solution has been defined in Eqs. (5.10), (5.17). The corresponding scaling function Jft(q) has been introduced in Eq. (9.4)  

Comparing to Eq. (5.54) we find the tree approximation in unrenormalized form  

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Racah parameters The parameters used to express quantitatively the inter-electronic repulsion between the various energy levels of an atom. Generally expressed as B and C. The ratios between B in a compound and B in the free ion give a measure of the nephelauxetic effect. ... 

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

A measure of the purity or cohereuce of a system is given by jpf = 1 for a pure state and S. for a mixed state the greater the degree of mixture the lower will be the purity. A general expression for the... 

By differentiating the defining equations for H, A and G and combining the results with equation (A2.T25) and equation (A2.T27) for dU and U (which are repeated here) one obtains general expressions for the differentials dH, dA, dG and others. One differentiates the defined quantities on the left-hand side of equation (A2.1.34), equation (A2.1.35), equation (A2.1.36), equation (A2.1.37), equation (A2.1.38) and equation (A2.1.39) and then substitutes die right-hand side of equation (A2.1.33) to obtain the appropriate differential. These are examples of Legendre transformations. ... 

The result is, of course, a case of the more general expression of Fourier s law, namely... 

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... 

All interactions associated with NMR can be expressed as tensors and may be represented by a general expression... 

Using equation (C2.15.24), we can derive a general expression for die absorjition coefficient for dris simple two-level system ... 

Returning to the more general expression, in the low temperature limit we find that the rate... 

These general expressions are quite cumbersome, but they are much simplified if attention is limited to a pure gas, when (A.1.11) reduces to... 

A general expression for the entropy of a system, involving any phase transitions, is... 

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... 

This contrasts with a limiting ratio of 2 for the case of termination by disproportionation. Since and can be measured, this difference is potentially a method for determining the mode of termination in a polymer system. In most instances, however, termination occurs by some proportion of both modes. Although general expressions exist for the various averages and their ratio when both modes of termination are operative, molecular weight data are generally not sufficiently precise to allow the proportions of termination modes to be determined in this way. 

In this section and the last, we have examined the lattice model of the Flory-Huggins theory for general expressions relating AHj and ASj to the composition of the mixture. The separate components can therefore be put together to give an expression for AGj as a function of temperature and composition ... 

We defined the equation of motion as a general expression of Newton s second law applied to a volume element of fluid subject to forces arising from pressure, viscosity, and external mechanical sources. Although we shall not attempt to use this result in its most general sense, it is informative to consider the equation of motion as it applies to a specific problem the flow of liquid through a capillary. This consideration provides not only a better appreciation of the equation of... 

Taking the attitude described in item (5) toward the previously developed equations for [r ] is an important step. Wliat this enables us to do is write a general expression for [r ] in solutions of flexible polymers ... 

Overvoltage. Overvoltage (ti. ) arises from kinetic limitations or from the inherent rate (be it slow or fast) of the electrode reaction on a given substrate. The magnitude of this value can be generally expressed in the form of the Tafel equation... 

The first term on the right represents scalar conduction and the second term the Hall effect. This is generally expressed in terms of the Hall parameter l3 = so that... 

On more severe thionation, a third thiamine ring is formed to give a sulfur black. However, if hydroxyl groups instead of amino groups are attached at positions 2 and 2, no ring closure would take place and the blue dye would be stable to heat. These formulas are general expressions for the nuclear stmctures of the blue-to-black sulfur dyes they do not take into consideration the quinonoid formation of each dye and other aspects. 

Erequenfly, the term compressed fluid, a more general expression than supercritical fluid, is used. A compressed fluid can be either a supercritical fluid, a near-critical fluid, an expanded Hquid, or a highly compressed gas, depending on temperature, pressure, and composition. 

Area. The SI unit of area is the square meter (m ). The hectare (ha) is a special name for the square hectometer (hm ). Large land or water areas are generally expressed in hectares or in square kilometers (km ). 

Chain transfer to monomer is the main reaction controlling molecular weight and molecular weight distribution. The chain-transfer constant to monomer, C, is the ratio of the rate coefficient for transfer to monomer to that of chain propagation. This constant has a value of 6.25 x lO " at 30°C and 2.38 x 10 at 70°C and a general expression of 5.78 30°C, chain transfer to monomer happens once in every 1600 monomer... 

As shown in Figure 10, all of the organics removed in the process are either oxidized to CO2 and H2O or synthesized to biomass generally expressed as volatile suspended soHds. As previously noted, a small portion of the organics removed results in SMP products. 

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