Original Derivation

We show here the derivation of the phase rule as done by Gibbs in his original paper [3, pp. 55-353]. In a multicomponent system for a homogeneous phase the 

We emphasize here that a prerequisite for the validity of the phase rule is that the thermodynamic system can be described fully by Eq. (7.1). This means in detail that entropy, volume, and mol numbers are sufficient as independent variables. The components should be independent as such, which implies no chemical reactions or other transformations. It is interesting to note that the occurrence of phases implies the existence of phase borders, which contribute to surface energy. This energy form is neglected. In the literature, explicitly situations are discussed, where the phase rule fails [4-6], 

If there are K components, then there are AT + 2 variables, i.e., all the fit and temperature T, and pressure p. However, Eq. (7.1) restricts the free variation of these variables to A + 1. So we have AT + 1 independent variables. For each phase there is one equation like Eq. (7.1), thus for P phases there are totally P K + l)-independent variables. The temperatures and the pressures of the individual phases must be equal in equilibrium, so we have in addition P - 1 equations relating the temperatures and P -1 equations relating the pressures. Also the chemical potentials must be equal in equilibrium, so for K components we have another K(P — 1) equations. Totally we have (P — l)(Af -I- 2) equations. The number of freedoms is the difference between variables and equations  

We emphasize once more that the foregoing arguments are based entirely on intensive variables. In other words, we do not worry about the size of the phases, they 

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A selection of important anionic surfactants is displayed in table C2.3.1. Carboxylic acid salts or tire soaps are tire best known anionic surfactants. These materials were originally derived from animal fats by saponification. The ionized carboxyl group provides tire anionic charge. Examples witlr hydrocarbon chains of fewer tlran ten carbon atoms are too soluble and tliose witlr chains longer tlran 20 carbon atoms are too insoluble to be useful in aqueous applications. They may be prepared witlr cations otlrer tlran sodium. 

This procedure is then repeated after each time step. Comparison with Eq. (2) shows that the result is the velocity Verlet integrator and we have thus derived it from a split-operator technique which is not the way that it was originally derived. A simple interchange of the Ly and L2 operators yields an entirely equivalent integrator. 

Values of the compound speciBc constants m and c were originally derived by Othmer et al. and greatly expanded to over 600 common organics by Danner and Daiibert. If constants are not available but any two vapor pressure data points are available, the constants m and C can be calculated using Eqs. (2-46) and (2-47). 

A kinetic model originally derived by Nyholm is distinguished from Monod s model by the fate of a hmiting substrate. Instead of immediate metabolism, the substrate in Nyholm s model is sequestered. The governing equations are ... 

The majority of published °Pb reports address the utility of °Pb as a geochronological tool rather than as an element that is involved in complex biogeochemical cycles. Nonetheless, some of these studies do provide insight into the geochemical behavior of °Pb and °Po. Nearly all of the lead in the world s surface oceans is believed to be of anthropogenic origin—derived from combustion... 

This approximation has been originally derived and extensively explored in the path-integral techniques (see the review [Leggett et al. 1987]). Most of the results cited in section 2.3 can be obtained from (5.62) and (5.63). Equation (5.62) makes it obvious that only when the integrand/(r) falls off sufficiently fast, can the rate constant be defined, and it equals... 

Carra and Forni (1974) derived the criteria that Carberry (1976) referred to in his book. These are equivalent to the original derivation of Aris and Amundson (1958). The notation is easier to understand and closer to the notation in this book. Eliminating some typographical errors, the criteria are ... 

Guinier s equation [19] was originally derived for a material with randomly positioned identical pores embedded in an uniform background. The intensity of small angle scattering, I(k), from pores with volume V, is... 

The sphere radii were deduced from Slater s (1965) table based on crystal data. The basic molecular HF-Xa equations were originally derived on the basis that the spheres did not overlap (Schwarz and Connolly, 1971). But the equations remain valid when the spheres are allowed to overlap, provided that each sphere does not contain more than one nucleus and that none of the nuclei lie outside the outer sphere. A 10% overlap seems to be normal practice, and our results are given in Table 12.2. 

Another commonly used method is Quadratic CISD (QCISD). It was originally derived from CISD by including enough higher-order terms to make it size extensive. 

Clausius-Clapeyron Equation. This equation was originally derived to describe the vaporization process of a pure liquid, but it can be also applied to other two-phase transitions of a pure substance. The Clausius-Clapeyron equation relates the variation of vapor pressure (P ) with absolute temperature (T) to the molar latent heat of vaporization, i.e., the thermal energy required to vajxirize one mole of the pure liquid ... 

The term drill collar originally derives from the short sub originally used to connect the bit to the drill pipe. Modern drill collars are each about 30 ft in length, and the total length of the string of drill collars may range from about 100 to 700 ft and more. 

Some potable supplies are treated surface waters from rivers, etc. These originally derive from any of the above, but will also contain the products of human activities, which lead to increased mineral contents and possibly some undesirable materials such as detergents. 

The temperature derivatives of G and A equations (3.35) to (3.38) can be seen to contain S. Because we usually do not know 5 as a function of p, V, and T, these derivatives are of limited value. However, expressions can be derived for the temperature derivatives of G/Tand A/T that do not involve 5, and they can often be used in applications where the original derivative was desired. We derive the equation for (d G/T)/OT)p to illustrate the process. 

What Eqs. (10-70) and (10-71) state is this One can predict the ET rate constant, given the other parameters. Or, a missing EE rate constant can be obtained by determining if the other factors are known. We shall return to this point shortly, after considering how to obtain these equations. The original derivation from statistical mechanics is outside the scope of this book, so we shall consider two others. 

The dyes on the group above are known as azo dyes, because they have the two nitrogen atoms together in the center (called an azo group). These dyes were originally derived from coal tar, but are now mostly made from petroleum. Azo dyes come in many colors besides those allowed in food (listed above). The other colors find use in fabrics, paper products, and plastics. 

This equation is always valid at steady state and has been originally derived for the case of NEMCA in alkaline solutions.37 In the present case of the acidic environment in the Nafion electrolyte the following two electrocatalytic (net charge transfer) reactions take place at the Pt working electrode, presumably at the three-phase-boundaries (tpb) Pt-Nafion-gas ... 

Marcus theory can be applied to any single-step process where something is transferred from one particle to another. It was originally derived for electron... 

Equation (24) is originally derived for a conical indenter. Pharr et al. showed that Eq (24) holds equally well to any indenter, which can be described as a body of revolution of a smooth function [67]. Equation (24) also works well for many important indenter geometries, which cannot be described as bodies of revolution. 

The above results for the three PID coefficients were originally derived by MacGregor ( ). The results of Prof. MacGregor s derivation have been published in reference (6). 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

The exchange part, ex, which represents the exchange energy of an electron in a uniform electron gas of a particular density is, apart from the pre-factor, equal to the form found by Slater in his approximation of the Hartree-Fock exchange (Section 3.3) and was originally derived by Bloch and Dirac in the late 1920 s ... 

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