Equivalent Equations

Equation X-17 was stated in qualitative form by Young in 1805 [30], and we will follow its designation as Young s equation. The equivalent equation, Eq. X-19, was stated in algebraic form by Dupre in 1869 [31], along with the definition of work of adhesion. An alternative designation for both equations, which are really the same, is that of the Young and Dupre equation (see Ref. 32 for an emphatic dissent). 

The equivalent equations for heterogeneous and quasi-heterogeneous systems (tire latter are small vesicles which can practically be handled as homogeneous systems, but which are nevertlieless large enough to possess a macroscopic solid-liquid interface) are dealt witli in section C2.14.7. 

The viscous or frictional loss term in the mechanical energy balance for most cases is obtained experimentally. For many common fittings found in piping systems, such as expansions, contrac tions, elbows and valves, data are available to estimate the losses. Substitution into the energy balance then allows calculation of pressure drop. A common error is to assume that pressure drop and frictional losses are equivalent. Equation (6-16) shows that in addition to fric tional losses, other factors such as shaft work and velocity or elevation change influence pressure drop. 

These two equations represent the assoeiative analogue of Eq. (14) for the partial one-partiele eavity funetion. It is eonvenient to use equivalent equations eontaining the inhomogeneous total pair eorrelation funetions. Similarly to the theory of inhomogeneous nonassoeiating fluids, this equiva-lenee is established by using the multidensity Ornstein-Zernike equation (68). Eq. (14) then reduees to [35]... 

The first derivative on the right-hand side of equation (2.53) must be zero because subsystems 1 and 2 have been defined to be independent of each other. Therefore, (50/5 ) in the second term of equation (2.53) must also equal zero in order for this equation to be true for all conditions. In a similar manner, starting with the equivalent equations involving the derivative of 0 /0, one can show that (50/5zi) = 0. If one substitutes this last result into equation (2.52), one gets (50 /5zi) = 0, and the conclusion that 0 is independent of ri. From a similar treatment, one can also show that (5 2/5c2) = 0, so that 02 is independent of r2. Thus, since the two ratios i/0 and 2/0 are independent of r neither 0, 0i, nor 0 can depend upon the c variables. 

In section 5.4.3 we have discussed the physical meaning and range of validity of the potential-work function equivalence equations of solid state electrochemistry ... 

The promotional kinetics described by equation (11.6) or by its equivalent equation (11.12) imply uniform distribution of the backspillover promoting species on the catalyst surface. This requires fast ion backspillover relative to its desorption or surface reaction. 

Dynamic mechanical measurements for elastomers that cover wide ranges of frequency and temperature are rather scarce. Payne and Scott [12] carried out extensive measurements of /a and /x" for unvulcanized natural mbber as a function of test frequency (Figure 1.8). He showed that the experimental relations at different temperatures could be superposed to yield master curves, as shown in Figure 1.9, using the WLF frequency-temperature equivalence, Equation 1.11. The same shift factors, log Ox. were used for both experimental quantities, /x and /x". Successful superposition in both cases confirms that the dependence of the viscoelastic properties of rubber on frequency and temperature arises from changes in the rate of Brownian motion of molecular segments with temperature. 

Traditionally, nutrient uptake from solution culture was taken to depend on the concentration of the external mineral nutrient, C , the amount of nutrientabsorbing surface, and the kinetics of uptake per unit surface area or unit length of root (22). The flux of nutrients into the roots, J, is described by one of two functionally equivalent equations. ... 

Here all components of the vector e,-, except the z-th one which is unity, are equal to zero. Because the molecular reaction (30) is induced by chemical interaction of groups Aj and A- its rate constant apparently equals k ia-a + a ja ). It is possible to write down an infinite set of kinetic equations corresponding to the scheme (30) for the concentrations C(l a f) of molecules of different composition and functionality which are present in the reaction system at the moment t. To solve these equations it is convenient to go over to the equivalent equation ... 

This equation says that the steepest path will be followed if the temperature is increased 1°K while the pressure is increased 14.7 atmospheres or 216 psia. If the pressure had been measured in mm Hg, the equivalent equation would have indicated that the pressure should increase 0.000374 psi for each degree Kelvin the temperature is increased. Figure 14-2 shows the results. 

Watch out In many physics textbooks on EPR (e.g., Abragam and Bleaney 1970), you will not hnd Equation 5.12 (or its equivalent Equation 5.13) but a different expression, namely ... 

Several important assumptions have been implicitly incorporated in Eqs. (15) and (16). First, these equations describe the release of a drug from a carrier of a thin planar geometry, equivalent equations for release from thick slabs, cylinders, and spheres have been derived (Crank and Park, 1968). It should also be emphasized that in the above written form of Fick s law, the diffusion coefficient is assumed to be independent of concentration. This assumption, while not conceptually correct, has been... 

The first reaction is typical of a basic metal oxide, whereas in the second Zn2+ is reacting as an acid. When written starting with the hydroxide, the equivalent equations are... 

Today, the Suzuki cross coupling of aryl halides and arylboronic acids is also carried out in aqueous-biphasic operation starting from chlorinated derivatives instead of their more costly bromo or iodo equivalents (Equation 5.6, [39]). 

The sulfamate ester variant of this chemistry has already been shown to be a very powerful protocol for the syntheses of 1,3-amino alcohols and related /3-amino acids (Equation (90)), as well as iminium ion equivalents (Equation (91)). The further showcases of this chemistry are the total syntheses of the bromopyrrole alkaloids, manzacidins A and C (Scheme 13).234 The cyclic sulfamidate 129 was obtained diastereospecifically from sulfamate 128 using intramolecular rhodium-catalyzed G-H insertion. It was then found to react with sodium azide in NfN-dimethylformamide at room temperature after introduction of the Boc-activating group to afford the 1,3-diamino precursor 130 in 78% yield over 3 steps. Four subsequent manipulations afford the target structure 131. 

Thus, the equivalent weight of sodium carbonate may be equal to either the formula weight divided by 1 (105.99 g per equivalent, Equation (4.6)) or the formula weight divided by 2 (52.995 g per equivalent, Equation (4.7)), depending on which reaction is involved. 

The equivalent equation for movement normal to a cylinder, such as a plant root, is... 

S.4.3.2 Model of Hillert and Jarl. In his original treatment, Inden (1976) used a complicated but closed expression for the enthalpy, but had to use a series expansion in order to calculate the entropy. Hillert and Jarl (1978) therefore decided to convert the Cp expression directly through a series expansion which substantially simplifies the overall calculation and leads to a maximum error of only 1-2 J/mol at the Curie temperature of Fe. The equivalent equations to those used by Inden (1976) are given by... 

An equivalent equation to Eq. (19) in terms of the G-matrix was first published in 1969 by Garrod and Rosina [39] and also later reported by Valdemoro et al. [35]. 

The equivalent equation in general gauge field theory is... 

Similarly, an equivalent equation for flop-down molecules would permit us to calculate N JM. Thus, from these two quantities, N and N, the variable Z becomes [see Eq.(6.150)],... 

It is convenient to express this on a per-mole basis, dividing through by the total number of moles (nA + nB) to obtain the equivalent equation in terms of mole fraction xB... 

Several significant pyrrole syntheses involve the formal tricomponent cyclization of type III ace (equation 126). The Hantzsch pyrrole synthesis involves a dicarbonyl compound, an a -halo ketone and ammonia or an amine. The mechanistic pattern is similar to that involved in the Knorr synthesis (Section 3.06.3.4.1). In addition to a-halo ketones and a-haloal-dehydes, compounds such as 1,2-dichloroethyl acetate, 1,2-dibromoethyl acetate and 1,2-dichloroethyl ethyl ether can serve as a -haloaldehyde equivalents (equation 127) (70CJC1689, 70JCS(C)285>. It is believed that the initial step in these reactions is the formation of a stabilized enamine from the amine and the /3 -dicarbonyl compound. A structural ambiguity... 

In formulating Eqn. (5.101) and the following flux equations we tacitly assumed that they suffer no restrictions and so lead to the individual chemical diffusion coefficients >(/). If we wish to write equivalent, equations for,/(A) and/(B), and allow that v(A) = = v(B), then according to Eqn. (5.103), /(A) /(B) since Ve(A) = ]Vc(B)j. However, the conservation of lattice sites requires that j/(A) j = /(B), which contradicts the previous statement. We conclude that in addition to the coupling of the individual fluxes, defect fluxes and point defect relaxation must not only also be considered but are the key problems in the context of chemical diffusion. Let us therefore reconsider in more detail the Kirkendall effect which was introduced qualitatively in Section 5.3.1. It was already mentioned that this effect played a prominent role in understanding diffusion in crystals [A. Smigelskas, E. Kirkendall (1947) L.S. Darken (1948)]. 

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