Equation verification

While Eq. III-18 has been verified for small droplets, attempts to do so for liquids in capillaries (where Rm is negative and there should be a pressure reduction) have led to startling discrepancies. Potential problems include the presence of impurities leached from the capillary walls and allowance for the film of adsorbed vapor that should be present (see Chapter X). There is room for another real effect arising from structural peiturbations in the liquid induced by the vicinity of the solid capillary wall (see Chapter VI). Fisher and Israelachvili [19] review much of the literature on the verification of the Kelvin equation and report confirmatory measurements for liquid bridges between crossed mica cylinders. The situation is similar to that of the meniscus in a capillary since Rm is negative some of their results are shown in Fig. III-3. Studies in capillaries have been reviewed by Melrose [20] who concludes that the Kelvin equation is obeyed for radii at least down to 1 fim. 

Fig. in-12. Verification of the Gibbs equation by the radioactive trace method. Observed (o) and calculated (line) values for for aqueous sodium dodecyl sulfate solutions. (From Ref. 108.)... 

The extensive use of the Young equation (Eq. X-18) reflects its general acceptance. Curiously, however, the equation has never been verified experimentally since surface tensions of solids are rather difficult to measure. While Fowkes and Sawyer [140] claimed verification for liquids on a fluorocarbon polymer, it is not clear that their assumptions are valid. Nucleation studies indicate that the interfacial tension between a solid and its liquid is appreciable (see Section K-3) and may not be ignored. Indirect experimental tests involve comparing the variation of the contact angle with solute concentration with separate adsorption studies [173]. 

Equation (A2.1.21) includes, as a special case, the statement dS > 0 for adiabatic processes (for which Dq = 0) and, a fortiori, the same statement about processes that may occur in an isolated system (Dq = T)w = 0). If the universe is an isolated system (an assumption that, however plausible, is not yet subject to experimental verification), the first and second laws lead to the famous statement of Clausius The energy of the universe is constant the entropy of the universe tends always toward a maximum. ... 

Table III-55 shows the verification of the Taft equation. The result is analogous to that obtained for 3-alkylpyridines.

TABLE m-55. VERIFICATION OF THE TAFT EQUATION FOR THE KINETICS OF OUATERNIZATION OF 5-ALKYLTHIAZOLES (254)... 

The work of adhesion was determined from the a versus P measurements (see Eq. 11). The work of adhesion between two rubber spheres was found to be 71 4 mJ/m. The work of adhesion reduced to 6.8 0.4 mJ/m in the presence of 0.01 M solution of dodecyl sulfate. Using these measurements of adhesion between rubber in air and a surfactant solution, Johnson et al. [6] provided the first direct experimental verification of the Young s equation (Eq. 40). They also measured... 

The plotting of Dixon plot and its slope re-plot (see 5.9.5.9) is a commonly used graphical method for verification of kinetics mechanisms in a particular enzymatic reaction.9 The proposed kinetic mechanism for the system is valid if the experimental data fit the rate equation given by (5.9.4.4). In this attempt, different sets of experimental data for kinetic resolution of racemic ibuprofen ester by immobilised lipase in EMR were fitted into the rate equation of (5.7.5.6). The Dixon plot is presented in Figure 5.22. 

Verification of the molecular weight of thiirene dioxides by mass spectrometry, employing the conventional electron-impact (El) ionization method, has been unsuccessful due to the absence or insignificant intensity of molecular ion peaks in their mass spectra. The base peak is rather characteristic, however, and corresponds to the formation of the disubstituted acetylene ion by loss of sulfur dioxide91 (equation 3). 

We shall use the following values of N for ions with the structures indicated He, 5 Ne, 7 Ar, Cu+, 9 Kr, Ag+, 10 Xe, Au+, 12. Averages of these values for the two ions concerned, given in the last column of Table III, are seen to correspond satisfactorily with the experimental results. There is no experimental verification of the values of n assumed for the eighteen-shell ions. The radii calculated with these values of n from the univalent radii R by means of Equation 13 are included in Table II.13... 

Formal verification that this result actually satisfies Equation (14.13) is an exercise in partial differentiation, but a physical interpretation will confirm its validity. Consider a small group of molecules that are in the reactor at position z at time t. They entered the reactor at time i = t — (zju) and had initial composition a t, z) = ai (t ) = ai (t — z/u). Their composition has subsequently evolved according to batch reaction kinetics as indicated by the right-hand side of Equation (14.14). Molecules leaving the reactor at time t entered it at time t — t. Thus,... 

A more complete discussion of the kinetics of isocyanate crosslinking in the presence of humidity along with experimental verification of the rate equations used above has been given by van der Ven, et al. (18). Equations 38 and 39 can be integrated numerically for different values of isocyanate to hydroxy ratio and H. Plots of the consumption of hydroxy and isocyanate functionality are shown in Figure 2 for H-0 and H-0.66 assuming equal Initial Isocyanate and hydroxy levels. High humidity increases the rate of consumption of isocyanate and reduces the consumption of hydroxy. 

Porra, R.J., Thompson, W.A., and Kriedemann, P.E., Determination of accurate extinction coefficients and simultaneous equations for assaying chlorophylls a and b extracted with four different solvents verification of the concentration of chlorophyll standards by atomic absorption spectroscopy, Biochim. Biophys. Acta, 975,384, 1989. 

Fig 3.21. Rate of oxygen removal from a ZnO film (a.u.) as a function of s of water-dioxane mixtures (verification of the equation (3.29)). 

Figure 3.22 shows the result of verification of equations (3.28) and (3.29) under conditions of interaction of oxygen adsorbed on ZnO film with molecules of polar and chemically analogous solvents for the same concentration of dissolved oxygen. The values s of these solvents vary from 17 to 81. Experimental points are fairly satisfactorily plotted against the straight line IgB - i/s which is in consistency with the requirements of equations (3.28) and (3.29). 

Fig. 4.25. The results of verification of the equation (3.29) in the case of ZnO film immersed in various solvents in a hydrogen atmosphere at room temperature ( / - 4) or in vacuum at 250 C (5) 1 water (liquid is= 78) and saturated vapour) 2 - methanol ( = 36) 3 ethanol (liquid ie= 26) and saturated vapour) 4 - butanol (f = 17).

The basic principle of heat-flow calorimetry is certainly to be found in the linear equations of Onsager which relate the temperature or potential gradients across the thermoelements to the resulting flux of heat or electricity (16). Experimental verifications have been made (89-41) and they have shown that the Calvet microcalorimeter, for instance, behaves, within 0.2%, as a linear system at 25°C (41)-A. heat-flow calorimeter may be therefore considered as a transducer which produces the linear transformation of any function of time f(t), the input, i.e., the thermal phenomenon under investigation]] into another function of time ig(t), the response, i.e., the thermogram]. The problem is evidently to define the corresponding linear operator. 

Equation (64) predicts correctly the increase in solid entrainment into thejet with increases in jet velocity and the decrease with increases in solid loading in a two-phase jet. Since neither the voidage nor the particle velocity inside thejet were measured, direct verification of Eq. (64) was not performed. 

The electron-transfer paradigm in Scheme 1 (equation 8) is subject to direct experimental verification. Thus, the deliberate photoactivation of the preequilibrium EDA complex via irradiation of the charge-transfer absorption band (/ vCT) generates the ion-radical pair, in accord with Mulliken theory (equation 98). 

Having now demonstrated that a moving electron can be considered as a wave, it remained for an equation to be developed to incorporate this revolutionary idea. Such an equation was obtained and solved by Erwin Schrodinger in 1926 when he made use of the particle-wave duality ideas of de Broglie even before experimental verification had been made. We will describe this new branch of science, wave mechanics, in Chapter 2. 

---