Valentiner equation

Forveiy thin hquids, Eqs. (14-206) and (14-207) are expected to be vahd up to a gas-flow Reynolds number of 200 (Valentin, op. cit., p. 8). For liquid viscosities up to 100 cP, Datta, Napier, and Newitt [Trans. In.st. Chem. Eng., 28, 14 (1950)] and Siems and Kauffman [Chem. Eng. Sci, 5, 127 (1956)] have shown that liquid viscosity has veiy little effec t on the bubble volume, but Davidson and Schuler [Trans. Instn. Chem. Eng., 38, 144 (I960)] and Krishnamiirthi et al. [Ind. Eng. Chem. Fundam., 7, 549 (1968)] have shown that bubble size increases considerably over that predic ted by Eq. (14-206) for hquid viscosities above 1000 cP. In fac t, Davidson et al. (op. cit.) found that their data agreed veiy well with a theoretical equation obtained by equating the buoyant force to drag based on Stokes law and the velocity of the bubble equator at break-off ... 

In 1924, John Alfred Valentine Butler derived an equation for the equilibrium potential of an electrode using the equations for the finite rate of anodic and cathodic steps. 

Butler, John Alfred Valentine — (Feb. 14,1899, Winch-combe, Gloucestershire, England - July 16,1977). Butler greatly contributed to theoretical electrochemistry, particularly, to connection of electrochemical kinetics and thermodynamics [i,ii]. The famous Butler-Volmer equation (1924) showing the exponential relation between current and potential was named after him (and... 

In equations (40) and (41), Xg " is the mole fraction solubility of the solid solute at an absolute temperature T, a is the apparent molar enthalpy of solution, b is the apparent molar entropy of solution, and c and c" are constants. The combined equation, attributed to Valentiner, has been used by Grant et al. (1984) in the... 

Enamines (e.g. 377) have been shown to react with conjugated nitroolefins 378 to give mainly dihydro-l,2-oxazine A -oxide derivatives 379 as products of kinetic control (sometimes a cyclobutane ring is formed in these reactions see Section II.B). The stability of these heterocycles is largely dependent on the parent enamine and the type of substituent used on the nitro olefin as has been extensively studied by Valentin and coworkers " . Usually they open into the corresponding nitroalkylated enamines 380 (equation 82), in particular in a solution of methanol or deuteriated chloro-and often an equilibrium between the two forms is established. Stable 1,2-oxazine A -oxides have been obtained in the reaction of 2-nitro-l,3-dienes with cyclic enamines . 

As described above, if this electrochemical reaction is driven out of equilibrium with the thermodynamic force ry, a current / will flow. One of the great successes of electrochemistry is its ability to provide a quantitative description of the charge transfer current characteristics. Max Volmer (1885—1965) and John Alfred Valentine Butler (1899—1977) proposed the following relation, which is known today as the Butler—Volmer equation, between the over potential 7j and the current density j (Fig. 3.4) ... 

The name of this gauge refers to the Dutch physicist Hendrik Antoon Lorentz (1853-1928), whereas the gauge was really introduced by the Danish physicist Ludvig Valentin Lorenz who not only introduced this gauge as well as their retarded solutions (91), but also independently developed Maxwell s equations [39]. 

A recent report by Valentin and coworkers191 has shown that the reaction of methanesulfonyl chloride with 4-tert-butylcyclohexanone enamines in the presence of the triethylamine gave a 1 1 mixture of two diastereomers (equation 72), indicating a complete lack of stereoselectivity. 

Temperature dependence equations for pK and pK values show the importance of good temperature control during pH meter calibration as well as in measurements. To a greater or lesser extent, all acid-base reactions vary with temperature. The effect of temperature changes for equilibrium reactions are closely described by the Valentiner-Lannung equation [80-82] ... 

The synthesis of a ferric-peroxo porphyrin eomplex, [(Porp)Fe (02)], was first reported by Valentine and co-workers. The ferrie-peroxo speeies was prepared by reacting Fe(TPP)Cl or Fe(OEP)Cl with two equivalents of superoxide in the presenee of erown ether in aprotie solvents sueh as aeetonitrile or toluene. The first equivalent of superoxide reduees the ferric porphyrin to the ferrous porphyrin, and the second superoxide eonverts the ferrous porphyrin to the ferrie-r/ -peroxo porphyrin complex (Equation (14)). The EPR speetrum of [(0EP)Fe (02)] eomplex... 

Despite detailed and repeated measurements by various techniques, almost no evidence can be found for the formation of (LFe=0) in the cyclidene systems, despite the fact that peroxo complexes form readily and by one route that constitutes the reverse of the reaction of equation 6. The results reported here also stand in contrast to recent studies based on non-heme oxygenase model systems that indicate the possibility of multiple pathways, implying that (hydroperoxo)iron(III) species might have catalytic activity e.g., a) Y.-D. Wu, K. N. Houk, J. S. Valentine, and W. Nam, Inorg, Chem, 31 718 (1992) b) W. Nam, R. Ho, and J. S. Valentine, J, Am,... 

Water vapor at room temperature will not penetrate well-defined crystallites but will be adsorbed in the amorphous regions. Consequently, moisture sorption measured gravimetrically at a given relative vapor pressure and temperature has been used to determine order in cellulosic materials. In the case of Valentine [252] and Jeffries [253], the fraction of ordered material was obtained by correlating moisture sorption with values obtained by the deuterium oxide method. Hailwood and Horrobin [254] developed an equation for water sorption of cellulose based on a solution theory that allowed the calculation of the fraction of the sample inaccessible to water. 

Lorentz-Lorenz equation A relation between the polarizability a of a molecule and the refractive index n of a substance made up of molecules with this polarizability. The Lorentz-Lorenz equation can be written in theforma= (3/4tcA/) [(n -l)/(n + 2)l, where JV is the number of molecules per unit volume. The equation provides a link between a microscopic quantity (the polarizability) and a macroscopic quantity (the refractive index). It was derived using macroscopic electrostatics in 1880 by Hendrik Lorentz and independently by the Danish physicist Ludwig Valentin Lorenz also in 1880. Compare ClAUSIUS-MoSSOTTr EQUATION. 

FIG. 13-1. Plots of J" ft against co/(Om fof four polymers black circles, 1,2-polybutadiene open circles, 1,4-polybutadiene, cis trans - 43 SO half black, styrene-butadiene copolymer crossed circles, polyisobutylene. 7 from equation 4, except for 1,4-polybutadiene (chosen to match the other curves) ti) from position of maximum on frequency scale. Solid curve is curve D of Fig. 10-7, with linear ordinate (Mooney modification of Rouse theory with most probable distribution of strand lengths). (Sanders, Valentine, and Ferry. )... 

The shear flowability index, n, was found, from past observations (Farley Valentin 1965, 67/68), to be independent of the bulk density of sheared compacted powder. Because of this independence of particle size from bulk density it is now realised that the shear flowability index, n, from the Warren Spring equation and the Jenike internal angle of friction may be the preferred parameters to eharacterise and quantify the flowability of powders. Jenike and others (Williams et al. 1970/71 Williams Birks 1965 Hill Wu 1996 Cox Hill 2004) selected the Jenike failure function to be one of the best indicators to predict the ease of powder movement and powder flowability. 

John Alfred Valentine Butler (1899-1977) was an English physical chemist, who greatly contributed to theoretical electrochemistry. Particularly, he contributed to developing a relationship between electrochemical kinetics and thermodynamics. He is best known for his contribution to the development of the famous Butler-Volmer equation. 

---