King equation

Liu and Ruckenstein [17] presented a semiempirical equation to estimate diffusivities under supercritical conditions that is based on the Stokes-Einstein relation and the long-range correlation, respectively. The parameter 20 was estimated from the Peng-Robinson equation of state. In addition, f = 2.72 — 0.3445 TcB/TcA for most solutes, but for C5 through C14 linear alkanes,/ = 3.046 — 0.786 l cBl l cA. In both cases Ta is the species critical temperature. They compared their estimates to 33 pairs, with a total of598 data points, and achieved lower deviations (5.7 percent) than the Sun-Chen correlation (13.3 percent) and the Catchpole-King equation (11.0 percent). 

The Underwood roots, 9a and 0b, are found from the feed equation (King, 1980). The vapour flowrate in the second column is calculated from Kings equation (King, 1980) for binary mixtures. Note that the effect of pressure on the separation is not included in these formulas. 

The useful fission products can be confuted from the Perkins-King " equations using the ACTICAy 2 program for the IBM-TO9O. 

Many different manipulations of these equations have been used to obtain solutions. As discussed by King (1971), many of the older approaches work in terms of V/L, which has the disadvantage of being unbounded and which, in the classical implementation, leads to poorly convergent iterative calculations. A preferable arrangement of this equation system for solution is based on the ratio V/F, which must lie between 0 and 1. If we substitute in Equation (7-1) for L from Equation (7-2) and for y from Equation (7-4), and then divide by F, we obtain... 

In the work of King, Dupuis, and Rys [15,16], the mabix elements of the Coulomb interaction term in Gaussian basis set were evaluated by solving the differential equations satisfied by these matrix elements. Thus, the Coulomb matrix elements are expressed in the form of the Rys polynomials. The potential problem of this method is that to obtain the mabix elements of the higher derivatives of Coulomb interactions, we need to solve more complicated differential equations numerically. Great effort has to be taken to ensure that the differential equation solver can solve such differential equations stably, and to... 

One of Che earliest examples of a properly conceived experimental investigation of the flux relations for a porous medium is provided by the work of Gunn and King [53] on the dusty gas model equations, and the following discussion is based largely on their work. Since all their experiments were performed on binary mixtures, the appropriate flux relations are (5.26) and (5,27). Writing... 

In summary, a combination of the plot based on equation (10.6), using any single substance, and determination of the asymptote (10.14), using any pair of substances, provides a sound means of evaluating the parameters K, tC and. Having found these, further experimental points on (10.6) and (10.15), and possibly also (10.7), provide a check on the adequacy of the dusty gas model. Provided attention is limited to binary mixtures, this check can be quite comprehensive. In their published paper Gunn and King... 

The value of the integration constant is determined by the magnitude of the displacement from the equilibrium position at zero time. King also gives a solution for Scheme IV, and Pladziewicz et al. show how these equations can be used with a measured instrumental signal to estimate the rate constants by means of nonlinear regression. 

In a special circumstance the rate equation for parallel reactions may be misleading.If two parallel reactions are catalyzed by a common catalyst, and if a significant fraction of the catalyst is tied up in the form of intermediates, then the two reactions are not independent, and the rate equation will not give the transition state composition. King has analyzed this case in terms of enzyme-catalyzed reactions. 

King has given another and more convenient approach.3 His method is useful not only in jump experiments but also if the reaction starts all on one side. Let us start with the system A +3 2P, imagining that the perturbation causes an instantaneous shift in [A] of So- With proper regard for stoichiometry and signs, at any point during the reaction the shift is 8, defined so as to be positive. These are the equations ... 

King and Harding296 have reported an interesting Sulfo-Cope rearrangement, and have presented evidence for the formation of the unstable sulfene (142) during the gas or liquid phase thermal [3,3]sigmatropic rearrangement of allyl vinyl sulfone (equation 88). 

This appendix illustrates the steps involved in deriving the reaction rate equation (Equation 17.11) from the reaction scheme given in Section 17.3.2 using the King and Altman method.41 This... 

For obtaining the value of the rate constant, it is desirable to determine the value of the diffusion coefficient of the metal ions or of one of the reactants (in the case of a redox couple) in the supporting electrolyte at an appropriate temperature. The value of the diffusion coefficient is experimentally determined using a McBain-Dowson cell and the King-Cathard equation, as described earlier.40... 

The diffusion coefficient of the depolarizer in any supporting electrolyte is determined using a McBain-Dowson cell and the King-Cathard equation. 

Formation of a bis-allylated product of 4-nitrobenzoyl chloride by the reaction with allyltrimethyltin in the presence of a benzylpalladium(ll) complex was observed by Stille and co-workers in 1983.4 Trost and King also reported allylation of aldehydes by allyltin reagents in 1990.456 However, the precise mechanism was unclear until the extended studies were performed by Yamamoto and co-workers since 1995.4S7,4S7a 4S7 Aldehydes and imines react with allyltin reagents in the presence of a palladium catalyst (Equations (95) and (96)), and imines are chemoselectively allylated in the presence of aldehydes (Equation (97)).4S7,4S7l 4 b Mechanistic studies using NMR spectroscopy proved that bis-7t-allylpalladium complex 203 is a key nucleophilic intermediate (Figure 3). 

XRD spectra were compared to JCPDS file Calculated using Scherrer equation (King and Alexander, 1974). 

The King and Altman Method. King and Altman developed a systematic approach for deriving steady-state rate equations, which has contributed to the advance of enzyme kinetics. The first step of this method is to draw an enclosed geometric figure with each enzyme form as one of the corners. Equation (5), for instance, can be rewritten as ... 

The Systematic Approach. The systematic approach for deriving rate equations was first devised by Fromm based on certain concepts advanced by Volkenstein and Goldstein. Its underlying principles, however, are more akin to the graphic method of King and Altman. The procedure to be described here is a modified method that includes the contributions from the aforementioned workers and from Wong and Hanes. ... 

Note. Noble gas radius from Zhang and Xu (1995). Molecular diffusivity from Jahne et al. (1987) except for Ar (Cussler, 1997). For SFe, the radius is based on S-F bond length of 1.56 A plus the radius of F- (1.33 A), and the diffusivity is from King and Saltzman (1995). The jumping distance is calculated from Equation 3-136e using pure water viscosity of 0.89 mPa s at 25°C. 

The overall rate equation of complex single-route reaction with the linear detailed mechanism was derived and analyzed in detail by many researchers. King and Altman (1956) derived the overall reaction rate equation for single-route enzyme reaction with an arbitrary number of intermediates... 

In this final section 1 shall attempt to identify what 1 regard as the central equations of physical chemistry, the equations that everyone should remember and appreciate their implications (they are summarized in Table 2). I will identify a handful of equations from each major region of physical chemistry, and finally identify what I regard as the king of all equations. 

King (8) points out that the salt effect has been successfully correlated by Set-schenow s equation in which... 

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