Jorgensen equation

The electronic transitions of silicalite and TS-1 in the UV-visible spectrum have provided significant information about the structure of TS-1. The diffuse reflectance spectra of the two materials (Fig. 11) show a strong transition at 48,000 cm-1 that is present in the spectrum of TS-1 and absent from that of silicalite. This transition must be associated with a charge-transfer process localized on Tiiv. The frequency of this transition is modified by the presence of H20 (Fig. 12). As the H20 partial pressure increases, the peak at 48,000 cm- is progressively eroded with formation of a lower-frequency absorption, which reaches a new stable maximum value at 42,000 cm. These frequencies come very close to those that can be calculated by the Jorgensen equation for Tiiv tetrahedrally and octahedrally coordinated to oxygen, respectively. Furthermore,... 

This empirical equation (when corrected for the effects of inter-electronic repulsion and for differences in the ligand field splitting) allows one to predict the frequencies of charge transfer bands for a great variety of complexes . Remarkable information on redox properties can be obtained from the parameters of the Jorgensen equation because they are roughly proportional to valence state ionization potentials of the metal and to the ionization potentials of the ligands. 

Aldehydes can be converted into optically active a-chloroaldehydes through the intermediacy of chiral enamines, as reported independently in work by MacMillan and Jorgensen (Equations 22 and 23, respectively) [133-135]. MacMillan developed a procedure using hexachloride 242 [130] and the chiral imidazolidinone catalyst 243 (Equation 22) [133]. Under the mild conditions described, chlorination of aldehyde 241 proceeded to give 244 in 94 % yield and 93 % ee. 

Upon doing so, the following set of equations is obtained (early referenees to the derivation of sueh equations inelude A. C. Wahl, J. Chem. Phys. 41,2600 (1964) and F. Grein and T. C. Chang, Chem. Phys. Lett. 12, 44 (1971) a more reeent overview is presented in R. Shepard, p 63, in Adv. in Chem. Phys. LXIX, K. P. Lawley, Ed., Wiley-Interseienee, New York (1987) the subjeet is also treated in the textbook Seeond Quantization Based Methods in Quantum Chemistry, P. Jorgensen and J. Simons, Aeademie Press, New York (1981))) ... 

Table 18.8 Selected Values for the fand g Parameters for Use in Jorgensen s Equation. ...

Jorgensen, W. L. 2000. Perspective on Equation of State Calculations by Fast Computing Machines Theor. Chem. Acc., 103, 225. 

In their analysis, which will form the basis of what follows here, Jorgensen and Aris chose to vary the Newtonian cooling time, keeping the residence time constant during any given experiment. Thus we may use tres as the timescale with which to make the rate equations dimensionless. The resulting forms, with the above simplifications, are... 

Calculations by Jorgensen and co-workers28 on primary systems assessed the size of the /1-effect by calculation of the energy change for the isodesmic reaction shown in equation 16. 

Molecular dynamics has been used to simulate water structures, wherein an accurate water potential function is used to enable solution of Newton s equations of motion for a small (e.g., 1000-10,000) number of molecules over time. In water and water structures, the SPC (Berendsen et al 1981) and the TIP4P (Jorgensen et al 1983) potential models are most often used. Reanalysis of extant diffraction data by Soper et al. (1997) has called both of these potentials into question. 

A->B- C has also been considered by Jorgensen (65). This last is a system of three equations for u and v, the concentrations of A and B, and w, the temperature, with seven parameters a, the Damkohler number for A->B 3, its dimensionless heat of reaction y, its Arrhenius number k, a dimensionless heat transfer coefficient v, the ratio of activation energies of the two reactions p, the ratio of the heats of reaction a, the ratio of the Damkohler numbers. They contain a characteristic non-linearity... 

This measure was based upon the ratio of the minimum necessary number of plates, A min (averaged over the reboiler composition) in a column to the actual number of plates in the given column, Nj. Christensen and Jorgensen assumed that the mixture has a constant relative volatility a and the column operates at total reflux using constant distillate composition (x o) strategy (section 3.3.2) and evaluated Nmin using the Fenske equation ... 

The necessity of the above discussion will now be realised (also see Christensen and Jorgensen, 1987). Figure 8.2 shows a quasi-steady state mode of operation, with off-cut recycle. A fresh charge BO, of composition xB0 is mixed with the off cut (Rl, xRi) from the previous batch to produce a mixed charge to the reboiler (BC, xBc) The main cut (Dl, x D]) is produced over the time // (Task 1), leaving a residue (B1, xbj). At this time the distillate is simply diverted to a second receiver, and further distillation in Task 2 for time t2 produces the off cut and the final bottom product (B2. x B2) where, B2 is the solution of Equations (8.1-8.4) as mentioned before. 

Tanabe and Sugano first calculated the required energy (44) matrices which have since been repeated by other authors (17, 34)- Simplified equations have been presented by Jorgensen (24). These latter equations are not always exact solutions to the Tanabe-Sugano secular determinants because the assumption is made that C = 4R to aid simplification. 

---