Formal shortness ratio

In order to compare Cr—Cr bond lengths with other bond lengths a formal shortness ratio (FSR) for a bond A—B has been defined ... 

The unique ligating behavior of the bridging 2,6-dimethoxyphcnyl ligand with respect to promoting a substantial decrease in the metal atom separation for molybdenum(II) dimers is even more prominent in the case of chromium. The chromium-chromium distance of 1.847(1) A in Cr2(DMP)4 (90) is more than 0.1 A less than the corresponding value in any other chromous dimer yet reported. To compare homonuclear multiple bonds among elements with inherently different atomic radii, Cotton, Koch, and Millar proposed a normalized value for intemuclear distances based on Pauling s atomic radius of the element in question (209). A simple definition of formal shortness as t/(M—M)/2r(M) then follows as a measure of the relative compactness of the attractive interaction (90). The formal shortness ratio of 0.778 for the quadruple bond in... 

Cr2(DMP)4 not only falls below comparable ratios for other multiply bound metal dimers but also is slightly less than the formal shortness ratio calculated for the strongest homonuclear bonds known, for example, N2 (0.786), C2 (0.783), and P2 (0.860). The utility of the formal shortness concept as a guide to the assessment of homonuclear attractive forces makes it a valuable addition to comparative chemistry of elements forming such bonds. 

Metal-metal multiple bonding can have dramatic effects on bond distances, as measured by X-ray crystallography. One way of describing the shortening of interatomic distances by multiple bonds is by comparing the bond distances in multiple bonds to the distances for single bonds. The ratios of these distances is sometimes called the formal shortness ratio. Values of this ratio are compared below for main group triple bonds and for some of the shortest of the measured transition metal quadruple bonds ... 

Dong et al. [35] analyzed the multiple bimetallic bond of Cr-Cr, V-Mn, V-Tc, Nb-Mn, Cr-Mo, Mo-Mo, and Nb-Tc using CASPT2. They determined the bimetallic bond length, NBO, EBO, formal shortness ratio (FSR) values, and Mulliken charge populations for these systems. The V-Mn molecule presents the shortest bond length value and the largest FSR value in the series. 

The length of an M-M multiple bond is judged by the formal shortness ratio, or FSR, defined as the M-M distance divided by the sum of the appropriate Pauling atomic radii. [Re2Cl8] has an FSR of 0.87, for... 

The pharmacoeconomics of the anxiety disorders has received litde attention. In the past drug costs were largely incurred by use of benzodiazepines, most of which are available in generic forms and are cheap. They are effective and acceptable in the short term. Long-term use is associated with the risk of physical dependence, with an adverse risk—benefit ratio and high cost terms to facilitate withdrawal. There is now a trend towards the use of antidepressants in the anxiety disorders. Clinical experience has been followed by formal trial evaluation. 

The formal criss-cross cycloaddition product of cyanogen with sulfur trioxide (molar ratio 1 2) has been shown to be l,2,3-oxathiazolo[5,4-d][l,2,3]oxathiazole 2,2,5,5-tetroxide (5). The S—O bond in the ring is extremely long at 1.71 A, while the exocyclic S=0 bonds are shortened to 1.39 A, suggesting a description of the molecule in terms of bond-nonbond resonance, the resonance formulae (5a) and (5b) also providing some rationalization for the short C—O bond (1.29 A) (79AG(E)223). 

In short, the time has undoubtedly come to present recent development in the field of carbon rich compounds compiled in the form of several up-to-date volumes with articles written by active practitioners of these arts. The term carbon rich is used here to refer to everything that has a carbon to hydrogen ratio of 1 (< 1), but it can also include formally saturated hydrocarbons of the general formula CnHn which frequently can serve as precursors to compounds with a higher C H ratio. 

In the following ptu-agraphs this approach will be applied to production of an alkaloid from plant cells cultured in a bioreactor. Regime analysis can be performed by comparison of characteristic parameters of the mechanisms involved in the process. Here the characteristic time concept will be used. The characteristic time is a measure for the rate of a mechanism. A fast mechanism has a short characteristic time. Other terms used are relaxation time, process time, or time constant. A time constant is formally only defined for first-order linear processes. Not all mechanisms involved in a plant cell production process are first order, therefore the term characteristic time is used. The characteristic time is defined as the ratio of a capacity and a flow for example, the characteristic time for oxygen transfer to the liquid phase in a aerated bioreactor q.l becomes... 

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