RRS equation

Cumulative data often are represented closely by the Rosin-Rammler-Sperling (RRS) equation... 

Figure 16.4. Several ways of recording the same data of crystal size distribution (CSD) (MulUn, 1972). (a) The data, (b) Cumulative wt % retained or passed, against sieve aperture, (c) Log-log plot according to the RRS equation P = exp[(—d/dm)"J off this plot, = 850, dm = 1000, n = 1.8. (d) Differential polygon, (e) Differential histogram.

The matter of rfi is of some importance to the estimation of solid interfacial tensions and has yet to be resolved. On one hand, it has been shown that the Giralfco and Good model carries the implication of rr being generally significant [10], and on the other hand, both Good [193] and Fowkes [145] propose equations that predict... 

In PPP-SCF calculations, we make the Bom-Oppenheimer, a-rr separation, and single-electron approximations just as we did in Huckel theor y (see section on approximate solutions in Chapter 6) but we take into account mutual electrostatic repulsion of n electrons, which was not done in Huckel theory. We write the modified Schroedinger equation in a form similar to Eq. 6.2.6... 

Iminoboianes have been suggested as intermediates in the formation of compounds derived from the pyrolysis of azidoboranes (77). The intermediate is presumed to be a boryl-substituted nitrene, RR BN, which then rearranges to the amino iminoborane, neither of which has been isolated (78). Another approach to the synthesis of amino iminoboranes involves the dehydrohalogenation of mono- and bis(amino)halobotanes as shown in equation 21. Bulky alkah-metal amides, MNR, have been utilized successfully as the strong base,, in such a reaction scheme. Use of hthium-/i /f-butyl(ttimethylsilyl)amide yields an amine, DH, which is relatively volatile (76,79). 

There are two work terms to consider when a nucleus forms from the liquid. Equations (6.1) and (6.2) show that work of the type AH (T, - T)/T, is available to help the nucleus form. If AH is expressed as the latent heat given out when unit volume of the solid forms, then the total available energy is (4/3)ot AH (T, - T)/T, . But this is offset by the work 4 rr ysL needed to create the solid-liquid interface around the crystal. The net work needed to form the crystal is then... 

Note that this equation has two energy terms that did not appear in eqn. (7.1). The first, Tir Cl - cos 0)Ycs> is the energy needed to create the new interface between the catalyst and the solid. The second, - rr (l - cos 0)Ycu is the energy released because the area of the catalyst-liquid interface is smaller after nucleation than it was before. 

RR C=NLi) . They can be prepared in high yield either by the addition of an organolithium compound across the triple bond of a nitrile (equation (1)) or by lithiation of a ketimine (equation (2)). 

This is the same high temperature limit predicted by the Einstein equation and is the limit approached by experimental results for monatomic solids.rr... 

Multiple regression on ar and parameters employs the dual substituent-parameter equation, which may be written as in equation 964. 

This subject has been associated with the development of the or and type scales almost from the start74 (see Section II.B), but the first paper in which sulfinyl and sulfonyl groups played a part appears to have been one by Taft and coworkers in 196367. The main object of this paper was to study the effect of solvent on the inductive order by 19F NMR measurements on a large number of mcta-substituted fluorobenzenes in a great variety of solvents. The relationship between the NMR shielding parameter and selected systems as equation 10 ... 

In the related paper on 19F screening parameters of para-substituted fluorobenzenes in relation to resonance effects, a few measurements for SOMe and S02Me were recorded but no use was made of them for calculation of years later, Sheppard and Taft113 used these data (carbon tetrachloride solution) to calculate nR values through equation 11 ... 

Enantiomerically pure /J-keto sulfoxides are prepared easily via condensation of a-lithiosulfinyl carbanions with esters. Reduction of the carbonyl group in such /J-keto sulfoxides leads to diastereomeric /J-hydroxysulfoxides. The major recent advance in this area has been the discovery that non-chelating hydride donors (e.g., diisobutylaluminium hydride, DIBAL) tend to form one /J-hydroxysulfoxide while chelating hydride donors [e.g., lithium aluminium hydride (LAH), or DIBAL in the presence of divalent zinc ions] tend to produce the diastereomeric /J-hydroxysulfoxide. The level of diastereoselectivity is often very high. For example, enantiomerically pure /J-ketosulfoxide 32 is reduced by LAH in diethyl ether to give mainly the (RR)-diastereomer whereas DIBAL produces exclusively the (.S R)-diastereomer (equation 30)53-69. A second example is shown in... 

Then the AH values of sulphoxides and amides were correlated with Taft s rr affording the following equation ... 

Wang JX, Springer TE, Adzic RR. 2006. Dual-pathway kinetic equation for the hydrogen oxidation reaction on Pt electrodes. J Electrochem Soc 153 A1732-A1740. 

Wang JX, Zhang JL, Adzic RR. 2007. Double-trap kinetic equation for the oxygen reduction reaction on Pt(lll) in acidic media. J Phys Chem A 111 12702-12710. 

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