General Formulas

1 General Formula We consider in this subsection the dynamic structure factor of a bead-spring model. For now we do not distinguish the three cases of the model. 

Because different modes are uncorrelated, the average of exp[ik-r (t)] for a given m and n is 

The last factor for i = N was incorporated into the product. The first factor represents the center-of-mass diffusion  

The dynamic structure factor of the chain is the average of the above equation with respect of m and n  

The long-time behavior is simple. At t Tj, 1 — exp(-t/Ti) = 1, and therefore the time-dependent factor is expC-Dcl O only. Then, ilk, t) decays with a decay rate of Dok indicating center-of-mass diffusion of the whole chain. This feature is common to all of the three models. 

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Bafmer series Frequencies of certain lines in the spectrum of hydrogen are simply related to each other, and can be expressed by a general formula. One group of lines is termed the Balmer series. Other series were later discovered in the spectrum of hydrogen by Lyman, Paschen, Brackett and Pfund. 

These consist of a chain of carbon atoms each carrying 0 to 3 hydrogen atoms except for the simplest molecule, methane CH. Each carbon atom is linked to four other atoms which can be either carbon or hydrogen. Their general formula is( 2 ... 

The general formula for cycloparaffins having a singie ring is There... 

Of the general formula, R - S — H, where R represents an aliphatic or cyclic radical, the thiols —also known as mercaptans— are acidic in behavior owing to their S—H functional group they are corrosive and malodorous. Their concentration in crude oils is very low if not zero, but they are created from other sulfur compounds during refining operations and show up in the light cuts, as illustrated in Table 8.6. 

The largest series is that of the alkanes or paraffins, which are open chain molecules with saturated bonds, and have the general formula C H2n+2-... 

Once the concept of discounting is accepted, the procedure becomes mechanical. The general formula for discounting a flow of money cooccurring in tyears time to its present value Cq assuming a discount rate r is... 

By the standard methods of statistical thermodynamics it is possible to derive for certain entropy changes general formulas that cannot be derived from the zeroth, first, and second laws of classical thermodynamics. In particular one can obtain formulae for entropy changes in highly di.sperse systems, for those in very cold systems, and for those associated, with the mixing ofvery similar substances. 

The general formula and the individual cases as presented in Eq. (97) indicate that indeed the number of conical intersections in a given snb-space and the number of possible sign flips within this sub-sub-Hilbert space are interrelated, similar to a spin J with respect to its magnetic components Mj. In other words, each decoupled sub-space is now characterized by a spin quantum number J that connects between the number of conical intersections in this system and the topological effects which characterize it. 

Other boron hydrides are known, most of them having the general formula B H + 4- for example pentaborane, B5H9, decaborane, BjqHi4. Each can be made by heating diborane in suitable conditions for example at 420 K, decaborane is obtained. Boron hydrides have been tried as rocket fuels. 

The structure of these solid compounds is not known with certainty but an approximate formula might be NaAlOj.xHjO. Many aluminates occur in minerals, for example the spinels of general formula M (A102)2 where M may be Mg, Zn or Fe these have a mixed oxide structure, i.e. consist essentially of M AF and O ions. 

These are double salts which have the general formula... 

Methane, CH4, is the first member of this series, all of which have the general formula C H2 + 2- Every carbon atom in any alkane molecule has a tetrahedral eonfiguration and is joined to four other atoms. Alkanes are resistant to attack, at room temperature, by... 

Silicon, unlike carbon, does notiorm a very large number of hydrides. A series of covalently bonded volatile hydrides called silanes analogous to the alkane hydrocarbons is known, with the general formula Si H2 + 2- I uf less than ten members of the series have so far been prepared. Mono- and disilanes are more readily prepared by the reaction of the corresponding silicon chloride with lithium aluminium hydride in ether ... 

The interstitial carbides These are formed by the transition metals (e.g. titanium, iron) and have the general formula M, C. They are often non-stoichiometric—the carbon atoms can occupy some or all of the small spaces between the larger metal atoms, the arrangement of which remains essentially the same as in the pure metal (cf. the interstitial hydrides). 

Double salts of general formula M2SO4 -FeS04.6H2O (M = alkali metal or ammonium) can be obtained by crystallisation of solutions containing the appropriate proportions of the two simple salts ... 

An extension of these calculations to cationic dialkylamide salts required an even more complex model [Adolf et al. 1995]. These molecules have the general formula (CH3)2N [(CH2) iCH3][(CH2)m-iCH3]CK and the isomer with m = m = 18 is one of the main active ingredients in commercial fabric softeners. The presence of two long alkyl... 

Identification of Amines. Picric acid combines with many amines to give crystalline picrates, of general formula B,(NO )aCeHjOH, where B is a molecule of a monacidic base. These picrates have usually sharp melting- or decomposition-points, and serve to characterise the amines concerned. They may be formed either by (a) direct union of the acid and the base in a suitable solvent, or (6) by the interaction of sodium picrate and a salt of the amine in aqueous solution. 

Note also that if another ester, of general formula R-COOCjHj, were used in place of benzaldehyde in the above reaction, a similar complex would be formed, and on acidification would give an unstable p-hydroxy-P-ethoxy ester, which would very readily lose ethanol with the formation of a 3-keto-ester. 

When an amine, or a solution of its hydrochloride, is added to an aqueous solution of chloroplatinic acid, a salt of the base with the cliloroplatinic acid, of general formula BjiHiPtCle (where B is one molecule of the base) is formed and usually crystallises out, for these chloroplatinates hai e normally a rather low solubility in cold water. The chloroplatinate can be filtered off, dried, and then analysed by direct ignition, when only the metallic platinum ultimately remains. Knowing the percentage of platinum in the chloroplatinate, the molecular weight of the latter, and hence of the constituent base, can readily be calculated. 

Therefore a = /nxl = hi, fl2i = hi, etc. (elements in the first column of a a,re the same as the elements in the first column of /) similarly multiplying rows of / by columns of u and equating the result with the corresponding element of a all of the elements of lower and upper triangular matrices are found. The general formula for obtaining elements of / and u can be expressed as... 

The formation of acyloins (a-hydroxyketones of the general formula RCH(OH)COR, where R is an aliphatic residue) proceeds best by reaction between finely-divided sodium (2 atoms) and esters of aliphatic acids (1 mol) in anhydrous ether or in anhydrous benzene with exclusion of oxygen salts of enediols are produced, which are converted by hydrolysis into acyloins. The yield of acetoin from ethyl acetate is low (ca. 23 per cent, in ether) owing to the accompanying acetoacetic ester condensation the latter reaction is favoured when the ester is used as the solvent. Ethyl propionate and ethyl ji-butyrate give yields of 52 per cent, of propionoin and 72 per cent, of butyroin respectively in ether. 

The hemicyanines constitute a particular class among streptocyanines in which one of the two nitrogen atoms belongs to a heterocycloammonium. Thiazolohemicyanine corresponds to the general formula in Scheme 31. When R = R, = H. the dyes are of aminovinyl type when R = H. R3 = Ph. they are aniiinovinyl compounds, and when (-CH=CH) is replaced by a benzene ring, they are aminophenyl. 

The general formula of the 2-(2- or 4-furyl. thienyl, or selenienyljselenazoles is shown in Scheme 24 (40, 105. 106). (Selenienyl-2), (furyl-2), (thienyl-2), and (chloromethyi-4)selenazoles may be prepared by Hantzsch s reaction from selenoamides as described in Scheme 25. Hydrolysis give hydroxymethyl derivatives. 

A series of meta- and para-bis (2-thiazolyI) benzenes and of meta- and para-bis(4-thiazolyl)benzenes of general formula 35 and 36 was prepared in higher yields (60-90%) from the appropriate bis-(haloacetyl)benzenes with a suitable thioamide (Scheme 18a and Table JI-8) (573, 574),... 

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