Statistical mixtures

Final state analysis is where dynamical methods of evolving states meet the concepts of stationary states. By their definition, final states are relatively long lived. Therefore experiment often selects a single stationary state or a statistical mixture of stationary states. Since END evolution includes the possibility of electronic excitations, we analyze reaction products in terms of rovibronic states. 

The synthetic utility of the alkene metathesis reaction may in some cases be limited because of the formation of a mixture of products. The steps of the catalytic cycle are equilibrium processes, with the yields being determined by the thermodynamic equilibrium. The metathesis process generally tends to give complex mixtures of products. For example, pent-2-ene 8 disproportionates to give, at equilibrium, a statistical mixture of but-2-enes, pent-2-enes and hex-3-enes ... 

For the regioselectivity similar rules as for the ester pyrolysis do apply. With simple, alkylsubstituted amine oxides a statistical mixture of regioisomeric olefins is obtained. On the other hand with cycloalkyl amine oxides the regioselectivity is determined by the ability to pass through a planar, five-membered transition state. This has been demonstrated for the elimination reaction of menthyl dimethylamine oxide 10 and neomenthyl dimethylamine oxide 11 ... 

The cyclotetramerization of suitable monopyrroles is the most simple approach to prepare porphyrins and it has been used in the synthesis of numerous porphyrin structures. But this method can only be used satisfactorily if the pyrroles 1 and 2 bear identical /i-substituents (R = R), otherwise the cyclotetramerization leads to a statistical mixture of four possible constitutional isomers. 

For the second method the threshold concentration of the filler in a composite material amounts to about 5 volume %, i.e. below the percolation threshold for statistical mixtures. It is bound up with the fact that carbon black particles are capable (in terms of energy) of being used to form conducting chain structures, because of the availability of functional groups on their surfaces. This relatively sparing method of composite material manufacture like film moulding by solvent evaporation facilitates the forming of chain structures. 

In order to understand the relationship between the mixture component, physical properties and consumer acceptance of the lipstick, various lipstick formulations have to be produced. The physical properties of each formulation should be studied. The consumer acceptance towards the product also should be investigated. However, only a part of this work will be discussed in this paper. Here, natural waxes, oils and solvent have been used to produce natural ingredient based lipstick formulations based on the formulation suggested by the statistical mixture design. Contour plot and response surface graph were formed in order to understand the relationship between the mixture component and physical characteristic of the lipstick. 

The statistical mixture design for 5-components was carried out by using Design Expert, D-Optimal criterion (Version 6.10, Stat-Easy Inc., Minneapolis USA). In this study, there are restriction on the component proportions Xj that take the form of lower Lj and upper Uj constraint as Lj experimental results of the previous study [2,5]... 

An important difference between the statistical mixture design techniques popular in HPLC and the PRISMA model is that the former yields a computed optimum solvent composition id>ile the latter relies on a structured trial and error approach, which is readily adaptable to TLC. Solvent changes and re-equilibration in HPLC can be quite time consuming, so that it becomes attractive to ainimize the number of experiments, while for TLC, experiments can be performed in parallel and time constraints are less significant. Changes in solvent strength are also more rapidly adjusted empirically within the PRISMA model when theoretical considerations are found inadequate or require modification due to differences in the experimental approach. 

When the reactant provides more than one kind of hydrogen for insertion, the catalyst can influence selectivity. For example, Rh2(acam)4 gives exclusively insertion at a tertiary position, whereas Rh2(02CC4F9)4 leads to nearly a statistical mixture.217aThe attenuated reactivity of the amidate catalyst enhances selectivity. 

It was synthesized from p-iodoanisole (65) by copper-catalyzed coupling with p-trifluoromethyliodobenzene (66) to give the expected statistical mixture from which unsymmetrical product 67 could be separated. Ether cleavage with HBr and HOAc gave 68 this was then alkylated with the aziridinium ion derived from N-(2-chloroethyl)pyrrolidine, using NaH as base, to complete the synthesis of boxidine (69). 

Cardounel AJ, Dumitrescu C, Zweier JL, and Lockwood SF. 2003. Direct superoxide anion scavenging by a disodium disuccinate astaxanthin derivative Relative efficacy of individual stereoisomers versus the statistical mixture of stereoisomers by electron paramagnetic resonance imaging. Biochemical and Biophysical Research Communications 307(3) 704-712. 

When a mixture of perdeuterio and protioazomethane in a ratio of 1 1 was photolyzed in the gas phase, the isolated ethane was found to be a statistical mixture of three possible dimers C2H6, C2H3D3, and C2D6 in the ratio 1 2 1 [83,84]. This is the result of the reactions... 

The two alkenes were so similar electronically and sterically, with the ester group too far away to have any affect on the double bond, that there was very little cross-/self-metathesis selectivity. An approximately statistical mixture of ester 13 and diester 14 was isolated. The high yield of the cross-metathesis product 13 obtained is due to the excess of the volatile hex-l-ene used, rather than a good cross-/self-metathesis selectivity. Although not as predominant as in the reactions involving styrene, trans alkenes were still the major products. 

Porphyrazines are typically synthesized by a templated cyclization of substituted dinitriles, Fig. 2 (2). The most common divalent metal used as the template for this reaction is Mg2+, usually as the butoxide or propoxide, although other group 1(1 A) and 2(IIA) metals have been reported (41). Mixed cyclizations, which utilize two different dinitriles, Fig. 3, in principal, would give a statistical mixture of six different products or isomers. The truly enabling synthetic foundation for modem pz chemistry is the development of strategies directed toward the synthesis... 

Other unsymmetrical aminoporphyrazines have been prepared accordingly. The base-catalyzed cross-condensation of bis(dimethylamino)maleonitrile (99) with 1,2-dicyanobenzene (132) produced a statistical mixture of six possible products that represent hybrids between a heterosubstituted aminoporphyrazine and pc (9). The individual yields of all six products depend on the molar ratio of both starting dinitriles. For instance, a 25 1 ratio of dinitriles 99 and 132 produced mainly pz 133 (35%) along with 100 (Scheme 23). 

An open system with a fluctuating number of particles is described by an ensemble or statistical mixture of pure states and the fractional electron number may arise as a time average. Thus, let No be an integer electron number, and N be an average in the interval Ai> < N < Nt) f 1. In this case Y is an ensemble or statistical mixture of the No -electron pure state with wave function T V 1 and probability 1 — t, and the (A 0 I l)-electron pure state with wave function Tv, i and probability t, where 0 < t < 1. 

If cyclopentene would react pair-wise with 2-pentene, only one product would form, namely 2,7-decadiene, and a similar result for cyclodimers etc. of cyclopentene. If somehow, the alkylidene species would be transferred one by one, we would obtain a mixture of 2,7-nonadiene, 2,7-decadiene, and 2,7-undecadiene in a 1 2 1 ratio. The latter turned out to be the case, which led the authors to propose the participation of metal-carbene (metal alkylidene) intermediates [6], Via these intermediates the alkylidene parts of the alkenes are transferred one by one to an alkene. The mechanism is depicted in Figure 16.4. In the first step the reaction of two alkylidene precursors (ethylidene -bottom- and propylidene -top) with cyclopentene is shown. In the second step the orientation of the next 2-pentene determines whether nonadiene, decadiene or undecadiene is formed. It is clear that this leads to a statistical mixture, all rates being exactly equal, which need not be the case. Sometimes the results are indeed not the statistical mixture as some combinations of metal carbene complex and reacting alkene may be preferred, but it is still believed that a metal-carbene mechanism is involved. Deuterium labelling of alkenes by Gmbbs instead of differently substituted alkenes led to the same result as the experiments with the use of 2-pentene [7],... 

Soybean oil is a statistical mixture of glycerol esters of palmitic acid (10%), stearic acid (3%), oleic acid (23%), linoleic acid (55%), and linolenic acid (9%). 

Astatohalobenzenes can be synthesized directly in At recoil (Table III) experiments either via hydrogen replacement by recoil astatine in monohalobenzenes or by halogen replacement in dihalo-benzenes. In the former, total yields range from 5 to 15%, with an almost statistical mixture of ortho-, meta, and para-astatohalobenzene products 145, 150). The production of AtC5H4F from the corresponding CIC6H4F isomers has been achieved with yields of 14% without noticeable isomerization of the products (19). 

Reduction of a mixture of two aryl halides is not generally a good route to the mixed biaryl. Either a statistical mixture of the three possible biaryls is formed or, if one aryl halide is more reactive, this forms a single biaryl after which, the second aryl halide reacts with itself. The principal exception to this generalisation involves the reduction of a 1 1 mixture of an aryl bromide and 1-chloropyridine. Oxidative-addition to Ni(o) is faster for the carbon-bromine bond. The second oxidative-addition to ArNi(i) is faster for the 2-chloropyridinc, possibly due to complexation from the pyridine nitrogen. Overall, the 1-aryipyridine is formed in 55-80 % yields [200]. 

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