Pairwise distribution

With a pairwise distribution of reagents it is impossible to find separately the parameters ve and ae from the kinetic curves in the majority of practically important situations. Only combinations of these parameters can be found. This is, first of all, connected with the fact that, for real systems, the form of the pair-wise distribution function is, as a rule, unknown. In addition, with the pair-wise distribution it is usually also impossible to determine the initial concentration n(0). 

The only known situation allowing the derivation of the parameters ve and ae from the kinetic curves with the pairwise distribution, is the recombination via electron tunneling 

Due to the rapid decrease in the process probability with increase of the distance between the reagents, it should be expected that reaction (13) will result in electron transfer primarily to the particle A which is nearest to the excited donor particle D. In this case, the condition n N is satisfied for reaction (13), where n is the concentration of the particles D and N is that of the particles A, and with the random initial distribution of the particles, A, the distribution function over the distances in the pairs D A formed, will have the same form [see Chap. 4, eqn. (13)] as with the non-paired random distribution under the conditions when n IV. In such a situation the kinetics of backward recombination of the particles in the pairs D A [reaction (12)] will be described by eqn. (24) of Chap.4 which coincides with eqn. (35) of Chap. 4 for electron tunneling reactions under a non-paired random distribution of the acceptor particles. Therefore, in the case of the pairwise recombination via electron tunneling considered here, the same methods of determining the parameters ve and ae can be applied as those described in the previous section for the case of the non-pair distribution. However, examples of the reliable determination of the parameters ve and ae for the case of the pairwise recombination using this method are still unknown to us. 

As noted above, the calculation of the parameters ve and ac, the main kinetic characteristics of electron tunneling reactions, from kinetic data is possible only if the spatial distribution of the reagents is known. For this reason experiments on the quantitative investigation of electron tunneling 

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For an outer-sphere reaction, given the translation mobility of the reactants, electron transfer may occur over a range of distances. The problem can be treated in a general way since from statistical mechanics the equilibrium distribution of intemuclear separations can be calculated based on pairwise distribution functions. Integration of the product of the distribution function and ket(r) over all space gives the total rate constant et-32b 48... 

The form of the initial spatial distribution of the reagents is determined by the structure and the phase state (solid or liquid) of the solution in which the reaction takes place, the method of generating the reagents (electron donors and acceptors), and also by the spatial distribution of the particles which are the precursors of the donors and acceptors. Two radically different forms of spatial distribution of donors and acceptors are possible the pairwise distribution, i.e. the distribution in isolated pairs, and the nonpair distribution. The difference between them is that, in the case of the pairwise distribution, the reaction occurs only in the isolated pairs of the reagents, i.e. the reaction between the donor and the acceptor from two different pairs is impossible. 

Pairwise distribution can occur, for example, in the case of recombination of the trapped electrons with the parent counterions or with the products of their transformation provided that the two reagents are produced in sufficiently low concentration (see Chap. 6). The pairwise distribution is also characteristic of the recombination processes in the reaction centre of photosynthesis (see Chap. 8). 

The simplest type of pairwise distribution is a rectangular distribution. It has the form... 

For pairwise distribution the kinetics of electron tunneling has the simplest form for the rectangular distribution function, i.e. when immobile reagents are located in isolated pairs with the distribution over the distance R of form... 

As it was shown above, the static kinetics of tunnelling luminescence is much simpler than the diffusion-controlled kinetics. The static tunnelling luminescence intensity often obeys the empirical Becquerel s law [87] I t) oc t ° , where the distinctive decay parameter a = —d ogI i)/ d ogt) is defined by the spatial distribution of defects only, usually considered to be either the isolated pairs of spatially well-correlated dissimilar defects (low dose irradiation) or the random mixture of dissimilar defects (high doses and/or high temperatures) [88]. Moreover, in the case of pairwise distribution, the partial lightsum method has been presented [88-91] in order to restorate the defect initial spatial distribution /(r) within geminate pairs -see equation (4.2.1) and below. What we have discussed here are the tran-... 

Scattering patterns can be translated by standard diffraction techniques to give these pairwise distribution functions. This is difficult to do with X-ray scattering patterns, although modem techniques such as anomalous X-ray scattering have helped. It is relatively easy to determine these using the nuclear diffraction isotope substimtion technique. 

Pig. 14.5 Illustrations of miscibility assessment using powder X-ray diffractometry (pXRD) data I. comparison of linear combination of drug and polymer pXRD patterns with measured pattern of ASD, II. comparison of pure component pXRD pattern extracted from multivariate curve resolution of ASD patterns with the measured patterns of pure components, and III. comparison of linear combination of dmg and polymer pairwise distribution function PDF) patterns with the PDF pattern derived from the measured pXRD data of ASD (residual plot with standard deviation is shown in inset). (Schemes are constructed based on Bates 2011 Ivanisevic et al. 2009 Moore and WUdfong 2011)... 

Boetker JP, Koradia V, Rades T, Rantanen J, Savolainen M (2012) Atomic pairwise distribution function analysis of the amorphous phase prepared by different manufacturing routes. Phatmaceutics... 

BordaUo HN, Zakharov BA, Boldyreva EV, Johnson MR, Koza MM, Seydel T, Fischer J (2012) Application of incoherent inelastic neutron scattering in pharmaceutical analysis relaxation dynamics in phenacetin. Mol Pharm 9 2434-2441 Bptker JP, Karmwar P, Strachan CJ, Cornett C, Tian F, Zujovic Z, Rantanen J, Rades T (2011) Assessment of crystalline disorder in cryo-milled samples of indomethacin using atomic pairwise distribution functions. Int J Pharm 417 112-119 Boutonnet-Fagegaltier N, Menegotto J, Lamure A, Duplaa H, Caron A, Lacabanne C, Bauer M (2002) Molecular mobihty study of amorphous and crystalline phases of a pharmaceutical product by thermally stimulated current spectrometry. J Pharm Sci 91 1548-1560 Bras AR, Noronha JP, Antunes AMM, Cardoso MM, Schdnhals A, Affouard Fdr, Dionfsio M, Correia NIT (2008) Molecular motions in amorphous ibuprofen as studied by broadband dielectric spectroscopy. J Phys Chem B 112 11087-11099... 

Of interest in the derivation of thermodynamic properties is the pairwise distribution function... 

This can be expressed in terms of the pairwise distribution function as follows ... 

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