Kinetics of Thallium (III) oxidation of aromatic ketones

The kinetics of oxidation of acetophenone and six substituted acetophenones by Thallium (III) acetate have been studied in 95% (v/v) aqueous acetic acid. The reaction is first order with respect to the oxidant as well as the organic substrate and acid catalysed. The probable nature of Tl (III) species in aqueous acetic acid has been indicated. The rate data fitted well with Brown’sσ ⁺ values withρ ⁺ = ?0.70. A mechanism for the Tl (III) oxidation of aromatic ketones has been proposed.     

Kinetics of oxidation of some α-hydroxy-carboxylic acids by peroxydisulphate catalysed by silver (I)

The kinetics of the reaction between peroxydisulphate and some α-hydroxy-carboxylic acids (i.e., lactic, mandelic, malic, glycollic, α-hydroxy isobutyric and citric) catalysed by silver (I) has been studied. The rate at which peroxydisulphate disappears was found to be proportional to the concentration of peroxydisulphate and of silver (I), but independent of the concentration of hydroxy acids, although the rate in presence of different hydroxy-acids is different. The silver catalysed decomposition of peroxydisulphate is considerably faster in presence of hydroxy acid than in its absence. The energy of activation and entropy of activation for these reactions have been estimated. A mechanism of the oxidation of hydroxy acids by peroxydisulphate catalysed by silver (I) has been suggested.     

Internal conversion coefficients of E2 transitions

An analysis of K-conversion coefficients of E2 transitions of the type 2⁺→0⁺, 0⁺→2⁺, 4⁺→2⁺, 6⁺→4⁺, and 8⁺→6⁺ together with their methods of determination has been carried out. The ratios of experimental to theoreticalα _k values are plotted against the atomic weights and it is seen that there is no similar behaviour among these transitions.     

Analysis of some allowed beta-transitions

Sakai’s method of analysing logft values has been extended to higher excited states in even-even nuclei. Within the context of the rather limited data an enhancement in the matrix element is observed for transitions to 2″⁺, 2?⁺ and 4′⁺ and 4″⁺ levels.     

Oxidation studies—III

The oxidation of aliphatic unsymmetrical ketones-n-propyl methyl ketone and isobutyl methyl ketone and cyclic ketone-cyclohexanone by ceric perchlorate in perchloric acid (≈0·5–2·0 M) was studied in the temperature range 10°–30° C. The kinetics of the reactions were followed by measuring rate of Ce⁴⁺ disappearance effecting variations in [Ce⁴⁺], [Ketone], [HClO₄], ionic strengthμ, temperature, etc., and the reactions were found to be 2nd order—first order each with respect to [Ce⁴⁺] and [ketone]. No evidence for complex formation between Ce⁴⁺ and ketone was obtained from kinetics or by an independent absorptiometric method. Ce⁴⁺ (aq) and enol of the ketones were the active species for oxidation. Correlations between rates of enolisation of ketones and oxidation of the latter by Ce⁴⁺ were observed. The rate constants and their thermodynamic data were calculated and discussed. A mechanism based on the concept of oxidation as a process of successive hydroxylation of organic molecules is proposed.     

Oxidation studies-VI

Uncatalysed oxidation of malonic, maleic, succinic, malic, mandelic and lactic acids by peroxydisulphate have been studied at 60°–80°C. Rates of oxidation and E values are reported. Probable mechanisms are proposed.     

Rotational analysis of some bands of C1 Σ u + -X1 Σ g + system of P2 molecule involving lowν′,ν″ values

Ten ultraviolet bands of the C¹ Σ _u ⁺ -X¹ Σ _g ⁺ system of P₂ involving lowv′ andv″ values have been photographed at dispersion of 0·38 and 0·56 Å/mm. and analysed for their rotational structure. While four of these bands were analysed earlier, six of them,viz., 0–10, 1–12, 2–7, 2–14, 4–8 and 6–9 have been analysed for the first time during the present studies. The rotational constants, B _v S with lowv″ quantum numbers are obtained from which value of B _θ ^″ has been derived. The value of B _θ ^″ is found to be in agreement with the value obtained by Douglas and Rao from their study of A¹ Π _g-X¹ Σ _g ⁺ bands of P₂. Earlier findings on the perturbation ofν′=2 level of the C¹ Σ _u ⁺ state have been confirmed from the analysis of the 2–7, 2–14 and 2–15 bands. Theν ₀₀ values of the bands show large deviations from the expected values.     

Systematics of some beta-transitions

0⁺ excited states of even-even nuclei can arise in several ways. In order to throw some light on the mechanism of excitation of these states, the log (ft) values for beta-transitions to the 0⁺ excited state and the 0⁺ ground state of even-even nuclei have been compared. In this analysis both allowed and forbidden beta-transitions have been considered. This survey is similar to that of beta-transitions by M. Sakai. Results of the present analysis are presented and compared to that of Sakai.     

Computing the number of ways of representing primes by a norm form

Formulae for the number of different integral solutions ofa ²+b²+c²+d²+ac+bd=p are given wherep is a prime and the solution satisfies certain natural congruence conditions. Similar formulae are given for the case of the quadratic forma ²+b²+2c²+2d²+ac+bd.     

Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities

In 1956, R. Penrose studied best-approximate solutions of the matrix equation AX = B. He proved that A⁺B (where A⁺ is the Moore-Penrose inverse) is the unique matrix of minimal Frobenius norm among all matrices which minimize the Frobenius norm of AX ? B. In particular, A⁺ is the unique best-approximate solution of AX = I. The vector version of Penrose's result (that is, the fact that the vector A⁺b is the best-approximate solution in the Euclidean norm of the vector equation Ax = b) has long been generalized to infinite dimensional Hilbert spaces.In this paper, an infinite dimensional version of Penrose's full result is given. We show that a straightforward generalization is not possible and provide new extremal characterizations (in terms of the Hermitian order) of A⁺ and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least-squares solutions. For a certain class of operators, we can phrase our characterizations in terms of a whole class of norms (including the Hilbert-Schmidt and the trace norms), thus providing new extremal characterizations even in the matrix case. We treat both operators with closed range and with not necessarily closed range. Finally, we characterize A⁺ as the unique inner inverse of minimal Hilbert-Schmidt norm if ∥A⁺∥₂ < ∞. We give an application of the new extremal characterization to the compensation problem in systems analysis in infinite-dimensional Hilbert spaces.     

Oxidation studies

Results on the oxidation of primary alcohols (methanol, ethanol,n-propanol,n-butanol and isobutanol) by peroxydisulfate in aqueous solution in the temperature range 50°–80°, under uncatalysed, Ag⁺ catalysed and deaerated conditions are presented. Oxygen was found to have a retarding effect on the rate of peroxydisulfate disappearance,\( - R_{S_2 O_3 ^ - } \), and the inhibition period was a maximum in the case of ethanol and isobutanol. Both under uncatalysed and Ag⁺ catalysed conditions\( - R_{S_2 O_3 ^ - } \) varied with [S₂O_8/=]^3/2 for all the alcohols;\( - R_{S_2 O_3 ^ - } \) was independent of alcohol concentration for all the alcohols except for methanol and ethanol where a one-half order with respect to [alcohol] was observed. In the Ag⁺ catalysed oxidation\( - R_{S_2 O_3 ^ - } \) varied as [Ag⁺]^1/2 with all the alcohols. The effect of [H⁺], ionic strength, temperature, initially added product of the reaction on the rate are discussed and the stoichiometries of the reactions determined. In the light of the observed results, reaction mechanisms are postulated with SO ₄ ^·? as the reactive species formed by thermal decomposition of S₂O ₈ ⁼ under uncatalysed conditions and by the interaction of Ag⁺ and S₂O_8/? under Ag⁺ catalysed conditions. The existence of Ag⁺⁺ in catalysed reactions is postulated. The values of rate constants, frequency factors, energies of activation entropies of activation, etc., were evaluated.     

Three-variable Mahler measures and special values of modular and Dirichlet L-series

In this paper we prove that the Mahler measures of the Laurent polynomials (x+x ^?1)(y+y ^?1)(z+z ^?1)+k ^1/2, (x+x ^?1)²(y+y ^?1)²(1+z)³ z ^?2?k, and x ⁴+y ⁴+z ⁴+1+k ^1/4 xyz, for various values of k, are of the form r ₁ L′(f,0)+r ₂ L′(χ,?1), where $r_{1},r_{2}\in \mathbb{Q}$ , f is a CM newform of weight 3, and χ is a quadratic character. Since it has been proved that these Mahler measures can also be expressed in terms of logarithms and ₅ F ₄-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results.     

Structure-reactivity relationship in the oxidation of aryl aliphatic ketones by CeIV

Kinetics of oxidation of aryl aliphatic ketones by Ce^IV in presence of HClO₄ in aqueous acetic acid medium has been studied. Reactivity has been found to be that substituents of electron withdrawing nature accelerate the reaction and the reverse is true with electron releasing substituents. A mechanism of radical intermediate is postulated to explain the observed phenomena. The unusual reactivity of the alkoxy derivatives may be traced to a change in the mechanism routed through a carbonium ion centre or alternatively it might be due to the hydrolysis of the alkoxy derivatives leading to the higher reactivity observed.     

A class of difference scheme for solving telegraph equation by new non-polynomial spline methods

In this paper, by using a new non-polynomial parameters cubic spline in space direction and compact finite difference in time direction, we get a class of new high accuracy scheme of O(τ⁴ + h²) and O(τ⁴ + h⁴) for solving telegraph equation if we suitably choose the cubic spline parameters. Meanwhile, stability condition of the difference scheme has been carried out. Finally, numerical examples are used to illustrate the efficiency of the new difference scheme.     

On the day airglow

The altitude variation of the intensity ofλ 6300 line of OI atom and that ofλ 3914 (0–0 band of the first negative system of N ₂ ⁺ ) in the day airglow have been calculated. It has been found that photodissociation of O₂ by the Schumann-Range continuum is mainly responsible for the emission ofλ 6300 line (intensity 65 KR). Emission produced by dissociative recombination of O ₂ ⁺ and NO⁺ ions with electrons is of less importance (intensity 22 KR). The low intensity ofλ 6300 line during a solar eclipse, when the photo dissociation process is absent, is in conformity with the intensity (<6 KR) obtained by Tanabe and Tohmatsu (1959). Assuming that the resonance fluorescence is responsible for the emission ofλ 3914 band, the intensity of this band is calculated. The calculated intensity (7 KR) is within the limits of the values (>1 KR and <70 KR) obtained from theoretical considerations by Chamberlain and Sagan (1960).     

Optical model analysis of (240–300) MeV K+-meson elastic scattering in nuclear emulsion

The differential elastic scatterings of (240–300) MeV K⁺-mesons in G-5 nuclear emulsions have been analysed in terms of exact diffuse-surface optical-model of the nucleus and the real nuclear volume potential integral per nucleon (I _v ) obtained. By making Chi square comparison between the experimental cross-sections and the theoretically calculated values for the optical model parameters R₀=1·07 F anda=0·57 F, I _v comes out to be (100±30) MeV F³. The analysis also confirms that for K⁺-nucleus scattering, repulsive potential is by far more probable than an attractive one. The K⁺-free nucleon cross-section has been estimated to lie in the range (13·5±1·7) mb.     

The non-existence of certain finite linear spaces

We study the existence of finite linear spaces with v points and n ²+n+2 lines, where n ²+1?v?n ²+n+1. For n?3, there is only one such linear space; it has ten points and fourteen lines.     

Finite linear spaces with two consecutive line degrees

Let L be a non-trivial finite linear space in which every line has n or n+1 points. We describe L completely subject to the following restrictions on n and on the number of points p: p≤n ²+n?1 and n≥3; n ²+n+2≤p≤n ²+2n?1 and n≥3; p=n ²+2n and n≥4; p=n ²+2n+2 and n≥3; p=n ²+2n+3 and n≥4.     

Spectrophotometric and potentiometric investigations of copper (II)-Ethanolamine complexes

Spectrophotometric and potentiometric investigations have been carried out on copper-diethanolamine system. Job plots at 900, 900 and 580 mμ have indicated the formation of CuD⁺⁺, CuD₂ ⁺⁺ and CuD₃ ⁺⁺. Then- pA curves obtained indicate the formation of CuD⁺⁺, CuD₂ ⁺⁺, CuD₃ ⁺⁺, CuDOH⁺, CuD₂OH⁺ and CuD₃OH⁺. Then- pA curves have been analyzed to obtain the stability constants of these complexes. Absorption curves of pure complexes have been computed by a graphical method. Gaussian analysis of the absorption curves of pure and hydroxy complexes show the presence of a second band, indicating that the structure is that of a distorted octahedron.     

The influence of isobar production on charge ratio and transverse momentum of secondary particles in high energy collisions

An isobar model in which collision between two particles leads to the creation of only two bodies which by subsequent decay give rise to the observed secondaries has been considered. On the basis of such a model, the charge ratios of pions, kaons andΣ-hyperons inp?p andπ?p collisions have been computed and compared with the available experimental data. Some features of transverse momentum of pions and protons in 24 GeV/cp?p collisions have also been studied. The main conclusions can be summarised as follows:

1. (1)
   The observed positive excess among pions produced in high energyp?p collisions leading toπ ⁺/π ^? andπ ⁺/π ⁰ ratios of ～3 and 1·6 respectively for high momentum pions can be explained on the basis of the isobar model. Further, the fast increase of K⁺/K^? ratio as the kaon momentum increases, the high ratio (～4) ofΣ ⁺/Σ ^? in 24 GeV/cp?p collisions and the existence of a strong positive (negative) excess amongΣ-hyperons produced inπ ⁺?p(π ^??p) collisions at various primary energies result, in a natural way, from such a model. The agreement results mainly from the restriction of only two bodies in the final states and does not critically depend on the isospins of produced isobars.