New approximate formula for the generalized temperature integral

A new procedure to approximate the generalized temperature integral $ \int_{0}^{T} {T^{m} {\text{e}}^{ - E/RT} } {\text{d}}T, $ which frequently occurs in non-isothermal thermal analysis, has been developed. The approximate formula has been proposed for calculation of the integral by using the procedure. New equation for the evaluation of non-isothermal kinetic parameters has been obtained, which can be put in the form: $$ \ln \left[ {{\frac{g(\alpha )}{{T^{(m + 2)0.94733} }}}} \right] = \left[ {\ln {\frac{{A_{0} E}}{\beta R}} - (m + 2)0.18887 - (m + 2)0.94733\ln {\frac{E}{R}}} \right] - (1.00145 + 0.00069m){\frac{E}{RT}} $$ The validity of the new approximation has been tested with the true value of the integral from numerical calculation. Compared with several published approximation, the new one is simple in calculation and retains high accuracy, which indicates it is a good approximation for the evaluation of kinetic parameters from non-isothermal kinetic analysis.     

Bounds for the Kirchhoff index via majorization techniques

Using a majorization technique that identifies the maximal and minimal vectors of a variety of subsets of ${\mathbb{R}^{n}}$ , we find upper and lower bounds for the Kirchhoff index K(G) of an arbitrary simple connected graph G that improve those existing in the literature. Specifically we show that $$K(G) \geq \frac{n}{d_{1}} \left[ \frac{1}{1+\frac{\sigma}{\sqrt{n-1}}} + \frac{(n-2)^{2}}{n-1-\frac{\sigma}{\sqrt{n-1}}}\right] ,$$ where d ₁ is the largest degree among all vertices in G, $$\sigma ^{2} = \frac{2}{n} \sum_{(i, j) \in E} \frac{1}{d_{i}d_{j}} = \left( \frac{2}{n}\right) R_{-1}(G),$$ and R _?1(G) is the general Randi? index of G for ${\alpha =-1}$ . Also we show that $$K(G) \leq \frac{n}{d_{n}}\left( \frac{n-k-2}{1-\lambda _{2}}+\frac{k}{2}+\frac{1}{\theta}\right) ,$$ where d _n is the smallest degree, ${\lambda _{2}}$ is the second eigenvalue of the transition probability of the random walk on G, $$k = \left \lfloor \frac{\lambda _{2} \left( n-1\right) +1}{\lambda _{2}+1}\right\rfloor {\rm and}\quad\theta = \lambda _{2} \left( n-k-2\right) -k+2.$$      

Non‐Isothermal Decomposition Kinetics,Specific Heat Capacity and Adiabatic Time‐to‐explosion of a Novel High Energy Material: 1‐Amino‐1‐Methylamino‐2,2‐Dinitroethylene (AMFOX‐7)

A novel high energy material, 1‐amino‐1‐methylamino‐2,2‐dinitroethlyene (AMFOX‐7), was synthesized by the reaction of 1,1‐diamino‐2,2‐dinitroethylene (FOX‐7) and methylamine aqueous solution in N‐methyl pyrrolidone at 80°C. The thermal behavior and non‐isothermal decomposition kinetics of AMFOX‐7 were studied with DSC and TG/DTG methods. The kinetic equation of thermal decomposition reaction can be expressed as: $ {\rm d\alpha /d}T = \frac{{10^{21.03}}}{{\rm \beta}}\frac{3}{2}\left({1 - {\rm \alpha}} \right)\left[{- 1{\rm n}\left({{\rm 1} - {\rm \alpha}} \right)} \right]^{\frac{1}{3}} \exp \left({- 2.292 \times 10^5 {\rm /}RT} \right) A novel high energy material, 1‐amino‐1‐methylamino‐2,2‐dinitroethlyene (AMFOX‐7), was synthesized by the reaction of 1,1‐diamino‐2,2‐dinitroethylene (FOX‐7) and methylamine aqueous solution in N‐methyl pyrrolidone at 80°C. The thermal behavior and non‐isothermal decomposition kinetics of AMFOX‐7 were studied with DSC and TG/DTG methods. The kinetic equation of thermal decomposition reaction can be expressed as: $ {\rm d\alpha /d}T = \frac{{10^{21.03}}}{{\rm \beta}}\frac{3}{2}\left({1 - {\rm \alpha}} \right)\left[{- 1{\rm n}\left({{\rm 1} - {\rm \alpha}} \right)} \right]^{\frac{1}{3}} \exp \left({- 2.292 \times 10^5 {\rm /}RT} \right) $. The critical temperature of thermal explosion of AMFOX‐7 is 244.89°C. The specific heat capacity of AMFOX‐7 was determined with micro‐DSC method and theoretical calculation method, and the standard molar specific heat capacity is 199.39 J·mol^?1·K^?1 at 298.15 K. Adiabatic time‐to‐explosion of AMFOX‐7 was also calculated to be 215.41 s. AMFOX‐7 has higher thermal stability than FOX‐7.     

Thermodynamic Properties of Ternary Ionic Liquid Mixture Containing a Common Ion: Excess Molar Volumes,Excess Isentropic Compressibilities,Excess Molar Enthalpies and Excess Heat Capacities

In the present investigations, the excess molar volumes, \( V_{ijk}^{\text{E}} \), excess isentropic compressibilities, \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), and excess heat capacities, \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \), for ternary 1-butyl-2,3-dimethylimidazolium tetrafluoroborate (i) + 1-butyl-3-methylimidazolium tetrafluoroborate (j) + 1-ethyl-3-methylimidazolium tetrafluoroborate (k) mixture at (293.15, 298.15, 303.15 and 308.15) K and excess molar enthalpies, \( \left( {H^{\text{E}} } \right)_{ijk} \), of the same mixture at 298.15 K have been determined over entire composition range of x _i and x _j . Satisfactorily corrections for the excess properties \( V_{ijk}^{\text{E}} \), \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), \( \left( {H^{\text{E}} } \right)_{ijk} \) and \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \) have been obtained by fitting with the Redlich–Kister equation, and ternary adjustable parameters along with standard errors have also been estimated. The \( V_{ijk}^{\text{E}} \), \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), \( \left( {H^{\text{E}} } \right)_{ijk} \) and \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \) data have been further analyzed in terms of Graph Theory that deals with the topology of the molecules. It has also been observed that Graph Theory describes well \( V_{ijk}^{\text{E}} \), \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), \( \left( {H^{\text{E}} } \right)_{ijk} \) and \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \) values of the ternary mixture comprised of ionic liquids.     

Synthesis,thermal behavior,and application of 4-amino-3,5-dinitropyrazole lead salt

Lead salt of 4-amino-3,5-dinitropyrazole (PDNAP) was synthesized from 4-amino-3,5-dinitropyrazole by the process of metathesis reaction, and its structure was characterized by IR, element analysis, TG, and DSC. The thermal decomposition kinetics and mechanism were studied by means of different heating rate differential scanning calorimetry (DSC) and thermolysis in situ rapid-scan FTIR simultaneous. The effects of PDNAP as an energetic combustion catalyst on the combustion performance of the solid propellant were studied. The results show that the peak temperature is 319.2 °C on DSC curve. The kinetic equation of major exothermic decomposition reaction is $ \frac{{\text{d}}\alpha}{{\text{d}}T} = \frac{{10^{15.45} }}{\beta }4(1 - \alpha )[ - \ln \left( {1 - \alpha } \right)]^{{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0pt} 4}}} \exp ({{ - 1.972 \times 10^{5} } \mathord{\left/ {\vphantom {{ - 1.972 \times 10^{5} } {RT}}} \right. \kern-0pt} {RT}}). $ The PDNAP is shown by IR spectroscopy to convert to PbO during the decomposition process. Combustion experiments show PDNAP can reduce the burning rate pressure exponent of the double-base or composite-modified double-base propellant.     

Berechnung der Madelungkonstante von Aragonit

Madelung's coefficientM _a of aragonite has been calculated considering the non-spherical shape of the CO ₃ ^2? -ions. As a result of the multipole expansionM _a has been found as a function of the C?O-distanced and the charge on the oxygen atomq _o to:

$$\begin{gathered} M_a = \frac{1}{4}\left\{ {10,4446---\left[ {0,65849 + \sum\limits_{n = 1}^{10} {A_n \left( {\frac{{d---0,8}}{a}} \right)^n } } \right]} \right\} \cdot q_o \hfill \\ \left. \begin{gathered} \hfill \\ ---\left[ {0,11066 + \sum\limits_{n = 1}^{12} {B_n \left( {\frac{{d---0,8}}{a}} \right)} ^n } \right] \cdot q_o^2 \hfill \\ \end{gathered} \right\}. \hfill \\ \end{gathered}$$     

Potential energy function based on the narcissus constant,its square and its cube

The narcissus constant, N = 2.3983843828..., is defined as a number that fulfills the narcissistic infinite nested radical equation

Dissolution kinetics of fluorapatite in the hydrochloric acid solution

A thermochemical study of hydrochloric acid attack of synthetic fluorapatite was performed by a DRC. The calculated thermogenesis curves show one peak. The plot of the heat quantity as a function of the dissolved mass undergoes only one straight segment, and the thermogenesis curves present a single peak, suggesting the occurrence of a one-step dissolution process. The dissolution kinetics was examined according to the heterogeneous reaction models and showed that the dissolution is controlled by the product layer diffusion process with a reaction rate expressed by the following semiempirical equation; \(\left[ {1 + 2(1 - X) - 3(1 - X)^{{\frac{2}{3}}} } \right] = 3195 \times 10^{ - 2} C^{0.145} \left( {\frac{S}{L}} \right)^{ - 0.628} e^{{ - \frac{2600}{\text T}}} t\). The activation energy was determined as 21.6 ± 1.5 kJ mol^?1     

Kinetics and mechanism of Am(III) extraction with TODGA

In the present paper, N,N,N’,N’-tetraoctyl diglycolamide (TODGA) as the extractant and n-dodecane as the diluent, the extraction kinetics behavior of Am(III) in TODGA/n-dodecane–HNO₃ system were studied, including stirring speed, the interfacial area, extractant concentration in n-dodecane, extracted ions concentration, acidity of aqueous phase and temperature. The results show that: the extraction process is controlled by diffusion mode under 130 rpm of stirring speed and by chemical reaction mode above 150 rpm. The extraction rate equation and the apparent extraction rate constant of Am(III) by TODGA/n-dodecane in 170 rpm and at 25 °C are followed as: $$ \begin{aligned} r_{0} = \left. {\frac{{{\text{d}}[{\text{M}}]_{{{\text{org}} .}} }}{{{\text{d}}{{t}}}}} \right|_{t = 0} & = k\,\frac{S}{V}\left[ {\text{Am}} \right]_{{{\text{aq}} . ,0}}^{0.94} \left[ {{\text{HNO}}_{3} } \right]_{{{\text{aq}} . ,0}}^{1.05} \left[ {\text{TODGA}} \right]_{{{\text{org}} . ,0}}^{1.19} \\ & \quad k = \left( {24.17 \pm 3.43} \right) \times 10^{ - 3} \,{\text{mol}}^{ - 2.18} \,L^{2.18} \,{ \hbox{min} }^{ - 1} \,{\text{cm}},\;E_{\text{a}} \left( {{\text{Am}}\left( {\text{III}} \right)} \right) = 25.94 \pm 0.98\;{\text{kJ/mol}} .\\ \end{aligned} $$      

Electron capture by protons from theK-shell of H and Ar

Whenever a collision takes place between charged particles, the first Born approximation for electron capture from hydrogenlike ions (Z _T ,e) by a bare nucleusZ _P , must be modified in order to account for the long-range Coulomb effects. One of the simplest ways to fulfill this requirement is provided by theT-matrix of the following form: $$T_{if}^{(1)} = \left\langle {\Phi _f exp\left\{ { - i\frac{{Z_T (Z_p - 1)}}{\upsilon } ln (\upsilon R + v \cdot R)} \right\}\left| {\frac{{Z_P }}{R} - \frac{{Z_P }}{{r_P }}} \right| exp\left\{ {i\frac{{Z_P (Z_T - 1)}}{\upsilon } ln (\upsilon R + v \cdot R)} \right\}\Phi _i } \right\rangle $$ where Φ's are the usual unperturbed channel states andZ's are the nuclear charges. In this transition amplitude, both initial and final scattering states satisfy the correct asymptotic boundary conditions in their respective channels. In the present paper, detailed computation of theK-shell cross sections is carried out for charge exchange in H⁺-H and H⁺-Ar collisions. The results are in good agreement with experimental data.     

Bohr Sommerfeld quantisation and molecular potentials

We combine, within the Bohr Sommerfeld quantization rule, a systematic perturbation with asymptotic analysis of the action integral for potentials which support a finite number of bound states with E < 0 to obtain an interpolation formula for the energy eigenvalues. We find interpolation formulae for the Morse potential as well as potentials of the form \({V=V_0 \left[ {\left( {\frac{a}{x}} \right)^{2k}-\left( {\frac{a}{x}}\right)^{k}} \right]}\). For k = 6 i.e. the well known Lennard Jones potential this yields results within 1 per cent of the highly accurate numerical values. For the Morse potential this procedure yields the exact answer. We find that the result for the Morse potential which approaches zero exponentially is the \({k\rightarrow\infty}\) limit of the Lennard Jones class of potentials.     

The Study of Solute–Solvent Interactions in 1-Butyl-3-Methylimidazolium Hexafluorophosphate + 2-Pyrrolidone from Volumetric,Acoustic, Optical and Spectral Properties

The density (ρ), speed of sound (u) and refractive index (n_D) of [Bmim][PF₆], 2-pyrrolidone and their binary mixtures were measured over the whole composition range as a function of temperature between (303.15 and 323.15)?K at atmospheric pressure. Experimental values were used to calculate the excess molar volumes \( \left( {V_{m}^{\text{E}} } \right) \), excess partial molar volumes \( \left( {\overline{V}_{m}^{\text{E}} } \right) \), partial molar volumes at infinite dilution \( \left( {\overline{V}_{m}^{{{\text{E}},\infty }} } \right) \), excess values of isentropic compressibility \( \left( {\kappa_{S}^{\text{E}} } \right) \), free length \( \left( {L_{\text{f}}^{\text{E}} } \right) \) and speeds of sound \( \left( {u^{\text{E}} } \right) \) for the binary mixtures. The calculated properties are discussed in terms of molecular interactions between the components of the mixtures. The results reveal that interactions between unlike molecules take place, particularly through intermolecular hydrogen bond formation between the C₂–H of [Bmim][PF₆] and the carbonyl group of pyrrolidin-2-one. An excellent correlation between thermodynamic and IR spectroscopic measurements was observed. The observations were further supported by the Prigogine–Flory–Patterson (PFP) theory of excess molar volume.     

Thermal analysis kinetics of thermal decomposition of nickel–viscose composite fiber

In this study, the thermal decompositions of nickel composite fibers (NCF) under different atmospheres of flowing nitrogen and air were investigated by XRD, SEM–EDS, and TG–DTG techniques. Non-isothermal studies indicated that only one mass loss stage occurred over the temperature regions of 298–1,073 K in nitrogen. The mass loss was from the decomposition. But after this decomposition, nickel was oxidized in air, when the temperature was high enough. In nitrogen media, the model-free kinetic analysis method was applied to calculate the apparent activation energy (E _a) and pre-exponential factor (A). The method combining Satava–?esták equation with one TG curve was used to select the suitable mechanism functions from 30 typical kinetic models. Furthermore, the Coats–Redfern method was used to study the NCF decomposition kinetics. The study results showed that the decomposition of NCF in nitrogen media was controlled by three-dimension diffusion; mechanism function was the anti-Jander equation, the apparent activation energy (E _a) and the pre-exponential factor (A) were 172.3 kJ mol^?1 and 2.16 × 10⁹ s^?1, respectively. The kinetic equation could be expressed as following: $$ \frac{{{\text{d}}\alpha }}{{{\text{d}}T}} = \frac{{ 2. 1 6\times 1 0^{ 9} }}{\beta }{ \exp }\left( {\frac{ - 2 0 7 2 4. 1}{T}} \right)\left\{ {\frac{ 3}{ 2}(1 + \alpha )^{2/3} [(1 + \alpha )^{1/3} - 1]^{ - 1} } \right\}. $$      

Effect of long-chain branching on the relation between steady-flow and dynamic viscosity of polyethylene melts

The steady-state viscosity η, the dynamic viscosity η′, and the storage modulus G′ of several high-density and low-density polyethylene melts were investigated by using the Instron rheometer and the Weissenberg rheogoniometer. The theoretical relation between the two viscosities as proposed earlier is:\documentclass{article}\pagestyle{empty}\begin{document}$ \eta \left( {\dot \gamma } \right){\rm } = {\rm }\int {H\left( {\ln {\rm }\tau } \right)} {\rm }h\left( \theta \right)g\left( \theta \right)^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} \tau {\rm }d{\rm }\ln {\rm }\tau $\end{document}, where \documentclass{article}\pagestyle{empty}\begin{document}$ \theta {\rm } = {\rm }{{\dot \gamma \tau } \mathord{\left/ {\vphantom {{\dot \gamma \tau } 2}} \right. \kern-\nulldelimiterspace} 2} $\end{document}; \documentclass{article}\pagestyle{empty}\begin{document}$ {\dot \gamma } $\end{document} is the shear rate, H is the relaxation spectrum, τ is the relaxation time, \documentclass{article}\pagestyle{empty}\begin{document}$ g\left( \theta \right){\rm } = {\rm }\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\left[ {\cot ^{ - 1} \theta {\rm } + {\rm }{\theta \mathord{\left/ {\vphantom {\theta {\left( {1 + \theta ^2 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \theta ^2 } \right)}}} \right] $\end{document}, and \documentclass{article}\pagestyle{empty}\begin{document}$ h\left( \theta \right){\rm } = {\rm }\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\left[ {\cot ^{ - 1} \theta {\rm } + {\rm }{{\theta \left( {1{\rm } - {\rm }\theta ^2 } \right)} \mathord{\left/ {\vphantom {{\theta \left( {1{\rm } - {\rm }\theta ^2 } \right)} {\left( {1{\rm } + {\rm }\theta ^2 } \right)^2 }}} \right. \kern-\nulldelimiterspace} {\left( {1{\rm } + {\rm }\theta ^2 } \right)^2 }}} \right] $\end{document}. Good agreement between the experimental and calculated values was obtained, without any coordinate shift, for high-density polyethylenes as well as for a low density sample with low n_w, the weight-average number of branch points per molecule. The correlation, however, was poor with low-density samples with large values of the long-chain branching index n_w. This lack of coordination can be related to n_w. The empirical relation of Cox and Merz failed in a similar way.     

Some observations on the scaled-particle theory of solutions and the surface tension of solvents

The energy necessary to form a cavity of appropriate size for a solute can be calculated by use of the scaled-particle theory if the effective hard-sphere diameter σ₁ of the solvent is known. A method is presented for obtaining σ₁ for solvents for which gas-solubility data are not available. The method is based on an empirical correlation between the surface tension of the solvent and the function $$\frac{1}{{\sigma _1^2 }}\left[ {\frac{{3y}}{{1 - y}} + \frac{1}{2}{\text{ }}\left( {\frac{{3y}}{{1 - y}}} \right)^2 } \right]$$ a function which appears in expressions for the surface tension and the cavity energy in the scaled-particle theory. Some general observations about the cavity energies for large solutes (≥6 Å diameter) are also made.     

The mechanism of thermal decomposition of ionic oxalates

A mechanism for the thermal decomposition of ionic oxalates has been proposed on the basis of three quantitative relationships linking the quantitiesr _c/r _i (the ratio of the Pauling covalent radius and the cation radius of the metal atom in hexacoordination) andΣI _i (the sum of the ionization potentials of the metal atom in kJ mol^?1) with the onset oxalate decomposition temperature (T _d) (Eq. 1) the average C-C bond distance (¯d) (Eq. 2), and the activation energy of oxalate decomposition (E _a) (Eq. 3): (1) $$T_d = 516 - 1.4006\frac{{r_c }}{{r_i }}(\sum I_i )^{\frac{1}{2}}$$ (2) $$\bar d = 1.527 + 5.553 \times 10^{ - 6} \left( {122 - \frac{{r_c }}{{r_i }}(\sum I_i )^{\frac{1}{2}} } \right)^2$$ (3) $$E_a = 127 + 1.4853 \times 10^{ - 6} \left( {\left( {\frac{{r_c }}{{r_i }}} \right)^2 \sum I_i - 9800} \right)^2$$ On the basis of these results it is proposed that the thermal decomposition of ionic oxalates follows a mechanism in which the C-O bond ruptures first. From Eq. 3 it is further proposed that strong mutual electronic interactions between the oxalate and the cations restrict the essential electronic reorganization leading to the products, thereby increasingE _a.     

Inner-sphere oxidation of a ternary dipicolinatochromium(III) complex involving a malonic acid co-ligand

The oxidation of a ternary complex of chromium(III), [Cr^III(DPA)(Mal)(H₂O)₂]^?, involving dipicolinic acid (DPA) as primary ligand and malonic acid (Mal) as co-ligand, was investigated in aqueous acidic medium. The periodate oxidation kinetics of [Cr^III(DPA)(Mal)(H₂O)₂]^? to give Cr(VI) under pseudo-first-order conditions were studied at various pH, ionic strength and temperature values. The kinetic equation was found to be as follows: \( {\text{Rate}} = {{\left[ {{\text{IO}}_{4}^{ - } } \right]\left[ {{\text{Cr}}^{\text{III}} } \right]_{\text{T}} \left( {{{k_{5} K_{5} + k_{6} K_{4} K_{6} } \mathord{\left/ {\vphantom {{k_{5} K_{5} + k_{6} K_{4} K_{6} } {\left[ {{\text{H}}^{ + } } \right]}}} \right. \kern-0pt} {\left[ {{\text{H}}^{ + } } \right]}}} \right)} \mathord{\left/ {\vphantom {{\left[ {{\text{IO}}_{4}^{ - } } \right]\left[ {{\text{Cr}}^{\text{III}} } \right]_{\text{T}} \left( {{{k_{5} K_{5} + k_{6} K_{4} K_{6} } \mathord{\left/ {\vphantom {{k_{5} K_{5} + k_{6} K_{4} K_{6} } {\left[ {{\text{H}}^{ + } } \right]}}} \right. \kern-0pt} {\left[ {{\text{H}}^{ + } } \right]}}} \right)} {\left\{ {\left( {\left[ {{\text{H}}^{ + } } \right] + K_{4} } \right) + \left( {K_{5} \left[ {{\text{H}}^{ + } } \right] + K_{6} K_{4} } \right)\left[ {{\text{IO}}_{4}^{ - } } \right]} \right\}}}} \right. \kern-0pt} {\left\{ {\left( {\left[ {{\text{H}}^{ + } } \right] + K_{4} } \right) + \left( {K_{5} \left[ {{\text{H}}^{ + } } \right] + K_{6} K_{4} } \right)\left[ {{\text{IO}}_{4}^{ - } } \right]} \right\}}} \) where k ₆ (3.65 × 10^?3 s^?1) represents the electron transfer reaction rate constant and K ₄ (4.60 × 10^?4 mol dm^?3) represents the dissociation constant for the reaction \( \left[ {{\text{Cr}}^{\text{III}} \left( {\text{DPA}} \right)\left( {\text{Mal}} \right)\left( {{\text{H}}_{2} {\text{O}}} \right)_{2} } \right]^{ - } \rightleftharpoons \left[ {{\text{Cr}}^{\text{III}} \left( {\text{DPA}} \right)\left( {\text{Mal}} \right)\left( {{\text{H}}_{2} {\text{O}}} \right)\left( {\text{OH}} \right)} \right]^{2 - } + {\text{H}}^{ + } \) and K ₅ (1.87 mol^?1 dm³) and K ₆ (22.83 mol^?1 dm³) represent the pre-equilibrium formation constants at 30 °C and I = 0.2 mol dm^?3. Hexadecyltrimethylammonium bromide (CTAB) was found to enhance the reaction rate, whereas sodium dodecyl sulfate (SDS) had no effect. The thermodynamic activation parameters were estimated, and the oxidation is proposed to proceed via an inner-sphere mechanism involving the coordination of IO₄ ^? to Cr(III).     

An eigenfunction expansion of the non-selfadjoint Sturm–Liouville operator with a singular potential

In this paper, we consider the operator $L$ L generated in $L^{2}\left( \mathbb{R }_{+}\right) $ L 2 R + by the differential expression $$\begin{aligned} l\left( y\right) =-y^{\prime \prime }+\left[ \frac{\nu ^{2}-\frac{1}{4}}{x^{2}}+q\left( x\right) \right] y,\,\,x\in \mathbb{R }_{+}:=\left( 0,\infty \right) \end{aligned}$$ l y = - y ' ' + ν 2 - 1 4 x 2 + q x y , x ∈ R + : = 0 , ∞ and the boundary condition $$\begin{aligned} \underset{x\rightarrow 0}{\lim }x^{-\nu -\frac{1}{2}}y\left( x\right) =1, \end{aligned}$$ lim x → 0 x - ν - 1 2 y x = 1 , where $q$ q is a complex valued function and $\nu $ ν is a complex number with $Re\nu >0$ R e ν > 0 . We have proved a spectral expansion of L in terms of the principal functions under the condition $$\begin{aligned} \underset{x\in \mathbb{R }_{+}}{Sup}\left\{ e^{\epsilon \sqrt{x}}\left| q(x)\right| \right\} <\infty , \epsilon >0 \end{aligned}$$ S u p x ∈ R + e ? x q ( x ) < ∞ , ? > 0 taking into account the spectral singularities. We have also investigated the convergence of the spectral expansion.     

Elektrometrische Untersuchungen über die Geschwindigkeit der Bildung von Cu2S in ammoniakalischen Lösungen von Cu+-Ionen mittels Thioacetamid, 4. Mitt.

Quantitative studies of the rate of Cu₂S-formation by thioacetamide (TAA) were made with the help of the polarographic method of continuous registration at constant potential, and the following equation for the reaction rate between Cu⁺-ions andTAA in ammoniacal solutions was derived: 1 $$ - \frac{{d[Cu^I ]}}{{dt}} = k \cdot \frac{{[Cu^I ] \cdot [CH_3 CSNH_2 ]}}{{[NH_3 H_2 O]^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} \cdot [H^ + ]}}\frac{{^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}} }}{{^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {10}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${10}$}}} }} \cdot \frac{{f_{Cu} }}{{f_{H^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {10}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${10}$}}} } }}$$ The value at 25.0° of the rate constantk is (1.6±0.2)·10^?2 mole^7/20·litre^?7/20·sec^?1. The validity of equation (1) has been proved over the pH range 8.5–9.5 and the ammonia concentration of 4.0·10^?2–4.0·10^?1 mole per litre, by only a small excess ofTAA and moderate reaction rates.     

Specific heat capacity and thermodynamic properties of CuTeO3 and HgTeO3

Tellurites of CuTeO₃ and HgTeO₃ are synthesized and their specific molar heat capacities are experimentally determined for the first time. The tellurites discussed in the present paper are used for preparation of optical glasses with special properties for optoelectronics, nuclear and power industries. The tellurites synthesized are prepared for chemical analysis, differential thermal analysis and X-ray analysis. The use of the tellurites studied is related to knowing their thermodynamic properties like specific molar heat capacity (C _p,m), enthalpy \( \left( {\Delta_{{{\text {T}}^{\prime}}}^{\text{T}} H_{\text{m}}^{0} } \right), \) entropy \( \left( {\Delta_{{{\text {T}}^{\prime}}}^{\text{T}} S_{\text{m}}^{0} } \right) \) and Gibbs energy \( \left( { - \Delta_{{{\text {T}}^{\prime}}}^{\text{T}} G_{\text{m}}^{0} } \right) \) . The temperature dependences of their molar heat capacities are determined using the least squares method. The thermodynamic properties are calculated: entropy, enthalpy and Gibbs function.