A mathematical approach to chemical equilibrium theory for gaseous systems—III: \hbox {Q}_\mathrm{p}, \hbox {Q}_\mathrm{c}, and \hbox {Q}_{\mathrm{x}}

The reaction quotient Q can be expressed in partial pressures as $\hbox {Q}_\mathrm{P}$ or in mole fractions as $\hbox {Q}_{\mathrm{x}}$ . $\hbox {Q}_\mathrm{P}$ is ostensibly more useful than $\hbox {Q}_{\mathrm{x}}$ because the related $\hbox {K}_{\mathrm{x}}$ is a constant for a chemical equilibrium in which T and P are kept constant while $\hbox {K}_{\mathrm{P}}$ is an equilibrium constant under more general conditions in which only T is constant. However, as demonstrated in this work, $\hbox {Q}_{\mathrm{x}}$ is in fact more important both theoretically and technically. The relationships between $\hbox {Q}_{\mathrm{x}}$ , $\hbox {Q}_\mathrm{P}$ , and $\hbox {Q}_{\mathrm{C}}$ are discussed. Four examples of applications are given in detail.     

Volumetric Properties of the Nucleoside Thymidine in Aqueous Solution at T = 298.15 K and p = (10 to 100) MPa

Sound speeds have been measured for aqueous solutions of the nucleoside thymidine at T = 298.15 K and at the pressures p = (10, 20, 40, 60, 80, and 100) MPa. The partial molar volumes at infinite dilution, $ V_{2}^{\text{o}} $ , the partial molar isentropic compressions at infinite dilution, $ K_{S,2}^{\text{o}} $ , and the partial molar isothermal compressions at infinite dilution, $ K_{T,2}^{\text{o}} $ $ \{ K_{T,2}^{\text{o}} = - (\partial V_{2}^{\text{o}} /\partial p)_{T} \} $ , have been derived from the sound speeds at elevated pressures using methods described in our previous work. The $ V_{2}^{\text{o}} $ and $ K_{T,2}^{\text{o}} $ results were rationalized in terms of the likely interactions between thymidine and the aqueous solvent. The $ V_{2}^{\text{o}} $ results were also compared with those calculated using the revised Helgeson–Kirkham–Flowers (HKF) equation of state.     

The principal measure and distributional (p, q)-chaos of a coupled lattice system with coupling constant \varepsilon =1 related with Belusov–Zhabotinskii reaction

We consider the following system coming from a lattice dynamical system stated by Kaneko (Phys Rev Lett, 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f\left( x_{n}^{m}\right) +\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f\left( x_{n+1}^{m}\right) \right] , \end{aligned}$$ where $m$ is discrete time index, $n$ is lattice side index with system size $L$ (i.e., $n=1, 2, \ldots , L$ ), $\varepsilon \ge 0$ is coupling constant, and $f(x)$ is the unimodal map on $I$ (i.e., $f(0)=f(1)=0$ , and $f$ has unique critical point $c$ with $0<c<1$ and $f(c)=1$ ). In this paper, we prove that for coupling constant $\varepsilon =1$ , this CML (Coupled Map Lattice) system is distributionally $(p, q)$ -chaotic for any $p, q\in [0, 1]$ with $p\le q$ , and that its principal measure is not less than $\mu _{p}(f)$ . Consequently, the principal measure of this system is not less than $\frac{2}{3}+\sum _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}$ for coupling constant $\varepsilon =1$ and the tent map $\Lambda $ defined by $\Lambda (x)=1-|1-2x|, x\in [0, 1]$ . So, our results complement the results of Wu and Zhu (J Math Chem, 50:2439–2445, 2012).     

Acid–Base Properties of the \mathrm{Fe}(\mathrm{CN})_{6}^{3-}/\mathrm{Fe}(\mathrm{CN})_{6}^{4-} Redox Couple in the Presence of Various Background Mineral Acids and Salts

The acid?Cbase behavior of $\mathrm{Fe}(\mathrm{CN})_{6}^{4-}$ was investigated by measuring the formal potentials of the $\mathrm{Fe}(\mathrm{CN})_{6}^{3-}$ / $\mathrm{Fe}(\mathrm{CN})_{6}^{4-}$ couple over a wide range of acidic and neutral solution compositions. The experimental data were fitted to a model taking into account the protonated forms of $\mathrm{Fe}(\mathrm{CN})_{6}^{4-}$ and using values of the activities of species in solution, calculated with a simple solution model and a series of binary data available in the literature. The fitting needed to take account of the protonated species $\mathrm{HFe}(\mathrm{CN})_{6}^{3-}$ and $\mathrm{H}_{2}\mathrm{Fe}(\mathrm{CN})_{6}^{2-}$ , already described in the literature, but also the species $\mathrm{H}_{3}\mathrm{Fe}(\mathrm{CN})_{6}^{-}$ (associated with the acid?Cbase equilibrium $\mathrm{H}_{3}\mathrm{Fe}(\mathrm{CN})_{6}^{-}\rightleftharpoons \mathrm{H}_{2}\mathrm{Fe}(\mathrm{CN})_{6}^{2-} + \mathrm{H}^{+}$ ). The acidic dissociation constants of $\mathrm{HFe}(\mathrm{CN})_{6}^{3-}$ , $\mathrm{H}_{2}\mathrm{Fe}(\mathrm{CN})_{6}^{2-}$ and $\mathrm{H}_{3}\mathrm{Fe}(\mathrm{CN})_{6}^{-}$ were found to be $\mathrm{p}K^{\mathrm{II}}_{1}= 3.9\pm0.1$ , $\mathrm{p}K^{\mathrm{II}}_{2} = 2.0\pm0.1$ , and $\mathrm{p}K^{\mathrm{II}}_{3} = 0.0\pm0.1$ , respectively. These constants were determined by taking into account that the activities of the species are independent of the ionic strength.     

Volumetric and Viscometric Studies in Binary Liquid Mixtures of 2-Methyl-2-propanol with Some Ketones at Different Temperatures

Densities, ??, and viscosities, ??, of binary mixtures of 2-methyl-2-propanol with acetone (AC), ethyl methyl ketone (EMK) and acetophenone (AP), including those of the pure liquids, were measured over the entire composition range at 298.15, 303.15 and 308.15?K. From these experimental data, the excess molar volume $V_{\mathrm{m}}^{\mathrm{E}}$ , deviation in viscosity ????, partial and apparent molar volumes ( $\overline{V}_{\mathrm{m},1}^{\,\circ }$ , $\overline{V}_{\mathrm{m},2}^{\,\circ }$ , $\overline{V}_{\phi ,1}^{\,\circ}$ and $\overline{V}_{\phi,2}^{\,\circ} $ ), and their excess values ( $\overline{V}_{\mathrm{m},1}^{\,\circ \mathrm{E}}$ , $\overline{V}_{\mathrm{m,2}}^{\,\circ \mathrm{ E}}$ , $\overline {V}_{\phi \mathrm{,1}}^{\,\circ \mathrm{ E}}$ and $\overline{V}_{\phi \mathrm{,2}}^{\,\circ \mathrm{ E}}$ ) of the components at infinite dilution were calculated. The interaction between the component molecules follows the order of AP > AC > EMK.     

Synergistic extraction of europium and americium into nitrobenzene by using hydrogen dicarbollylcobaltate and bis(diphenylphosphino)methane dioxide

Extraction of microamounts of europium and americium by a nitrobenzene solution of hydrogen dicarbollylcobaltate (H⁺B^?) in the presence of bis(diphenylphosphino)methane dioxide (DPPMDO, L) has been investigated. The equilibrium data have been explained assuming that the species $ {\text{HL}}^{ + } $ , $ {\text{HL}}_{2}^{ + } $ , $ {\text{ML}}_{2}^{3 + } $ , $ {\text{ML}}_{3}^{3 + } $ and $ {\text{ML}}_{4}^{3 + } $ (M³⁺ = Eu³⁺, Am³⁺) are extracted into the organic phase. The values of extraction and stability constants of the species in nitrobenzene saturated with water have been determined. It was found that the stability constants of the corresponding complexes $ {\text{EuL}}_{n}^{3 + } $ and $ {\text{AmL}}_{n}^{3 + } $ , where n = 2, 3 and L is DPPMDO, in water–saturated nitrobenzene are comparable, whereas in this medium the stability of the cationic species $ {\text{AmL}}_{4}^{3 + } $ (L = DPPMDO) is somewhat higher than that of $ {\text{EuL}}_{4}^{3 + } $ with the same ligand L.     

Photoelectrochemistry of silicon in HF solution

Photoelectrochemical, photoelectrocatalytic, and electrochemical processes of silicon anodic oxidation and hydrogen evolution in aqueous HF solution are discussed in terms of thermodynamic stability of Si, oxides SiO, SiO₂, and Si surface hydrides. It is shown that photoelectrochemical oxidation of n-type low-resistivity silicon to SiO₂ is catalyzed by Si $^{+}$ photo-hole formation, whereas in the case of p-type Si, the feasibility of this reaction is predetermined by p-type conductivity. It is suggested that anodic oxidation of Si goes through the stage of SiO oxide formation and its subsequent oxidation to SiO₂. Such mechanism accounts for chemical inertness of Si phase in HF solutions as well as for selective, anisotropic, and isotropic etching of Si within E ranges from $-0.5$ to 0.35 V, $0.35-0.8~V,$ and $E > 0.8$ V, respectively. Hydrogen evolution reaction on Si surface proceeds at very large overpotential ( $\geq 0.5$ V) through the stage of surface Si hydride formation: $\mathrm {Si + H_{2}O + e^{-} \rightarrow (SiH)_{surf} + OH^{-}}$ (the rate determining step) and $\mathrm {(SiH)_{surf} + H_{2}O + e^{-} \rightarrow Si + H_{2} + OH^{-}}$ . Illumination-related effects of surface reactions relevant to selective and anisotropic etching and nano/micro-structuring of Si surface are discussed.     

A CASPT2//CASSCF study of the quartet excited state \tilde{a}^{4}A^{\prime\prime} of the HCNN radical

Complete active space self-consistent field and second-order multiconfigurational perturbation theory methods have been performed to investigate the quartet excited state ${\tilde{a}}^{4}{A^{\prime\prime}}$ potential energy surface of HCNN radical. Two located minima with respective cis and trans structures could easily dissociate to CH $({\tilde{a}}^{4}\Sigma^{-})$ and $N_{2} ({\tilde{X}}^{1}\Sigma_{\rm g}^{+})$ products with similar barrier of about 16.0 kcal/mol. In addition, four minimum energy crossing points on a surface of intersection between ${\tilde{a}}^{4}A^{\prime\prime}$ and X ( $X={\tilde{X}}^{2}A^{\prime\prime}$ and ${\tilde{A}}^{2}A^{\prime}$ ) states are located near to the minima. However, the intersystem crossing ${\tilde{a}}^{4}A^{\prime\prime} \rightarrow X$ is weak due to the vanishingly small spin–orbit interactions. It further indicates that the direct dissociation on the ${\tilde{a}}^{4}{A^{\prime\prime}}$ state is more favored. This information combined with the comparison with isoelectronic HCCO provides an indirect support to the recent experimental proposal of photodissociation mechanism of HCNN.     

Study of Malachite Green Fading in Water–Ethanol–Ethylene Glycol Ternary Mixtures

The rate constant of malachite green (MG⁺) alkaline fading was measured in water–ethanol–ethylene glycol ternary mixtures. This reaction was studied under pseudo-first-order conditions at 283–303 K. In each series of experiments, the concentration of ethanol was kept constant and the concentration of ethylene glycol was changed. It was shown that due to hydrogen bonding and hydrophobic interaction between MG⁺ and alcohol molecules the observed reaction rate constant, $ k_{\text{obs}} $ , increased in the water–ethanol–ethylene glycol ternary mixtures. The fundamental rate constants of MG⁺ fading in these solutions ( $ k_{1} $ , $ k_{ - 1} $ and $ k_{2} $ ) were obtained by the SESMORTAC model. Analysis of $ k_{1} $ and $ k_{2} $ values in solutions containing constant ethanol concentrations show that in low concentrations of ethylene glycol, hydrogen bonding formed between ethanol and ethylene glycol molecules and in high concentrations of ethylene glycol, ethanol as a solvent for ethylene glycol affected the reaction rate.     

Free ions Pr IV (4\text{ f }^{2}) and Tm IV (4\text{ f }^{12}): intermediate versus LS coupling scheme

The intermediate and LS-coupling schemes for the free lanthanide ions $\text{ Pr }^{3+}$ Pr 3 + and $\text{ Tm }^{3+}$ Tm 3 + have been compared by the matrix elements of the tensor operator ${{\varvec{U}}}^{({\varvec{k}})}, \text{ k } = 2, 4, 6$ U ( k ) , k = 2 , 4 , 6 . The necessary eigenvectors and eigenvalues have been computed with the aid of four parameters, $\text{ F }_{2}, \text{ F }_{4}, \text{ F }_{6}$ F 2 , F 4 , F 6 , and $\zeta _{4\mathrm{f}}$ ζ 4 f , known from free-ion spectra of the same ions. It has been found that both coupling types for each ion lead to close values of ${\vert }{{\varvec{U}}}^{({\varvec{k}})}{\vert }^{2}$ | U ( k ) | 2 only for transitions from the ground level to certain lower-lying energy levels within the $4\text{ f }^\mathrm{N}$ 4 f N configuration.     

Thermodynamic Analysis of Homologous α-Amino Acids in Aqueous Potassium Fluoride Solutions at Different Temperatures

Partial molal volumes ( $V_{\phi} ^{0}$ ) and partial molal compressibilities ( $K_{\phi} ^{0}$ ) for glycine, L-alanine, L-valine and L-leucine in aqueous potassium fluoride solutions (0.1 to 0.5?mol?kg^?1) have been measured at T=(303.15,308.15,313.15 and 318.15) K from precise density and ultrasonic speed measurements. Using these data, Hepler coefficients ( $\partial^{2}V_{\phi} ^{0}/\partial T^{2}$ ), transfer volumes ( $\Delta V_{\phi} ^{0}$ ), transfer compressibilities ( $\Delta K_{\phi} ^{0}$ ) and hydration number (n _H) have been calculated. Pair and triplet interaction coefficients have been obtained from the transfer parameters. The values of $V_{\phi} ^{0}$ and $K_{\phi} ^{0}$ vary linearly with increasing number of carbon atoms in the alkyl chain of the amino acids. The contributions of charged end groups ( $\mathrm{NH}_{3}^{+}$ , COO^?), CH₂ group and other alkyl chains of the amino acids have also been estimated. The results are discussed in terms of the solute?Ccosolute interactions and the dehydration effect of potassium fluoride on the amino acids.     

Compound-specific isotope analysis of benzotriazole and its derivatives

Compound-specific isotope analysis (CSIA) is an important tool for the identification of contaminant sources and transformation pathways, but it is rarely applied to emerging aquatic micropollutants owing to a series of instrumental challenges. Using four different benzotriazole corrosion inhibitors and its derivatives as examples, we obtained evidence that formation of organometallic complexes of benzotriazoles with parts of the instrumentation impedes isotope analysis. Therefore, we propose two strategies for accurate $\delta^{13}$ C and $\delta^{15}$ N measurements of polar organic micropollutants by gas chromatography isotope ratio mass spectrometry (GC/IRMS). Our first approach avoids metallic components and uses a Ni/Pt reactor for benzotriazole combustion while the second is based on the coupling of online methylation to the established GC/IRMS setup. Method detection limits for on-column injection of benzotriazole, as well as its 1-CH $_{3}$ -, 4-CH $_{3}$ -, and 5-CH $_{3}$ -substituted species were 0.1–0.3 mM and 0.1–1.0 mM for δ¹³C and δ¹⁵N analysis respectively, corresponding to injected masses of 0.7–1.8 nmol C and 0.4–3.0 nmol N, respectively. The Ni/Pt reactor showed good precision and was very long-lived ( $>$ 1000 successful measurements). Coupling isotopic analysis to offline solid-phase extraction enabled benzotriazole-CSIA in tap water, wastewater treatment effluent, activated sludge, and in commercial dishwashing products. A comparison of $\delta ^{13}$ C and $\delta ^{15}$ N values from different benzotriazoles and benzotriazole derivatives, both from commercial standards and in dishwashing detergents, reveals the potential application of the proposed method for source apportionment.     

The Formation and Stability of the [FePy3Cl3]· Py Clathrate in the Pyridine–Iron(III) Chloride System: Phase Diagram and Solid–Gas Equilibria Study

The phase diagram of the pyridine–iron(III) chloride system has been studied for the 223–423 K temperature and 0–56 mass-% concentration ranges using differential thermal analysis (DTA) and solubility techniques. A solid with the highest pyridine content formed in the system was found to be an already known clathrate compound, [FePy₃Cl₃]·Py. The clathrate melts incongruently at 346.9 ± 0.3 K with the destruction of the host complex: [FePy₃Cl₃]·Py_(solid)=[FePy₂Cl₃]_(solid) + liquor. The thermal dissociation of the clathrate with the release of pyridine into the gaseous phase (TGA) occurs in a similar way: [FePy₃Cl₃]·Py_(solid)=[FePy2Cl₃]_(solid) + 2 Py_(gas). Thermodynamic parameters of the clathrate dissociation have been determined from the dependence of the pyridine vapour pressure over the clathrate samples versus temperature (tensimetric method). The dependence experiences a change at 327 K indicating a polymorphous transformation occurring at this temperature. For the process ${1 \over 2}[\hbox{FePy}_{3}\hbox{Cl}_{3}]\cdot \hbox{Py}_{\rm (solid)} = {1 \over 2}[\hbox{FePy}_{2}\hbox{Cl}_{3}]_{\rm (solid)} + \hbox{Py}_{\rm (gas)}$ in the range 292–327 K, ΔH $^{0}_{298}$ =70.8 ± 0.8 kJ/mol, ΔS $^{0}_{298}$ =197 ± 3 J/(mol K), ΔG $^{0}_{298}$ =12.2 ± 0.1 kJ/mol; in the range 327–368 K, ΔH $^{0}_{298}$ =44.4 ± 1.3 kJ/mol, ΔS $^{0}_{298}$ =116 ± 4 J/(mol K), ΔG $^{0}_{298}$ =9.9 ± 0.3 kJ/mol.     

Oxygen dissolution in polycrystalline palladium at O2 pressures of 0.1 to 100 Pa

Oxygen dissolution in polycrystalline palladium Pd(poly) at O₂ pressures ( $P_{O_2 } $ ) of 0.1 to 100 Pa and a temperature of 600 K has been investigated by temperature-programmed desorption. The dissolution process under these conditions includes O₂ chemisorption on the oxide film surface, the insertion of Oads atoms under the oxide layer, and their diffusion into the subsurface layers of palladium. During chemisorption, a structure ensuring that the O_ads coverage of the surface increases with increasing $P_{O_2 } $ forms on the surface of the oxide film. This is favorable for O_ads penetration through the oxide film and increases the amount of absorbed oxygen. The O_ads coverage of the surface calculated via the Langmuir equation at an O₂ desorption activation energy of E _des = 125 kJ/mol correlates with the number of absorbed oxygen monolayers (n). At n ≥ 1, oxygen absorption by Pd(poly) is due to the diffusion of O atoms in the palladium lattice. After the accumulation of 14–18 oxygen monolayers in the subsurface layers of palladium, oxygen absorption practically stops depending on $P_{O_2 } $ . Thus, the acceleration of oxygen dissolution in palladium is due to the formation of the surface oxide film and the increase in the O_ads coverage of this film, which facilitates the insertion of O_ads atoms into the subsurface layers of palladium.     

A note on Li–Yorke chaos in a coupled lattice system related with Belusov–Zhabotinskii reaction

This paper is concerned with the following system which comes from a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) and is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f(x_{n}^{m})+\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f(x_{n+1}^{m})\right] , \end{aligned}$$ x n m + 1 = ( 1 ? ε ) f ( x n m ) + 1 2 ε [ f ( x n ? 1 m ) + f ( x n + 1 m ) ] , where $m$ m is discrete time index, $n$ n is lattice side index with system size $L$ L (i.e., $n=1, 2, \ldots , L$ n = 1 , 2 , … , L ), $\varepsilon $ ε is coupling constant, and $f(x)$ f ( x ) is the unimodal map on $I$ I (i.e., $f(0)=f(1)=0$ f ( 0 ) = f ( 1 ) = 0 and $f$ f has unique critical point $c$ c with $0<c<1$ 0 < c < 1 and $f(c)=1$ f ( c ) = 1 ). It is proved that for coupling constant $\varepsilon =1$ ε = 1 , this CML (Coupled Map Lattice) system is chaotic in the sense of Li–Yorke for each unimodal selfmap on the interval $I=[0, 1]$ I = [ 0 , 1 ] .     

Symmetry-itemized enumeration of quadruplets of RS-stereoisomers: I—the fixed-point matrix method of the USCI approach combined with the stereoisogram approach

The RS-stereoisomeric group $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ is examined to characterize quadruplets of RS-stereoisomers based on a tetrahedral skeleton and found to be isomorphic to the point group $\mathbf{O}_{h}$ of order 48. The non-redundant set of subgroups (SSG) of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ is obtained by referring to the non-redundant SSG of $\mathbf{O}_{h}$ . The coset representation for characterizing the orbit of the four positions of the tetrahedral skeleton is clarified to be $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$ , which is closely related to the $\mathbf{O}_{h}(/\mathbf{D}_{3d})$ . According to the unit-subduced-cycle-index (USCI) approach (Fujita in Symmetry and combinatorial enumeration in chemistry. Springer, Berlin, 1991), the subdution of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$ is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). The fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ . After the subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.     

Heat capacity and density of solutions of calcium and cadmium nitrates in N-methylpyrrolidone at 298.15 K

The heat capacity and density of solutions of calcium and cadmium nitrates in N-methylpyrrolidone (MP) at 298.15 K are studied by calorimetry and densimetry. The obtained data are discussed in relation to certain features of solvation and complex formation in solutions of these salts. The standard partial molar heat capacities and volumes ( $\overline {C_{p^2 }^0 }$ and $\overline {V_2^0 }$ ) of the electrolytes in MP are calculated. The standard heat capacities $\overline {C_{p^i }^0 }$ and volumes $\overline {V_i^0 }$ of Ca²⁺ and Cd²⁺ ions in MP at 298.15 K were determined, along with the contribution from specific interactions to the values of $\overline {C_{p^i }^0 }$ and $\overline {V_i^0 }$ of Cd²⁺ ions in MP solution.     

A note on the principal measure and distributional ({\varvec{p, q}})-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction

García Guirao and Lampart in (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this paper, we prove that for any non-zero coupling constant $\varepsilon \in (0, 1)$ , this coupled map lattice system is distributionally $(p, q)$ -chaotic for any pair $0\le p\le q\le 1$ , and that its principal measure is not less than $(1-\varepsilon )\mu _{p}(f)$ . Consequently, the principal measure of this system is not less than $$\begin{aligned} (1-\varepsilon )\left( \frac{2}{3}+\sum \limits _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}\right) \end{aligned}$$ for any non-zero coupling constant $\varepsilon \in (0, 1)$ and the tent map $\Lambda $ defined by $$\begin{aligned} \Lambda (x)=1-|1-2x|,\quad x\in [0, 1]. \end{aligned}$$      

Correlation consistent, Douglas–Kroll–Hess relativistic basis sets for the 5p and 6p elements

The diatomic carbon molecule has a complex electronic structure with a large number of low-lying electronic excited states. In this work, the potential energy curves (PECs) of the four lowest lying singlet states ( $X^{1} \Sigma^{ + }_{g}$ , $A^{1} \Pi_{u}$ , $B^{1} \Delta_{g}$ , and $B^{\prime1} \Sigma^{ + }_{g}$ ) were obtained by high-level ab initio calculations. Valence electron correlation was accounted for by the correlation energy extrapolation by intrinsic scaling (CEEIS) method. Additional corrections to the PECs included core–valence correlation and relativistic effects. Spin–orbit corrections were found to be insignificant. The impact of using dynamically weighted reference wave functions in conjunction with CEEIS was examined and found to give indistinguishable results from the even weighted method. The PECs showed multiple curve crossings due to the $B^{1} \Delta_{g}$ state as well as an avoided crossing between the two $^{1} \Sigma^{ + }_{g}$ states. Vibrational energy levels were computed for each of the four electronic states, as well as rotational constants and spectroscopic parameters. Comparison between the theoretical and experimental results showed excellent agreement overall. Equilibrium bond distances are reproduced to within 0.05 %. The dissociation energies of the states agree with experiment to within ~0.5 kcal/mol, achieving “chemical accuracy.” Vibrational energy levels show average deviations of ~20 cm^?1 or less. The $B^{1} \Delta_{g}$ state shows the best agreement with a mean absolute deviation of 2.41 cm^?1. Calculated rotational constants exhibit very good agreement with experiment, as do the spectroscopic constants.     

Free-radical chain oxidation of 1,4-dioxane and styrene in the presence of fullerene C60

The antiradical activity of fullerene C₆₀ was studied for the oxidation of 1,4-dioxane and styrene initiated by azobisisobutyronitrile and benzoyl peroxide as model reactions. The effective rate constants of the reaction of peroxyl radicals with fullerene C₆₀ (k ₇) and the stoichiometric inhibition factor (f _eff) were determined in air ( $P_{O_2 }$ = 0.21 atm) and oxygen ( $P_{O_2 }$ = 1.0 atm). The rate of the liquid-phase oxidation of 1,4-dioxane does not depend on $P_{O_2 }$ , and the effective rate constant of inhibition is k ₇ = (2.4 ± 0.2) × 10⁴ L mol^?1 s^?1. Chain termination in the oxidation of styrene occurs when C₆₀ reacts with both the peroxyl radicals (k ₇ = (1.2 ± 0.1) × 10³ L mol^?1 s^?1) and alkyl (k ₈ = 1.07 × 10⁷ L mol^?1 s^?1) radicals.