On Two Different Objectives of the Concepts of Ionic Radii

Experimentally and theoretically derived interatomic distances (D) and ionic radii (R) of more than a hundred monomeric (AX), dimeric (A₂X₂, ABXY), and crystalline ([AX]) alkali halide species (A=Li, Na, K, Rb, Cs, Fr; X=H, F, Cl, Br, I, At) have been analyzed. Chemists use the word “atomic radius” for two antithetic concepts. Let D_CiEE′jj′ be the “billion” of distances i between two adjacent atoms in the millions of known compounds C from a hundred different elements E in bonding states j. The common chemical aim is partitioning D approximately into increments R_Ej+R_E′j′. This can be achieved with a few (say <thousand) predictive constants R_Ej. An antipodal aim is specifying in hindsight an electron density feature in the “billion” of different bonds i, by partitioning them into “two billions” of exact bonded radii ${R{{{\rm C}i\hfill \atop {\rm E}\hfill}}}$ +${R{{{\rm C}i\hfill \atop {\rm E}{^\prime}\hfill}}}$ . The constant incremental and the variable bonded radii concepts with the same generic name are useful in different fields of research. Different concepts should be well distinguished, since they have different meaning, different numerical values, and different purposes.     

Reaction of O(1D) with N2O

O(¹D), produced from the photolysis of N₂O at 2139 Å, reacts with N₂O in accord with: We have used the method of chemical difference to obtain an accurate measure of k₂/k₃ = 0.59 ± 0.01. Furthermore, the quantum yield of production of O(³P), either on direct photolysis or on deactivation of O(¹D) by N₂O, is less than 0.02 and probably zero.     

Configuration-Dependent properties of the poly(dimethylsilmethylene) chain in the third-order interaction approximation

On the basis of rotational isomeric state theory, first-order, second-order, and third-order conformation energies E_σ, E_ω and E_φi respecively, are calculated for poly(dimethylsilmethylene) (CH₂—Si(CH₃)₂)_x using the Lennard–Jones potential function. With the third-order interaction included, the characteristic ratios and temperature coefficients 〈R²〉₀ and 〈μ²〉₀ are obtained: These results are in satisfactory agreement with the experimental data previously reported. © 1993 John Wiley & Sons, Inc.     

On the Randić Index of Quasi-tree Graphs

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertex u and v in G. A graph G is called quasi-tree, if there exists such that G − u is a tree. In the paper, we give sharp lower and upper bounds on the Randić index of quasi-tree graphs. Mei Lu: Partially supported by NSFC (No. 10571105).     

A möbius inversion of the ulam subgraphs conjecture

The generalized Eichinger matrices are defined asE = _j ⁿ ₁( _j S ^T S)^–1, where _j M denotes the matrixM withj th row and column deleted.S is the incidence matrix andM ^T is the transposed matrix. The conjectureS ^T SE = S _K ^TS _K , where SK is the incidence matrix of the complete graph, is proven for trees, simple cycles and complete graphs. The consequence of the conjecture isS _G ^T S _G (E _G -I) = S _G ^TS _G , whereG is the complementary graph ofG. It leads to graphs with imaginary arcs as the complements of graphs with multiple arcs.     

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.     

On Unicycle Graphs with Maximum and Minimum Zeroth-order Genenal Randić Index

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as R _α⁰(G) = ∑ _v∈V(G) d _v ^α where α is an arbitrary real number. In this paper, we obtained the lower and upper bounds for the zeroth-order general Randić index R _α⁰(G) among all unicycle graphs G of order n. We give a clear picture for R _α⁰(G) of unicycle graphs according to real number α in different intervals.     

On the Randić Index of Unicyclic Conjugated Molecules

The Randić index R(G) of a graph G is the sum of the weights of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on the Randić index of unicyclic graphs is given in terms of the order and given size of matching.     

Harmonic oscillations in a polymerization

A simple radical polymerization is proposed in this paper, with step‐by‐step chain growth (R_i + M → R_i+1), and termination by transfer to a third body (R_i + S → polymer) such as the solvent. It is assumed that, for a certain critical degree of polymerization n, the propagator R_n reacts with substrate H to produce a deactivator (V) of the first propagator (H + R_n → R_n + V; V + R₁ → P₁) R₁. Assuming that monomer, M, and precursor concentrations are constant, and assuming that the deactivator reaches fast a steady state, the resulting kinetic equations are formally linear, but they admit, perturbations r_j(t) of the steady‐state concentrations of the propagators R₁, R₂, …, R_n, which are periodic functions of time. Even more, they can be purely sinusoidal functions (which have been called “harmonic,” in analogy to the solutions of the well‐known classical harmonic oscillator) with phase shift between perturbations r_j(t) = R_j? (R_j)₀ and r_j+1(t) = R_j+1 ? (R_j+1)₀. Based on these periodic solutions and aiming to a model as simple as possible, a theoretical analysis is made, resulting in that the minimum value for n would be n = 3. Of course, these harmonic oscillations “driven by trimer” are equally found in the group of all the remaining propagators with polymerization degree higher than 3 (variable Y = ∑ R_j). © 2009 Wiley Periodicals, Inc. Int J Chem Kinet 41: 507–511, 2009     

Kirchhoff index of linear hexagonal chains

The resistance distance r_ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this work, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of linear hexagonal chain L_n consists of the Laplacian spectrum of path P_2n+1 and eigenvalues of a symmetric tridiagonal matrix of order 2n + 1. By applying the relationship between roots and coefficients of the characteristic polynomial of the above matrix, explicit closed‐form formula for Kirchhoff index of L_n is derived in terms of Laplacian spectrum. To our surprise, the Krichhoff index of L_n is approximately to one half of its Wiener index. Finally, we show that holds for all graphs G in a class of graphs including L_n. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Background elimination in three-dimensional spectroscopy using the intensian method

The method to eliminate background in the case of quantitative multidimensional spectroscopy, chromatography or any analytical 3-dimensional technique is shown. The 3-dimensional signal is required to be proportional to the concentration of determined substance and the additivity of signals should be obeyed. Eliminated background is assumed to be a low-order polynomial of two variables. The intensian method [1] is a generalization of the Beer-Lambert law, where a certain determinant called intensian replaces absorption and absorptivity. In practice there will be no need to use determinants, since usually they are replaced by expressions of few terms. Some details on the practical use of the method are given.Index of used symbols x, y UV, IR, GC, NMR or other scale. - A (x, y) intensity (absorption) of the 3-dimensional band of interest. - a (x, y) standard intensity (absorption) of the 3-dimensional band of interest. - B(x, y) intensity (absorption) of 3-dimensional background. - S(tx, y) intensity (absorption) of 3-dimensional multicomponent spectrum. - f (x, y) auxiliary function:f (x, y) =A (x, y), B(x, y), a(x, y), S(x, y). - (x _i, Yi) selected point,i positive integer number. - f(xi, yi) value off (x, y) in point (xi, yi). - S _i value ofS(x, y) in point (x _i, y_i). - b pathlength, measurement coefficient,c concentration. - , , , real numbers. - ij, _ij real coefficients of power expansions. - x ⁱy^j monomial of degree (i +j). - F(·) linear functional acting on 3-dimensional spectral functions. - J^3-dim(·) 3-dimensional intensian acting on 3-dimensional spectral functions. - J _n ^3-dim (·) 3-dimensionaln-points intensian. - d _i ith intensian coefficient, cofactor of expansion of J _n ^3-dim (f(x, y)) according to its first row (eq. (10)). - (.) absolute error. - r _i,, R random variables: eqs. (13) and (14). - G(.) (normal) distribution function. - z ordinate axis. - a - f , h abreviations for some arguments. - d _ijk,D _mnp abreviations defined in eq. (22).     

Temperature and shear dependencies of rheology of poly(acrylonitrile‐co‐itaconic acid) dope in DMF

Poly(acrylonitrile‐co‐itaconic acid) (poly(AN‐co‐IA)) precursor required for carbon fiber production is made into a dope and spun into fibers using a suitable spinning technique. The viscosity of the resin dope is decided by the polymer concentration, polymer molecular weight, temperature, and shear force. The shear rheology of concentrated poly(AN‐co‐IA) polymer solutions in N,N‐dimethylformamide (DMF), in the range of 1 × 10⁵–1 × 10⁶ g mol^?1, has been investigated in the shear rate (γ′) range of 1 × 10¹–5 × 10⁴ min^?1. The zero shear viscosity (η₀) has been evaluated at different temperatures. The temperature dependence of zero shear viscosity conformed to the Arrhenius–Frenkel–Eyring model. The free energy of activation of viscous flow (ΔG_V) values were in the range 5–32 kJ mol^?1 and this value increased with increase in polymer concentration and molecular weight. A master equation for the ΔG_V value of the polymer solution of any and concentration (c) is suggested. The power law fitted well for the shear dependency of viscosity of these polymer solutions. The pseudoplasticity index (n) diminished with increase in polymer concentration and molecular weight. An empirical relation between viscosity (η) and was found to exist at constant shear rate, concentration and temperature. For each , the equation relating n, c, and T was established. Copyright © 2008 John Wiley & Sons, Ltd.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

Kinetics of Oxygen Exchange in the System BrO - H2O (D2O)

The kinetics of oxygen exchange between water (H₂O, D₂O) and ¹⁸O-labelled bromate ion has been investigated over the range of 1.7 ≤ pH ≤ 14.3 and 20 ≤ °C ≤ 95. At 60° and ionic strength I ? 1.0M (NaNO₃), the experimental results were consistent with the rate laws (R in moll^?1 s^?1): From the temperature dependence of the rate constants the activation parameters ΔH^≠, ΔS^≠ and ΔC^≠ were derived. In the acid-catalysed region the form of the rate law and the direction of the solvent isotope effect were the same as previously found, but the numerical values of ΔH^≠ and k_2H/k_2D differ considerably. For the spontaneous and the OH^?-catalysed exchange reactions bimolecular displacement mechanisms are proposed.     

Relation between refractive index and density of polymer solutions

The relation between the fractive index n and the density ρ of a liquid mixture is formulated as where w_i and Rs_i are the weight fraction and the specific refraction, respectively, of component i. The calculation of the specific refraction of straight-chain polyethylene and polystyrene from data for pure compounds of low molecular weight is discussed. The result is applied to dilute solutions of polystyrene in toluene. The calculated values of (dp/dw)₀ and (d²ρ/dw²)₀ at three different temperatures are compared with measured values. The agreement is satisfactory.     

Theoretical treatment of predissociation of the CO (3sσ) B and (3pσ) C1Σ+ Rydberg states based on a rigorous adiabatic representation

Ab initio multireference configuration interaction calculations for adiabatic potential curves, nonadiabatic couplings 〈φ _i (R,r)|d/dR|φ _j (R,r)〉 and 〈φ _i (R,r)|d²/dR ²|φ _j (R,r)〉, and nuclear kinetic energy corrections 〈dφ _i (R,r)/dR|dφ _i (R,r)/dR〉 for the (3sσ) B and (3pσ) C¹Σ⁺ Rydberg states of the CO molecule have been carried out. The energy positions and predissociation linewidths for the observed vibrational levels of these two states have been determined in a rigorous adiabatic representation by the complex scaling method employing a basis of complex scaled harmonic vibrational functions in conjunction with the Gauss-Hermite quadrature method to evaluate the complex Hamiltonian matrix elements. The present treatment correctly reproduces the observed trends in energies and line broadening for vibrational levels of the B¹Σ⁺ state and represents an improvement over the previous treatment in literature. The errors in the determined spacings of the v = 0–4 vibrational levels of the C¹Σ⁺ state are less than 2% compared with measured data. The predissociation linewidths for the v=3,4 levels of the C¹Σ⁺ state are found to be 4.9 and 8.9 cm⁻¹, respectively, in good agreement with the observed values. Received: 23 March 1998 / Accepted: 27 July 1998 / Published online: 9 October 1998     

Graphical Models of Characters of Groups

A graphical method of generating one- and (some) two-dimensional characters () has been developed on the basis of a reduced homomer set, which has been derived from a new concept of negative graphs. Thus, a homomer set H[G(/G _i )]={h₁,...,h _d–1,h _d } (d=|G|/|G _i |) has been generated from a regular body of G so that it has been governed by the coset representation G(/G _i ). The homomer set has been reduced into a reduced homomer set []={h₁,...,h _d–1}, where we have placed h _d –(h₁++h _d–1) in terms of negative graphs. The action of the symmetry operations of G on the reduced homomer set [] has graphically generated a one- or (some) two-dimensional character (). The versatility of the graphical method has been tested by using C _3v , D _2h , C _2h , C _2v , D _3h , and C _3h as examples. The graphical method has been compared with an alternative algebraic generation using marks (or markaracters), i.e., =G(/G _i )–G(/G).     

Non-linear transformations of factors in the generalized standard addition method

Summary Experimental designs and non-linearly transformed factors (concentrations of analytes) have been applied in the Generalized Standard Addition Method (GSAM). The relationship betweenN measured analytical signals,R ⁽¹⁾,...,R su(n), and concentrations of standard additions, c1,..., cn, ofn analytes has been approximated by the following polynomial models:The regression coefficientsB in models (1) are calculated from thesimple formulae^{1, 3, 4}.Functionsf _i ^(l) (e.g. linear, parabolic, exponential, hyperbolic or logarithmic) are selected on the basis of the experimentally determined relationshipsR ^(l) (l=1, ...,N) vs.c _i (i=1, ...,n). An example concerns the flame emission photometric determination of Na and Ca. 2² factorial and uncomplete second-degree modelsR ^(l) have been applied expressed by linear and hyperbolic functionsf _i ^(l). The results of determination of Na by GSAM reveal significant improvement of accuracy as compared with the conventional single-component standard addition method.

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Nicht-lineare Transformation von Faktoren bei der verallgemeinerten Standardadditio nsmethode

Zusammenfassung Statistische Versuchsplanung und nicht-linear transformierte Faktoren (Konzentrationen der zu bestimmenden Substanzen) wurden für die verallgemeinerte Standardadditionsmethode eingesetzt. Die Bezeichnung zwischen N gemessenen analytischen Signalen,R ⁽¹⁾,...,R ^(N) und den Konzentrationen der Standardadditionszugaben C_i,..., C ^N , vonn zu bestimmenden Substanzen wurde durch Polynome des folgenden Typs approximiert: (wobei ). Die Funktionenf _i ^(l) (z. B. lineare, parabolische, exponentielle, hyperbolische oder logarithmische) werden auf Basis experimentell bestimmter Abhängigkeiten festgelegtR ^(l) (l=1,...,N) gegen C _i (l=1,...,N). Als Beispiel wird die Flammenemissionsbestimmung von Na und Ca herangezogen. Dabei wurden ein 2² faktorieller Versuchsplan und ein unvollständiges Polynom der Ordnung 2R ^(l) mit linearen und hyperbolischen Funktionenf _i ^(l) angesetzt. Die Resultate der Na-Be-stimmung zeigen durch die Verwendung der verallgemeinerten Standard-addition signifikante Verbesserungen der Resultate gegenüber der Einkomponenten-Standardaddition.

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The author thanks Doc. dr. hab. A. Parczewski for his helpful discussion and editorial comments during the preparation of this paper.     

Sterisch behinderte Rotation in monosubstituierten Ansa-Verbindungen

The barriers to rotation about the pivot bonds in with n = 8 or 10 and R = CH₂C₆H₅, CH₂COOH, CH₂COOCH₃ or CH(CH₃)₂ were studied. ΔG^≠ values were found to be > 24 kcal. mol^?1. The solvent, the temperature, and the concentration dependence of the double magnetically non-equivalent methylenic protons and methyl groups were studied.