Volume 60, Issue 1-2

A Riccati equation rational expansion (RERE) method is presented by a general ansatz, which is a direct and unified algebraic method for constructing multiple and more general travelling wave solutions of nonlinear partial differential equations. It can be implemented in a computer algebraic system. The proposed method is applied to consider the shallow long wave approximation equation and leads to a rich variety of exact solutions, including rational form solitary wave solutions, triangular periodic wave solutions and rational wave solutions.

In this paper, we find travelling wave solutions of some nonlinear partial differential equations (PDEs) by using the extended tanh-function method. Some illustrative equations are investigated by this method and new travelling wave solutions are found. In addition, the properties of these nonlinear PDEs are shown with some figures.

Several new nonclassical potential symmetry generators to the Burgers equation are derived. Some explicit solutions, which cannot be derived from the Lie symmetry group of Burgers or its adjoined equation, are obtained by using these nonclassical potential symmetry generators.

The thermodynamic quantities of Lennard-Jones gases, evaluated till the fourth virial coefficient, are investigated for an isobaric process. A partition function in the T-P grand canonical ensemble Y(T,P,N) may be defined by the Laplace transform of the partition function Z(T,V,N) in the canonical ensemble. The Gibbs free energy is related with Y(T,P,N) by the Legendre transformation G(T,P,N) = −kT logY(T,P,N). The volume, enthalpy, entropy, and heat capacity are analytically expressed as functions of the intensive variables temperature and pressure. Some critical thermodynamic quantities for Xe are calculated and drawn. At the critical point the heat capacity diverges to infinity, while the Gibbs free energy, volume, enthalpy, and entropy are continuous. This suggests that a second-order phase transition may occur at the critical point.

The transformation group theoretic approach is applied to the problem of the flow of an electrically conducting incompressible viscoelastic fluid near the forward stagnation point of a heated plate. The application of one-parameter transformation group reduces the number of independent variables, by one, and consequently the basic equations governing flow and heat transfer are reduced to a set of ordinary differential equations. These equations have been solved approximately subject to the relevant boundary conditions by employing the shooting numerical technique. The effect of the magnetic parameter M, the Prandtl number Pr and the non-dimensional elastic parameter representing the non- Newtonian character of the fluid k on velocity field, shear stress, temperature distribution and heat flux are carefully examined.

The electronic structure and strength of acid-base bonding in the protonated boron-Lewis complexes BH n=2,4 X + (X = NH 3 , PH 3 , H 2 O, H 2 S) and their neutral parents were studied. The results showed that in the H2BX+ monocations the electron releasing from the donor atom in X to BH 3 is significant and more complete than in the other studied complexes. Therefore the bonding between the donor and acceptor in H2BX+ is strongest. To obtain these data, the electric field gradient (EFG) at the quadrupolar nuclei in each complex was calculated. The EFG’s of the boron atom and other quadrupolar nuclei were calculated by the Gaussian 98 program, using the MP2/6-31G** method.

The results of the interaction of two 60-fullerene molecules interaction under external pressure, studied by a semi-empirical PM3 quantum chemical method, are reported. A set of 15 space structures of 60-fullerene dimers from the simplest one up to partially graphitized material has been simulated. Calculated pressures referring to the dimers’ formation reproduce the experimental order of magnitude rather well. The dependences of the heat of formation and the force ballancing the applied pressure versus compression have been determined. A mechanism of the dimerization is proposed.

MINDO-Forces calculations with complete geometry optimization have been performed on nitromethane, aci-nitromethane and X-substituted nitromethane and aci-nitromethane (X = F, OH, NH 2 , CH 3 , CN, CF 3 , NO 2 , CHO). It is found that nitromethane is more stable than aci-nitromethane by 9.337 kcal/mol. This agrees with theoretical calculations. Thermodynamically, substituted aci-nitro tautomers are more stable than the corresponding nitromethane, except in case of the substituent F. Geometrical parameters, heats of formation, electron densities, Gibbs free energies and isodesmic reactions are reported.

In the present paper we report on an analysis of the fine structure of the first excited quartet 4 T 2g of Mn 4+ ions which occupy the octahedral site in the Cs 2 GeF 6 host crystal. The dynamic 4 T 2g ⊗(e g +t 2g ) Jahn-Teller effect is considered in details, including the Ham effect of the reduction of the spin-orbit splitting and displacements of the ligands due to the combined effect of the a1g and eg normal modes of the [MnF 6 ] 2− octahedral complex. The electron-phonon coupling constants are evaluated using the experimental spectroscopic data. The value of the Jahn-Teller stabilization energy EJT = 438 cm −1 for the considered complex is estimated from both the Ham effect and the potential energy surface of the 4 T 2g excited state.

A thermodynamic model for the partition of a solute (pesticide) between two immiscible phases, such as the stationary and mobile phases of supercritical fluid chromatography with CO 2 , is developed from first principles. A key ingredient of the model is the result of the calculation made by Liptay of the energy of interaction of a polar molecule with a dielectric continuum, which represents the solvent. The strength of the interaction between the solute and the solvent, which may be considered a measure of the solvent power, is characterized by a function g = (ε − 1)/(2ε +1), where ε is the dielectric constant of the medium, which is a function of the temperature T and the pressure P. Since the interactions between the nonpolar supercritical CO 2 solvent and the slightly polar pesticide molecules are considered to be extremely weak, a regular solution model is appropriate from the thermodynamic point of view. At constant temperature, the model predicts a linear dependence of the logarithm of the capacity factor (lnk) of the chromatographic experiment on the function g = g(P), as the pressure is varied, with a slope which depends on the dipole moment of the solute, dispersion interactions and the size of the solute cavity in the solvent. At constant pressure, once the term containing the g (solvent interaction) factor is subtracted from lnk, a plot of the resulting term against the inverse of temperature yields the enthalpy change of transfer of the solute from the mobile (supercritical CO 2 ) phase to the stationary (adsorbent) phase. The increase in temperature with the consequent large volume expansion of the supercritical fluid lowers its solvent strength and hence the capacity factor of the column (or solute retention time) increases. These pressure and temperature effects, predicted by the model, agree excellently with the experimental retention times of seven pesticides. Beyond a temperature of about 393 K, where the liquid solvent densities approach those of a gas (and hence the solvent strength becomes negligible), a dramatic loss of the retention times of all pesticides is observed in the experiments; this is attributed to desorption of the solute from the stationary phase, as predicted by Le Châtelier’s principle for the (exothermic) adsorption process.

Ultrasonic velocities and densities of binary mixtures of acetonitrile (AN) and n-butyronitrile (n-BTN) with Bu 4 NBPh 4 , Bu 4 NClO 4 , Bu 4 NI, Bu 4 NBr, Pr 4 NBr, Et 4 NI, Et 4 NBPh 4 , NaBPh 4 , NaClO 4 and CuClO 4 have been measured in the concentration range 0.0045 - 0.2 mol kg −1 over the entire composition range at 298.15 K. Isentropic compressibilities (Ks) and apparent molal isentropic compressibilities (Ks,ø ) have been calculated. Limiting apparent molal isentropic compressibilities (K o s ,ø ) have been evaluated and split into the contribution of individual ions, i. e. into (K o s ,ø )± values. (K o s ,ø )± for Cu + , Na + and Br − in all cases is negative and large, for Bu 4 N + and Ph 4 B − positive and large, and for I − and ClO 4 − negative only in AN, and becomes positive in binary mixtures of AN and n-BTN for all compositions. The negative (K o s ,ø )± values for Cu + , Na + and Br − indicate strong ionsolvent interactions involving electrostatic ion-dipole interactions. The large and positive (K o s ,ø )± values for Bu 4 N + and Ph 4 B − show some special type of ion-solvent interactions (hydrophobic or dispersion interactions) with AN and n-BTN. The small negative value changing to a small positive value for I − and ClO 4 − indicates the change of a weak electrostatic ion-solvent interaction to another weak special interaction. A comparison of (K o s ,ø )± for Cu + in some solvent systems shows that the ion-solvent interaction behaviour of Cu + in AN + n-BTN is similar to that in AN + NM, AN + TEP, BN + TEP and PY + TEP mixtures but differs from that observed in AN+DMF and AN+DMSO mixtures.

The electrical conductivity of liquid binary LaBr 3 -alkali metal bromide mixtures was measured as function of the temperature over the whole composition range. Prior to these measurements, LaBr 3 was reinvestigated because of the discrepancies in the literature values. The classical Arrhenius equation did not stand for any individual mixture. These results were discussed in terms of complex formation in the melts.

The local structure of molten CdBr 2 was investigated by high temperature X-ray absorption fine structure (XAFS) analysis. The quartz cell designed for hygroscopic high temperature molten salts was successfully used in the measurement. At room temperature the nearest neighbor Cd 2+ -Br − distance decreased from 2.71 Å in solid state to 2.60 Å in the molten state. The coordination number decreased from 6 to 4 on melting. The obtained structural parameters showed that (CdBr 4 ) 2− is predominant in molten CdBr 2 .

A The principal components of the g - and -tensor of 63 Cu 2+ obtained from single crystal ESR spectra of copper doped KHCO 3 are g zz = 2.2347(1), g yy = 2.0474(1), g xx = 2.0468(1), A zz = 20.43(2) mT, A yy = 3.22(1) mT, A xx = 2.63(2) mT. The unique axes of both tensors are parallel and lie in the mirror plane of the point group of the crystal (2/m). The direction of this axis conforms with the expectation from the structure of the first coordination shell of the K + ions. This provides strong evidence for copper entering the K + positions without major lattice disturbance. Several of the results presented here are at variance with those of an earlier report on the same subject.

The EPR parameters (zero-field splitting D and g factors g ‖ , g ⊥ ) of Cr 3+ in a NaInS 2 crystal are calculated from high-order perturbation formulas based on the two spin-orbit coupling parameter model for the EPR parameters of 3d 3 ions in trigonal octahedral sites. In the calculations, both the contribution to EPR parameters from the spin-orbit coupling parameter of the central 3d 3 ion and that of ligands are considered. From the calculations it is found that, to explain reasonably the EPR parameters, the local structure (in particular the local trigonal distortion angle θ ) in the vicinity of the Cr 3+ impurity is different from the corresponding structure in the host crystal. The change of the local angle θ with temperature is also obtained from the temperature dependence of zero-field splitting. The results are discussed.

The electron paramagnetic resonance of VO 2+ doped trisodium citrate dihydrate (Na 3 C 6 H 5 O 7 ·2H 2 O) single crystals and powder has been examined at room temperature. The angular variation of the EPR spectra has shown that two different VO 2+ complexes are located in different chemical environments, each environment containing two magnetically inequivalent VO 2+ sites in distinct orientations occupying substitutional positions in the lattice and showing a very large angular dependence. The crystalline field around the VO 2+ ion is nearly axial. The optical absorption spectrum shows three bands, centered at 24631, 16892, and 12195 cm −1 . The spin Hamiltonian parameters and the molecular orbital coefficients are calculated from the EPR and optical data, and result are discussed. Keywords EPR; Trisodium Citrate; Absorption the Spectrum; Vanadyl Ion

The EPR parameters (zero-field splitting D and g factors g ‖ and g⊥) and the local structure for the tetragonal Fe + center in KTaO 3 are theoretically studied by using the perturbation formulas of the EPR parameters for a 3d 7 ion in tetragonally distorted dodecahedra. Based on these studies, we find that the impurity Fe + may not locate on the regular dodecahedral K+ site but suffer a large off-center displacement ΔZ (≈ 0.43 Å) along one of the 100 〈or C 4 〉 axes, which is responsible for the large tetragonal distortion of the impurity center. The displacement ΔZ obtained in this work is comparable with that (≈ 0.46 Å ) of a similar monovalent Li + on K + site of KTaO 3 obtained from the nuclear quadrupole shift and can be regarded as reasonable. The calculated g factors, particularly the anisotropy Δg (= Kg ⊥ −g ‖ ) based on the above displacement, agree with the observed values.

Twenty three N-(2-/3-substituted phenyl)-4-substituted benzenesulphonamides of the general formula, 4-X’C 6 H 4 SO 2 NH(2-/3-XC 6 H 4 ), where X’ = H, CH 3 , C 2 H 5 , F, Cl or Br and X = CH 3 or Cl have been prepared and characterized, and their infrared spectra in the solid state, 1 H and 13 C NMR spectra in solution were studied. The N-H stretching vibrations, νN−H, absorb in the range 3285 - 3199 cm−1, while the asymmetric and symmetric SO 2 vibrations vary in the ranges 1376 - 1309 cm −1 and 1177 - 1148 cm −1 , respectively. The S-N and C-N stretching vibrations absorb in the ranges 945 - 893 cm −1 and 1304 - 1168 cm −1 , respectively. The compounds do not exhibit particular trends in the variation of these frequencies on substitution either at ortho or meta positions with either a methyl group or Cl. The observed 1 H and 13 C chemical shifts of are assigned to protons and carbons of the two benzene rings. Incremental shifts of the ring protons and carbons due to -SO 2 NH(2-/3-XC 6 H 4 ) groups in C 6 H 5 SO 2 NH(2-/3-XC 6 H 4 ), and 4- X’C 6 H 4 SO 2 - and 4-X’C 6 H 4 SO 2 NH- groups in 4-X’C 6 H 4 SO 2 NH(C 6 H 5 ) are computed and employed to calculate the chemical shifts of the ring protons and carbons in the substituted compounds, 4-X’C 6 H 4 SO 2 NH(2-/3-XC 6 H 4 ). The computed values agree well with the observed chemical shifts.

Fifty four N-(i,j-disubstituted phenyl)-4-substituted benzenesulphonamides of the general formula 4-X’C 6 H 4 SO 2 NH(i,j-X 2 C 6 H 3 ), where X’ = H, CH 3 , C 2 H 5 , F, Cl or Br; i,j = 2,3; 2,4; 2,5; 2,6 or 3, 4; and X = CH3 or Cl, are prepared and characterized and their infrared, 1 H and 13 C NMR spectra in solution are studied. The N-H stretching vibrations νN−H absorb in the range 3305 - 3205 cm −1 , while the asymmetric and symmetric SO 2 vibrations vary in the ranges 1377 - 1307 cm −1 and 1184 - 1128 cm −1 , respectively. The N-(i,j-disubstituted phenyl)-4-substituted benzenesulphonamides show C-S, S-N and C-N stretching vibrations in the ranges 844 - 800 cm −1 , 945 - 891 cm −1 and 1309 - 1170 cm −1 , respectively. The compounds do not exhibit particular trends in the variation of these frequencies on substitution either at ortho or meta positions with either a methyl group or Cl. The observed 1 H and 13 C chemical shifts of are assigned to protons and carbon atoms of the two benzene rings. Incremental shifts of the ring protons and carbon atoms due to -SO 2 NH(i,j-X 2 C 6 H 3 ) groups in C 6 H 5 SO 2 NH(i,j-X 2 C 6 H 3 ) and 4-X’C 6 H 4 SO 2 NH- groups in 4-X’C 6 H 4 SO 2 NH(C 6 H*) are computed and employed to calculate the chemical shifts of the ring protons and carbon atoms in the substituted compounds 4-X’C 6 H 4 SO 2 NH(i,j-X 2 C 6 H 3 ). The different methods of calculation lead to almost the same values in most cases and agree well with the observed chemical shifts, indicating the validity of the principle of additivity of the substituent effects with chemical shifts in these compounds.

CaHPO 4 and SrHPO 4 were investigated using Raman, 1 H NMR and 31 P NMR techniques to study the environment of their PO 3− 4 tetrahedra and the percentage of mobile protons. 1 H NMR spectra at room temperature suggest the presence of three types of protons, in agreement with RX investigation. The percentage of mobile protons in SrHPO 4 is greater than in CaHPO 4 because Sr 2+ is bigger than Ca 2+ . 31 P NMR spectra at room temperature show two lines in the spectrum of SrHPO 4 , revealing an equal environment of two sets of pairs of PO 3− 4 . The 31 P NMR spectrum of CaHPO 4 , however, exhibits three lines. This result was confirmed using a cross polarization (CP) sequence program. The first peak is attributed to the first set of pairs of P(1)O 4 units and the two other ones to P(2)O 4 and P(2)O 4 units.