Coefficients A2 Research Articles

Mixing coefficients for d orbitals in copper(II) complexes with lower symmetry. Part 2.—Ionic radii of copper(II) ions in complexes

The ionic radii of low-symmetry copper(II) compounds have been investigated in the xy plane and are shown to be reasonably represented by linear equations. The theoretical expression is ri=K(r)i〈ψ21〉/E|(z2)〉+r0i(i=x, y). When evaluated for six compounds whose structure is known this becomes ry/A= 1.00556(a1–b1/[graphic omitted])2/E|(z2)〉+ 0.6706 and rx/A= 0.5832(–a1–b1/[graphic omitted])2/|(z2)〉+ 0.6787, where a1 and b1 are the mixing coefficients of the dx2–y2 and dz2 orbitals in the ground-state wavefunction, and E|(z2)〉 is the energy of the 2Ag|(Z2)〉â†�2Ag|(X2–Y2)〉 transition in kK. The values for the mixing coefficients a1 and b1 can be calculated for compounds of copper(II) if the radii and E|(z2)〉 are known. The least-squares equations for all compounds are ry/A= 1.0464/E|(z2)〉+ 0.6706 and rx/A= 0.4446/E|(z2)〉+ 0.6787. The average error is ca. 3–4%. An equation was also developed for the average radius in the xy plane.

Estimation of the Interaction Parameter between Polystyrene and Poly(p-chlorostyrene) from Osmotic Pressure Measurements

The composition dependences of the osmotic second virial coefficients A2 for the mixtures of polystyrene (PSt) and poly(p-chlorostyrene) (PClSt) and for the block copolymers of ABA and BAB types (A, PSt; B, PClSt) were studied in toluene, 2-butanone (MEK), and cumene solutions. All the samples were prepared by anionic polymerization and they are close in molecular weight which is ca. 35×104. It was found that in toluene and MEK, plots of A2 vs. composition obtained for the polymer mixtures and for the block copolymers of ABA and BAB types fell on a single curve in the respective solvents, whereas in the selective solvent, cumene, A2 value of the polymer mixtures was smaller than that of the block copolymers with the same composition. The thermodynamic interaction parameter χAB between PSt and PClSt was estimated from A2 data, according to the theories of dilute polymer solutions; the χAB values obtained from A2 for the ternary system of PSt–PClSt–solvent were 0.086 (in toluene) and 0.063 (in MEK), and those from A2 for the block copolymer solutions were 0.076 (in toluene), 0.064 (in MEK) and 0.063 (in cumene). χAB=0.070±0.016 was obtained as the average.

Non-linear magnetisation effects in the spin glass Fe0.3Mg0.7Cl2

The authors present a study of the non-linear magnetisation in the vicinity of the freezing temperature Tf of the insulating spin glass Fe0.3Mg0.7Cl2. The magnetisation of the sample was measured in the range 1 or=1.70K and for a slow cooling rate of 0.5 mK s-1 the equilibrium magnetisation is obtained. In these experimental conditions, non-linear effects are studied and a rapid increase of the non-linear part mod M- chi 0H mod of the magnetisation is observed when decreasing the temperature down to 1.7K. The present the temperature dependence of the two first non-linear coefficients a3 and a5 (b3 and b5) in the expansion of (M- chi 0H) in function of odd powers of H( chi 0H). Coefficients b3 and b5 vary by one and two orders of magnitude in the temperature range 1.70-2.50K. Comparison with both simple superparamagnetic and typical spin glass systems is made.

Small angle neutron scattering (SANS) of cyclic and linear polystyrene in toluene

AbstractSamples of ring and open chain polystyrene in dilute perdeuterated toluene solution were measured by small angle neutron scattering up to a value of the scattering vector Q = 2 nm−1. The molar masses 12000 ≤ M/(g/mol) ≤ 22000, the mean square radii of gyration 〈R2〉 and the second virial coefficients A2 of the samples were determined. The results are compared to theories which describe the dependencies 〈R2〉 = f(M) and A2 = f(M) for both cyclic and linear chain molecules. A qualitative agreement between theory and experiment is obtained.

Random walk models for particle displacements in inhomogeneous unsteady turbulent flows

First a small time analysis is developed for the first and second moments of the velocity (W) and displacement (Z) in one direction of particles marked at a given point in an inhomogeneous unsteady turbulent flow, in terms of the local energy dissipation rate, and the local derivatives of the second and third moments of the vertical component of the velocity field, ∂∼(u23)/∂z and ∼(∂u33)/∂z. Then the appropriate form of a Langevin equation in inhomogeneous turbulence is suggested, namely, dW=(−W/TL+a1)dt+a1/22 dωt where a1, a2, and TL are functions of the particle position and time, and dωt is a random Gaussian velocity increment with ∼(dωt)=0 and ∼((dωt))2=dt. For simplicity, only one component of the particle motion, W(t), is considered. The functions a1 and a2 are determined by relating the random walk model to the Eulerian conservation equations for the mass of the contaminant and volume of the flow (i.e., the continuity equation), using the Fokker–Planck equation and the Eulerian equations for the moments of a vertical velocity. The coefficients a1 and a2 reduce to the same form as that obtained by the statistical analysis, namely a1=∂∼(u23)/∂z, (≊dW̄/dt, when t → 0) and a2=2∼(u23)/TL +d∼(u23)/dt (≊2∼(W′2)/TL+d∼(W′2)/dt, when t → 0). It is shown that the random walk model has the correct behavior as t/TL → 0. The theory is shown to agree reasonably well with the measurements of mean height and mean vertical displacement of particles released in a convective boundary layer [Q. J. R. Meteorol. Soc. 102, 427 (1976)]. Yaglom’s [Isv. Atmos. Oceanic Phys. 8, 333 (1972)] surface similarity result is recovered as a special case. For t ≫ TL, and in a zero-skewness steady turbulence, the random walk model reduces to the familiar K-diffusion equation. Some examples are presented to show that mean and mean square particle displacements from the random walk model are virtually identical to those obtained from the analytical solution of the corresponding Eulerian moment equations. Careful analysis is still required when concentration distributions in turbulent flows near a boundary are evaluated using random walk models.

Theory of the second virial coefficients in polymeric solutions below the theta temperature

Recent experimental measurements on the temperature variation of the second virial coefficients A2 in polymer solutions below the Θ point are analyzed theoretically. In contrast to the existent theories we incorporate the ternary collision integral between monomers into our theory in order to recover the stability of the system beyond the Θ region. A scaling law A2=f(τ(M)1/2)/(M)1/2 is derived for the virial coefficient, where τ ≡ T/Θ−1 is the temperature deviation and M is the molecular weight of the polymer. Based on this scaling law, observed molecular weight independence is explained. It is also pointed that the tendency of A2 to increase through a minimum is expected in connection with the coil–globule transition.

Bifurcation in gravity waves

A new method is proposed for the calculation of gravity waves on deep water. This is based on some recently discovered quadratic identities between the Fourier coefficients an in Stokes's expansion. The identities are shown to be derivable from a cubic potential function, which in turn is related to the Lagrangian of the motion. A criterion for the bifurcation of uniform waves into a series of steady waves of non-uniform amplitude is expressed by the vanishing of a particular determinant with elements which are linear combinations of the coefficients an. The critical value of the wave steepness for the symmetric bifurcations discovered by Chen & Saffman (1980) are verified. It is shown that a truncated scheme consisting of only the coefficients a0, a1 and a2 already exhibits Class 2 bifurcation, and similarly for Class 3. Asymmetric bifurcations are also discussed. A recent suggestion by Tanaka (1983) that gravity waves exhibit a Class 1 bifurcation at the point of maximum energy is shown to be incorrect.

Mixing coefficients for d orbitals in copper(II) complexes of lower symmetry

A study has been made of the effect of D2h symmetry on the wavefunctions of copper(II) complexes. The wavefunction becomes an extended orbital with the form ψn=an|(x2–y2)+〉+bn|(z2)+〉+cn|(xy)+〉+dn|(xz)–〉+en|(yz)–〉. Mixing coefficient (a–e) have been calculated in two different ways from spectral and e.s.r. data and are compared. The relative potential and equations have been derived and can reproduce spectral and e.s.r. g values within experimental error. The mixing coefficients a1 and b1 for the ground state [2Ag, |(X2–Y2)〉] are related to the radii in the xy plane, and the b2 mixing coefficient [2Ag, |(Z2)〉] for the excited state is related to the radius in the z direction by a simple linear equation of the form r=K(r)〈ψ*|ψ〉+r0. The ratio of the radii in the xy plane has been calculated. The radius in the z direction has been calculated by assuming that the radius is proportional to the energy of the |(Z2)〉 excited state, giving the equation rz/A= 0.01384(2b2)2E|(Z2)〉+ 0.68. A most likely electronic configuration can be determined if the symmetry is approximately correct, although the symmetry used in this paper is probably too high for many compounds.

Driving complex linear systems to zero in given time

The coefficients a1, i = 1, 2, …, n of the control function u(t) = a1E(t)+a2E(t—α1)+…+an+1E(t—αn) where an+1 = − (a1+a2+…an), 0<α1<α2<…<αn, are found so that the linear system [xdot]= Ax + bu, x(0)=x0, is driven exactly to zero in an arbitrary time T = αn. The switching times α1, α2,…, αn are assumed known. In contrast to an earlier paper, this paper does not assume that the eigenvalues of A are real.

The decay $$W^ - \to b\bar t$$ of the W-boson produced with high transverse momentum in $$\bar pp$$ collisionscollisions

We study the mass effects of the top-quark t on the angular distributions of the b-quark jet arising from the decays of the W−-boson produced with high transverse momentum in\(\bar pp \to \)Open image in new windowhigh-QTjet+anything. We point out that the angular coefficients α1 in the helicity frame and β1 in the Gottfried-Jackson as well as Collins-Soper frames can provide useful information in determining the t-quark mass.

Characterisation and antimicrobial activity studies of the mixed ligand complexes of Cu(II) With 8-hydroxyquinoline and salicylic acids

Mixed ligand complexes of copper(II) with 8-hydroxyquinoline and various salicylic acids were isolated. Elemental analysis, conductometric andIR data of these complexes were recorded. The esr and electronic spectral data of these complexes indicate distorted square planar coordination for central copper ion. The σ bond and inplane π bond coefficients α2 and β2 respectively, of these complexes indicate partial covalency in the metal ligand bonding. The antimicrobial activity studies show that the mixed ligand complexes act as more effective toxic agents thanbis (8-hydroxyquinolinato) copper(II) against certain bacteria and fungi. The lipophilic tendency of these complexes is determined and its influence on their antimicrobial activity is critically examined. A probable mechanism for the antimicrobial activity of these complexes is discussed

Theta point (‘‘tricritical’’) region behavior for a polymer chain: Transition to collapse

The conformational space renormalization group method is generalized further to describe excluded volume effects in finite molecular weight polymers in the theta point region where contributions from effective three-body interactions become appreciable. The theory builds upon our previous description of the good solvent region and uses t’Hooft-Veltman style dimensional regularization along with the renormalization group (RG) to determine the general analytic structure of measurable quantitites of interest. Our formalism is compared in detail with that employed in field theory to describe tricritical behavior. Although many results are in qualitative agreement between the two approaches, there are some numerical differences. Our method considers renormalization for chains with fixed length, and this introduces differences into the renormalization scheme from that used in field theory, differences dictated by the physical differences in the types of systems considered. Our treatment of finite length chains contrasts with the field theoretical expansions about the unphysical limit of infinitely long polymers; thus it entirely avoids the use of the method of insertions. The RG theory is used to calculate the mean square end-to-end distance 〈R2〉 and the second and third virial coefficients A2 and A3 in the theta region as a function of both the two- and three-body interactions and of the chain length. We compute the temperature difference between the point at which A2 vanishes and at which 〈R2〉 is the pure Gaussian limiting value. The virial coefficients are used to evaluate the dilute solution portion of the coexistence curve, providing an estimate of the collapse transition temperature. The presence of essential logarithmic dependences of the coexistence curve on chain length implies that mean field theory is not valid for describing measurable quantities in this region.

Study on the molecular weight dependence of dilute solution properties of narrowly distributed polystyrene in toluene and in the unperturbed state

AbstractIntrinsic viscosities [η], second and third virial coefficients A2 and A3, respectively, and mean square radius of gyration 〈RG2〉 of a polymer homologous series of narrow‐molecular‐weight‐distribution polystyrene samples in a molecular weight range of 2000 up to 24 · 106 were determined in toluene at 25°C by light scattering and viscometry measurements. The results are (3 · 104 &lt; Mw &lt; 24 · 106, Mw/Mn ≤ 1,3): These results were obtained by consideration of the limit of the dilute solution regime, the determination of zero‐shear rate intrinsic viscosities, and the molecular weight dependence of the refractive index increment dn/dc. It was found that dn/dc increases slightly up to Mw = 1,8 · 106. A comparison of the [η]−Mw‐relationship with literature data is given. In addition the unperturbed (theta) dimension parameters K0, 〈h2〉0/M, the characteristic ratio C∞, the steric factor σ, and the thermodynamic interaction parameter B were calculated from the modified Burchard‐Stockmayer‐Fixman procedure using polymolecularity correction and the results compared with experimental data from the literature.

A New Method for the Determination of Equilibrium Constant and Molar Extinction Coefficient of 1 : 1 Type Complex

Abstract Job’s method of continuous variation has been reinvestigated with respect to the formation of 1 : 1 type complex from the view point of (i) geometrical significance of the Job curve, (ii) developing a method for determining the equilibrium constant and molar extinction coefficient of the complex. A mathematical analysis of the method of continuous variation as applied to the formation of strictly 1 : 1 type complex shows that the Job curve is a hyperbola obeying the equation \frac(d–β)2a2−\frac(CM°⁄λ–1⁄2)2b2=1 where d is the corrected optical density, CM° is the initial concentration of the metal ion in solution, λ is a constant equal to the sum of the concentrations of the metal and the ligand in solution, β is the ordinate of the local origin of the hyperbola referred to the conventional framework of the Job curve, a and b represent respectively the semi-major and semi-minor axes of the hyperbola. The equation has been recast into the form x=A3d2+A2d+A1, where x=(CM°/λ–1⁄2)2 and A3, A2, and A1 are the appropriate coefficients. A parabolic fit of this equation between x and d by the least square deviation method using the computer B-6800 system enables one to find the coefficients A3, A2, and A1 from which K and ε are evaluated. This equation has been tested with literature data on a variety of 1 : 1 type metal complexes whose formation constants and extinction coefficients as calculated by the present method agree closely with those determined by other methods. The promising features of the equation have been discussed.

Second osmotic virial coefficient revisited: Variation with molecular weight and temperature from endothermal to exothermal conditions

Light scattering measurements for six samples of polystyrene (M = 20 400 to 2.5×106) were performed in tert-butylacetate from 0–140 °C. Within this T interval, the second virial coefficients A2 change from negative to positive, pass through a maximum at intermediate temperatures, and get negative again. The M dependence vanishes when A2 = 0 at ϑ+ = 15.5 °C for endothermal, and at ϑ− = 109.5 °C for exothermal conditions. The temperature of athermal behavior increases with M from 42 °C (M = 20 400) to 62 °C (M = 2.5×106). For sufficiently high temperatures A2 grows larger as M is raised, in contrast to the normal situation. There exist strong indications that limM→∞ A2 is generally different from zero, except for theta conditions. The enthalpy part of the second virial coefficient A2,H becomes independent of M at Ξ+ = 31 °C for endothermal solutions and at Ξ− = 90 °C for exothermal. Between these temperatures A2,H decreases as M rises, else it increases. The unperturbed dimensions are the same at ϑ+ and ϑ−. The observations concerning A2(T;M) can be explained theoretically on the basis of the corresponding states principle. For a better understanding of the influence of M, a modified excluded volume treatment is presented; it accounts for the differences in intra- and intermolecular contacts between segments. The interrelation of the chain expansion coefficient and A2 is in agreement with the Zimm–Stockmayer–Fixman equation.

A class of D-extreme Minkowski-reduced forms

AbstractBarnes (1978, 1979) introduced the concept of a -extreme form, which is a Minkowski-reduced positive definite quadratic form having prescribed diagonal coefficients α1, α2, …, αn and providing a local minimum of the determinant of the form over all such forms. Here a class of forms which are -extreme for all α and all n is described.

Nuclear spin relaxation by translational diffusion in liquids and solids: high- and low-frequency limits

Nuclear spin relaxation rates due to magnetic dipole coupling between spins of a single species undergoing relative diffusion depend on two-spin correlation functions J(p)( omega ). The high- and low-frequency limits of J(p)( omega ) are discussed for liquids, cubic crystals and two- and one-dimensional systems. It is shown that the low-frequency form of J(p) ( omega ) is J(p)(0)-A3 omega 1/2 in liquids and crystals, A2ln omega -1 in two-dimensional systems and A1 omega -1/2 in one-dimensional systems and expressions for the coefficients A1, A2 and A3 are derived which are valid for any diffusive model whose long-range, long-time behaviour may be described by the diffusion equation. The high-frequency form of J(p)( omega ) is derived for a simple-hopping model of spins on one-, two- and three-dimensional lattices and in each case J(p)( omega ) is of the form B omega -2. Expressions are derived for the coefficient B, for each dimension, which correctly include the effect of correlations between the hops of the spins.

Second Virial Coefficients for Solutions of Polystyrene Mixtures Obtained With Low‐Angle Laser Light Scattering

AbstractLow‐angle laser light scattering measurements were performed on practically binary mixtures of standard polystyrenes dissolved in toluene. Second virial coefficients A2 were obtained as a function of the composition of the mixed polymer. No solid evidence was found for the existence of maxima in A2 vs composition. Existing theories predict a maximum. The possible existence of minima such as reported by Suzuki5 could not be confirmed with the present system.

Activity coefficient of hydrochloric acid in HCl/MgCl2 mixtures and HCl/NaCl/MgCl2 mixtures from 5 to 45°C

Magnesium, the dominant bivalent cation in natural seawater, exerts a substantial influence on the patterns of ion interactions in this saline medium. Mean activity coefficients of hydrochloric acid in mixtures of this acid with magnesium chloride at four ionic strengths, namelyI=0.1, 0.3809, 0.6729, and 0.8720 mole-kg−1, were obtained from emf measurements of cells without liquid junction at nine temperatures from 5 to 45°C. The three highest ionic strengths correspond to seawater of salinities 20, 35, and 45‰, respectively. In addition, mixtures of HCl, NaCl, and MgCl2 were studied atI=0.6729, the molal ratio of NaCl to MgCl2 being maintained at 7.202 as in natural seawater. The Harned coefficients α1 2 were found to decrease slowly with increase in temperature. The trace activity coefficient of HCl in solutions of MgCl2 as well as in NaCl MgCl2 mixtures was found to be nearly identical with that measured earlier in synthetic seawater of the same ionic strength but containing NaCl, MgCl2, KCl, and CaCl2.

Measurements of Hydrodynamic Parameters for Nematic 5CB

Abstract A nematic liquid crystal, pentyl-cyano-biphenyl (5CB) is studied in two different flow situations: Poiseuille flow and torsional shear flow, in both cases with and without the simultaneous application of an electric field. The Poiseuille flow set-up gives an accurate determination of the Miesowicz viscosity ηc and a less precise estimation of ηb. With ηc known it is possible to separately determine the elastic constants K 11 and K 33 and the viscous coefficients α2, α3, and thereby γ1, γ2 from the torsional shear flow experiment. Using the data we have also been able to calculate the actual director profile in the liquid crystal as a function of applied shear and electric field. Curves are given of K 11, K 33, K 11/K 33, α2, α3, γ1, γ2, γ1/γ2, ηb and ηc for 5CB as a function of temperature through the whole nematic range from 22.0 °C to 35.1 °C. After the completion of our manuscript we have noticed the recent article by J. Wahl, Z. Natur-forsch. 34a, 818 (1979). Wahl discusses shear flow in the ...