Three-parameter Correlation Research Articles

Structures in the fundamental plane of early-type galaxies

The fundamental plane of early-type galaxies is a rather tight three-parameter correlation discovered more than twenty years ago. It has resisted a both global and precise physical interpretation despite a consequent number of works, observational, theoretical or using numerical simulations. It appears that its precise properties depend on the population of galaxies in study. Instead of selecting a priori these populations, we propose to objectively construct homologous populations from multivariate analyses. We have undertaken multivariate cluster and cladistic analyses of a sample of 56 low-redshift galaxy clusters containing 699 early-type galaxies, using four parameters: effective radius, velocity dispersion, surface brightness averaged over effective radius, and Mg2 index. All our analyses are consistent with seven groups that define separate regions on the global fundamental plane, not across its thickness. In fact, each group shows its own fundamental plane, which is more loosely defined for less diversified groups. We conclude that the global fundamental plane is not a bent surface, but made of a collection of several groups characterizing several fundamental planes with different thicknesses and orientations in the parameter space. Our diversification scenario probably indicates that the level of diversity is linked to the number and the nature of transforming events and that the fundamental plane is the result of several transforming events. We also show that our classification, not the fundamental planes, is universal within our redshift range (0.007 - 0.053). We find that the three groups with the thinnest fundamental planes presumably formed through dissipative (wet) mergers. In one of them, this(ese) merger(s) must have been quite ancient because of the relatively low metallicity of its galaxies, Two of these groups have subsequently undergone dry mergers to increase their masses. In the k-space, the third one clearly occupies the region where bulges (of lenticular or spiral galaxies) lie and might also have formed through minor mergers and accretions. The two least diversified groups probably did not form by major mergers and must have been strongly affected by interactions, some of the gas in the objects of one of these groups having possibly been swept out. The interpretation, based on specific assembly histories of galaxies of our seven groups, shows that they are truly homologous. They were obtained directly from several observables, thus independently of any a priori classification. The diversification scenario relating these groups does not depend on models or numerical simulations, but is objectively provided by the cladistic analysis. Consequently, our classification is more easily compared to models and numerical simulations, and our work can be readily repeated with additional observables.

A novel continuous re-extraction procedure of penicillin G by a micro-extractor based on ceramic membrane

A ceramic membrane micro-extractor (CME), employed for continuous re-extraction of penicillin G (PEN G), was studied systematically. The droplet size was investigated in the re-extraction process. Re-extraction efficiency ( RE) and degradation rate of PEN G were extensively examined by superficial Reynolds number ( Re h), phase ratio ( R), initial PEN G concentration and membrane pore size. To further understand and quantitatively analyze the re-extraction behavior, resistance-in-series (RIS) model was developed to predict the re-extraction efficiency at steady-state conditions. First of all, a three-parameter correlation was defined for quantitative description of droplet size. The main factors affecting the re-extraction efficiency of PEN G were superficial Reynolds number and phase ratio. The re-extraction efficiency of PEN G reached 99% or above at Re h > 934 and R > 0.08. Meanwhile, the degradation rate of PEN G ( D) decreased to 0.5% when Re h was 934. Compared with the re-extraction performance of the batch extractor, less residence time was needed and the degradation rate decreased 50%. The theoretical re-extraction efficiency ( RE T) was in good agreement with the experimental data.

Preparation of hollow polylactide microcapsules through premix membrane emulsification—Effects of nonsolvent properties

Hollow polylactide microcapsules that can be used as ultrasound contrast agents were prepared using premix membrane emulsification. Polylactide/dichloromethane and dodecane solutions were emulsified together with a nonsolvent phase (water or a water–alcohol mixture) by repeated passage through a glass fibre membrane. The solvent, dichloromethane, diffuses out of the droplets and the polylactide solidifies around a droplet of dodecane. To investigate the effect of the nonsolvent properties on the size and span of the microcapsules, different methanol–water, ethanol–water and 2-propanol–water mixtures were used as nonsolvents. The alcohol lowers the interfacial tension and increases the viscosity of the nonsolvent, and therewith it decreases the size and the span of the microcapsules. It was remarkable that 2-propanol yields the smallest size (0.35 μm) followed by ethanol (0.8 μm) and methanol (1.4 μm). In contrast, the smallest span was obtained with methanol (0.7), whereas 2-propanol gave the largest span (1.5). The results further show that the size and the span of the microcapsules decreases with increasing number of emulsification passes and transmembrane flux. The presence of alcohol in the nonsolvent phase increases the efficiency of the emulsification process and decreases the optimum number of passes required to obtain the minimum average size of the droplets. A three-parameter correlation was defined that could quantitatively describe the effects of all the aforementioned parameters on the size of the microcapsules.

Effect of the medium on the reaction of tert-butyl hypochlorite with hydrocarbons

The selectivity in the reaction of tert-butyl hypochlorite with 2,3-dimethylbutane in various solvents may be described by a three-parameter correlation equation taking into account the ability of solvents to nonspecific solvation and their cohesion energy density.

Comparative study of the performance of three- and four-parameter correlation equations for the temperature dependence of the enthalpy of vaporization for pure substance refrigerants

A few commonly used correlation equations of the enthalpy of vaporization which is essential to the analysis of refrigeration cycles are reviewed. A new four-parameter correlation equation is proposed assuming that the enthalpy of vaporization could be represented by a function having a linear term to the temperature and an additional term which slowly decreases as the temperature increases. It is not a common practice to measure the enthalpy of vaporization by experiment; therefore, performance of the new correlation equation for the temperature dependence of the enthalpy of vaporization is examined using numeric data from the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) tables for 22 pure substance refrigerants. The new correlation equation and other existing ones are fitted to the data optimizing the root mean squared deviation. All data points are weighted equally and NBP (normal boiling point) is used as a fixed point since the NBP is important for refrigeration applications. The new four-parameter equation yields an average absolute deviation of 0.05% for 22 refrigerants which is smaller than those of other four-parameter equations, such as Guermouche–Vergaund (0.08%), Aerebrot (0.13%), Radoz–Lyderson (0.08%), and Somayajulu four-parameter equation (0.08%). After the optimization, it is found that a specific exponent in the new equation has approximately the same value for almost every substance tested so that the original four-parameter correlation could be reduced to a three-parameter one. The new three parameter correlation yields an average absolute deviation of 0.14% for the same 22 refrigerants, which also is smaller than those of other three-parameter correlations, such as Xiang (0.18%), Majer–Svoboda–Pick (0.18%), and Somayajulu three-parameter equation (0.27%).

Kinetics and mechanism of reactions of arenes in the HVO3-sulfuric acid system: Extended compensation effect

Kinetics, activation parameters, and the kinetic isotope effect (KIE) of hydrogen have been determined for the first rate-limiting step of the reactions of benzene, monoalkylbenzenes, dimethylbenzenes, and trimethylbenzenes in HVO3-sulfuric acid (54–70 wt % H2SO4) solutions. The effects of solution acidity on the rate constants (k) and KIE have been studied. The reactions begin with electron transfer from the arene to VO 2 + and end with oxidative coupling: $$PhCH_3 \xrightarrow{{VO_2^ + }}PhCH_3^{ + \cdot } \xrightarrow{{ - H^ \cdot }}PhCH_2^ + \xrightarrow{{PhCH_3 }}PhCH_2 C_6 H_4 CH_3 .$$ logk is a linear function of the ionization potential of the arene (I ArH). A three-parameter correlation is for the first time established between the activation energy (E), the logarithm of the preexponential factor (log A), and I ArH: logA = 0.15E − 0.058I ArH + 49.1 (E and I ArH are expressed in kJ/mol). The reactions of arenes with VO 2 + , NO 2 + , and Co3+ are intercompared.

Liquid Extraction of Inorganic Acids with α-Aminophosphoryl Compounds

Liquid extraction of inorganic acids, such as hydrochloric, hydrobromic, hydroiodic, and nitric, with four α-aminophosphonates with different-size hydrocarbon substituents in the aminomethylphosphoryl skeleton was studied. The extraction effectiveness increases with increasing number of carbon atoms in the molecules of organophosphorus extractants, that is with increase in their lipophilicity, and is also affected by the hydration enthalpy of the inorganic acid anion. The extractive ability of phosphorylated amines is 3–4 orders of magnitude lower compared with corresponding amines, on account of the fact that the basicity of the former is decreased by the electron-acceptor phosphoryl group. The absence of satisfactory linear two- and three-parameter correlations between the extraction constants and the number of carbon atoms in and the basicity of phosphorylated amines is evidently connected with the disregard for other structural and medium factors that may influence the extraction coefficients.

Three-parameter correlation functions for the calculation of thermal conductivity of gases, liquids, and refrigerants over wide temperature-presseure ranges

Two new correlation functions with three adjustable parameters for the calculation of thermal conductivity of gases, liquids, and refrigerants over wide temperature-pressure ranges are presented. It is shown that by usingsuitable mathematical expressions for the correlation functions, the number of adjustable parameters can be limited to three, which is less than for similar functions. The average and maximum deviation of the correlation compared with the measured data is about 1% and 8%, respectively.

Kinetic study of hydrolysis of benzoates. Part XXII.† Variation of the ortho inductive, resonance and steric terms with temperature in the alkaline hydrolysis of substituted phenyl benzoates in aqueous 2.25 M Bu4NBr

The second order rate constants k2 (M−1 s−1) for the alkaline hydrolysis of meta-, para-, and ortho-substituted phenyl benzoates C6H5CO2C6H4–X (X = H, 3-Cl, 3-CH2, 3-NH2, 3-NO2, 4-F, 4-NO2, 2-Cl, 2-F, 2-CH3, 2-OCH3, 2-NO2) in 2.25 M aqueous n-Bu4NBr have been measured spectrophotometrically at various temperatures (15, 25, 30, 40, 50 °C). The log k values were treated using both the modified Fujita–Nishioka and the Charton three-parameter correlation equations. The σ°, σI, σR°, EsB scales were used as the measure of the ordinary polar, inductive, resonance, and steric components of the substituent effects. EsB = (log kX − log kH)H+, where kX and kH are the rate constants of the acidic hydrolysis for ortho-substituted and unsubstituted phenyl benzoates in water at 50 °C. The modified Fujita–Nishioka and Charton treatments gave the excellent and identical results. The susceptibility to the induction effect of ortho-substituents was found to be about 0.2 units of the ρ scale smaller [ρI)ortho = 1.80] than that for meta- and para-substituents [(ρ°)m,p = 2.03], in the alkaline hydrolysis of substituted phenyl benzoates in 2.25 M aqueous n-Bu4NBr at 50 °C. Also in this solution, the variation of the susceptibility to the inductive and the resonance effects with temperature for the ortho-derivatives appeared slightly smaller than that for para-derivatives. During the transition from water to 2.25 M aqueous Bu4NBr the susceptibility of the meta- and para-substituent polar effect (ρ°)m,p was found to increase by about 1.0 units, but the susceptibility to the inductive effect of ortho-substituents was enhanced only by about 0.5 units of the ρI scale at 50 °C. The variation of the inductive term of ortho-substituents with both solvent and temperature was found to be half that for para-substituents. Variation of the inductive term of ortho-substituents with the solvent and temperature by the same extent had earlier been found for the alkaline hydrolysis of substituted phenyl tosylates, though the ratio of the susceptibilities to the polar influence of meta-, para-, and ortho-substituents in water differed two-fold. The steric term of ortho-substituents for 2.25 aqueous n-Bu4NBr was found to be independent of temperature and nearly equal to the steric term for water.

Three-Parameter Corresponding-States Correlations for Joule–Thomson Inversion Curves

In the present work, the Lee–Kesler (LK) and Boublik–Alder–Chen–Kreglewski (BACK) equations of state were used to compute Joule–Thomson inversion curves for nonsimple fluids. Comparisons with available data showed that predictions were quite reliable and could be used in place of experimental values. Two sets of corresponding-states correlations were developed, giving reduced inversion pressures and densities as functions of reduced temperature and acentric factor. The LK-based correlations are valid for Tr≤4.0, giving an average absolute deviation (AAD) of 4.5% for pressures. The BACK-based correlations are valid up to the maximum inversion temperature and give a 6.7% AAD for pressures. Respective volume AADs are 12.0 and 8.0% in the high-density region.

Extended molecular connectivity: Prediction of boiling points of alkanes

It is shown that extended molecular connectivity may be used for “structure-property” studies. The statistical parameters of the correlation relations of boiling points with extended molecular connectivity indices found for a sample of alkanes (n = 74) are the following: the best two-parameter correlation r = 0.9948, s = 4.7; the best three-parameter correlation r = 0.9971, s = 3.5.

A model of molecular fluids: a three-parameter direct correlation function

A previous two-parameter model of a molecular fluid is extended to three parameters and applied to a fluid of hard ellipsoids with restricted orientations. The direct correlation function of the fluid is written as a linear combination of three basic functions derived from the geometry of two molecules and the coefficients determined from algebraic equations also satisfied by the Percus–Yevick (PY) solution of the problem. The theory yields the PY solution and PY equation of state in the special case of hard spheres. For ellipsoids the equation of state agrees better with the results of computer simulation than the two-parameter version although the discrepancy increases with increasing anisotropy of the molecules. According to the model the fluid makes a transition to a nematic state although the predicted densities at which the transition takes place are too low.

Correlations of indices calculated from the matrix of maximal topological distances with boiling points of alkylbenzenes

The matrix of maximal topological distances of a graph can serve as a basis for constructing new topological indices of ringcontaining structures. A comparative study of the structureproperty correlations of the indices of the matrix of maximal topological distances and known Wiener, Horary, and Schultz indices (instructive sample of 29 alkylbenzenes) showed that the best two- and three-parameter correlations with boiling points include the indices of the maximal distance matrix. The two-parameter (r = 0.0992, s = 3.5) and three-parameter (r = 0.994, s = 3.1) correlations may be used for quantitative predictions of the boiling points of alkylbenzenes.

Kinetics and mechanism of acid hydrolysis of peroxyketals

The kinetics of hydrolysis of aliphatic ketone di-tert-butylperoxyketals R1R2C=O, R1, R2=CH3, CH3; CH3, C2H5; CH3, n-C3H7; CH3, n-C6H13; CH3, i-C5H10; CH3, i-C4H9; C2H5, i-C3H7; n-C4H9, n-C4H9; CH3, C6H5-CH2, in dioxane in the presence of H2SO4 were investigated by IR spectroscopy. It was found that the reaction is reversible and takes place according to the equation R1R2C· (OOC(CH3)3)2 + H2O;H+ R1R2C=O + 2HOOC(CH3)3. The proposed mechanism of hydrolysis includes the fast, quasiequilibrium formation of protonated peroxyketal and subsequent formation of the alkylperoxycarbenium ion. A three-parameter correlation equation is proposed for describing the initial rates of hydrolysis of R1R2C(oo-t-Bu)2 peroxyketals.

The pulse-height correction technique for improving γ-ray spectra from coaxial Ge detectors

We demonstrate that the pulse-height correction technique can be used for coaxial as well as planar Ge γ-ray detectors. Pulses from the detectors are analyzed according to their rise-times, and an improved spectrum is obtained by correcting the incompleteness of charge collection, using the relation between rise-time and pulse-height defect. Geometrical effects in coaxial detectors require at least a three-parameter correlation, in which the rise-times of the pulses are analyzed for two time segments. We have been able to improve the energy resolution of neutron-damaged Ge detectors by more than 50% and their peak-to-Compton ratios by almost as much, all without significant loss in detector efficiency.

The ortho effect in ortho-substituted 2-(X-benzal)-1,3-indanediones, 3-(2-X-benzal)phthalides and 3-(2-X-benzal)-5,6-dihydro-4,7-dithiaphthalides: A 1H-NMR study

The 1H-NMR spectra of ortho-substituted 2-(X-benzal)-1,3-indanediones, 3-(2-X-benzal)-phthalides and 3-(2-X-benzal)-5,6-dihydro-4,7-dithiaphthalides have been studied. The chemical shifts observed for the methine proton were correlated with various constants of the substituents. In addition to simple correlations, two- and three-parameter correlations were also made. Statistically most important one-parameter correlations were observed for σp-constants. Apart from the common electronic effects, in the transfer of effects of the substituents from the ortho-position upon the chemical shift of the methine proton the field effect plays also an important role. The deviations observed in the case of substances substituted in both ortho-positions confirm the assumed considerable distortion of the coplanarity of the studied systems.

Cluster-variational calculations for extended nuclear systems

The energy as a function of density is calculated for neutron matter and for symmetrical nuclear matter, based on Jastrow trial wave functions. The energy expectation value is truncated in low cluster order. A detailed analysis of the two- and three-body cluster contributions and a special portion of the four-body contribution is given. Variation of a parameterized two-body correlation function is subjected to constraints designed to confine the trial wave function to the domain corresponding to rapid cluster convergence. Results are presented for a variety of model central potentials containing hard cores, for different sets of constraints, and for two- and three-parameter correlation functions. Calculations constrained by the “average Pauli condition” are found to yield results very close to those constrained by the “normalization” or “unitarity” condition, and the two-parameter correlation function appears to be quite adequate. The convergence of the cluster expansion, as reflected in the low orders, is good except at the highest densities considered. The three-body cluster contribution displays, in all cases, a remarkable internal cancellation between its “two-correlation-line” addends.

The Correlated Basis Function Method and its Application to Liquid 3He, Nuclear Matter and Neutron Matter

Abstract The method of correlated basis functions is studied and applied to the Fermi systems: liquid 3 He, nuclear matter and neutron matter. The reduced cluster integrals xijkl... and so the sub-normalization integrals Iijkl... are generalized to coinciding quantum numbers out of the set {i, j, k, I,...}. This generalization has an important consequence for the radial distribution function g (r) (and then for the liquid structure function) ; g(r) has no contributions of the order O (A-1). For 3 He the state-independent two-body correlation function g(r) is calculated from the Euler-Lagrange equation (in the limit of r → 0) for the unrenormalized two-body energy functional. For nuclear matter and neutron matter we adopt the three-parameter correlation function of Bäckman et al. Then the energy expectation values are calculated by including up to the three-body terms in the unrenormalized and renormalized version of the correlated basis functions method. The experimental ground-state energy and density of liquid s He can be well reproduced by the present method with the Lennard-Jones-(6 -12) potential. The same method is applied to the nuclear matter and neutron matter calculations with the OMY-potential. The results of the energy expectation values indicate a practical superiority of the unrenormalized cluster expansion method over the renormalized one.

Generalized f/P, ZT and ZP values for liquids at low reduced temperatures

AbstractThe three‐parameter correlation of Pitzer et al. for fugacity coefficients (f/P) has been extended to liquids in the 0.5 ≤ Tr ≤ 0.8 and 0.2 ≤ Pr ≤ 9.0 region. In addition, values of the derivative compressibility factors ZT and ZP have also been obtained in the same region. This extension is based on the generalized correlation of the compressibility factor Z recently reported. The isothermal coefficient of bulk compressibility βT compiled by Rowlinson has been employed with the pressure effect on βT taken into consideration. The correlation for f/P obtained in this work is compared with that of Curl and Pitzer at Tr = 0.8, Chao et al. at Tr from 0.50 to 0.70, and that of Carruth and Kobayashi in the 0.50 ≤ Tr ≤ 0.64 region. Using the tabulated ZP and ZT values \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{H^* - H}}{{T_c }},{\rm }\left ({\frac{{\partial V}}{{\partial P}}} \right)_T $\end{document}, and \documentclass{article}\pagestyle{empty}\begin{document}$ \left ({\frac{{\partial V}}{{\partial T}}} \right)_P $\end{document} have been calculated and compared satisfactorily with the corresponding values derived from literature data. In addition, a correlation of vapor pressures of both hydrocarbons and nonhydrocarbons is also presented. The agreement between this correlation and those reported by Pitzer et al. and Carruth and Kobayashi is very satisfactory.

Energy and Angular Distribution of Alpha Particles Emitted in Thermal-Neutron Fission ofU235

The energy and angular distributions of $\ensuremath{\alpha}$ particles emitted in thermal-neutron fission of $^{235}\mathrm{U}$ have been studied in a three-parameter correlation experiment for $\ensuremath{\alpha}$ angles between 50 and 130\ifmmode^\circ\else\textdegree\fi{}. These distributions are compared with the results of trajectory calculations to get a "best fit" set of initial scission conditions. The derived initial fragment separation of 24 fm and initial average kinetic energy of 31 MeV are in good agreement with liquid-drop calculations, while the initial most probable $\ensuremath{\alpha}$ energy was found to be 3.4 MeV. The results indicate a shift in the scission point of 1-2 fm towards the light fragment, for a variation in mass ratio between 1.1 and 2. The effect of $\ensuremath{\alpha}$ recoil on the experimental distributions is discussed in detail.