Four-parameter Equation Research Articles

Adsorption from solutions of nonelectrolytes on heterogeneous solid surfaces: A four-parameter equation for the excess adsorption isotherm

A four-parameter single-solute adsorption isotherm equation is generalized to the adsorption of binary liquid mixtures of nonelectrolytes. This equation comprises all isotherms being a simple extension of theEverett isotherm to liquid adsorption on heterogeneous solid surfaces. The benzene—cyclohexane excess adsorption data on silica gel are studied by using this new equation.

Toluene blood level following subcutaneous injection of toluene in the rat

A model of toluene level in blood following subcutaneous injection of toluene mixed with polyoxyethylated-vegetable-oil vehicle was developed. The purpose was to provide a means of predicting dose received for subsequent toxicologic studies for any time and dose combination. The pharmacokinetics were of secondary interest. Using data from 111 rats, a four-parameter equation was devised to predict the time course of toluene blood level from 20–480 min post dosing, for dose levels of 50–1000 mg/kg. Blood concentrations rose at a rate which was independent of dose level. Maximum blood levels were uniquely determined by dose level. Blood levels fell at different rates depending upon dose level. Injection exposure, when compared to inhalation, has the advantages of (a) low expense, (b) low equipment requirements, and (c) simplicity. The disadvantage is, for some experiments, poor temporal simulation of the normal route of administration, inhalation.

Thermal degradation of polystyrene—5. Effects of initiator residues

The thermal degradation at sub-volatilisation temperatures of polystyrenes prepared with the radical initiators benzoyl peroxide and azoisobutyronitrile was studied. The plots of degree of degradation vs heating time show marked curvature in contrast to those of anionically and thermally polymerised polystyrenes. These plots were well simulated by a four-parameter equation which indicates that chain-scission occurs by at least three concurrent processess of markedly differing rates—rapid “weak link” scission, slow normal bond scission and scission at an intermediate rate. The last-mentioned appears to occur at ca one bond per polymer molecule. Neither these bonds nor the “weak links” have been identified but neither appears to be associated with end-groups.

Diffusion and convection across heteroporous membranes: A simple macroscopic equation

Here the problem of identifying a set of macroscopic parameters which describe the transport of a neutral solute across a heteroporous membrane by convection and diffusion is considered. An approximate equation for the solute flux is developed in terms of the membrane boundary conditions and the macroscopic parameters L p (hydraulic conductivity), σ s (reflection coefficient), P s (permeability), and q s (index parameter). The unique feature of this model is that the four macroscopic parameters are (1) expressed in terms of the set of single-pore parameters { L pi, σ si, P si} and the pore-size distribution only, (2) invariant to conditions which alter the volume ( J v) and solute ( J s) fluxes, and (3) consistent with simultaneous measurements of J v and J s. The four-parameter equation for J s is shown to be in good agreement with the exact solution for the extreme case of a membrane composed of two distinct sets of pores which differ greatly in radii.

Solvent effect - Classification of parameters describing influence of solvents

On the basis of factor analysis of data matrices containing 35 empirical and non-empirical parameters for 85 solvents (or a limited matrix of 20 parameters for 51 solvents), an orthogonal set of parameters with normed normal distribution has been suggested which characterizes influence of the solvents on chemical processes. A four-parameter equation has been suggested for description of the solvent effect in the form A = A0 + aAP + bBP + eEP + pPP, where AP, BP, EP, PP are newly introduced parameters expressing individual types of interactions between solvent and solute. A comparison has been carried out of these parameters with those defined earlier. The new set of parameters was tested with 22 examples taken from literature. The correlations obtained show higher or comparable agreement as compared with those carried out earlier. Values of the regression coefficients make it possible to discuss the reaction mechanisms from the point of view of solvent effects.

A four-parameter equation of state

A new equation of state which contains four parameters and remains cubic in terms of volume has been developed and successfully applied to the representation of properties of pure components (compressibility factor, fugacity coefficient, enthalpy departure, vapor pressure, and saturated liquid and vapor volumes) and mixtures (vapor-liquid equilibria and compressibility factor). The pure component parameters have been evaluated on the basis of optimal representation of the critical isotherm, and correlated in terms of critical temperature, critical pressure and the acentric factor. The mixing rules used include only one binary interaction parameter, the values of which have been obtained for twenty-six binary mixtures.

Quantitative Radiation Dose-Response Relationships for Normal Tissues in Man: I. Gustatory Tissue Response during Photon and Neutron Radiotherapy

Quantitative radiation dose-response curves for normal gustatory tissues in man were studied. Taste function, expressed as taste loss, was evaluated in 84 patients who were given either photon or neutron radiotherapy for tumors in the head and neck region. Patients were treated to average tumor doses of 6600 cGy (photon) or 2200 cGy (neutron) using standard treatment techniques. Taste loss was determined in patients using a standard forced-choice, three-stimulus drop technique prior to radiotherapy and at 1000-cGy intervals for photon patients and 320-cGy intervals for neutron patients during radiotherapy. The dose-response curves for photons and neutrons were analyzed by fitting a four-parameter logistic equation to the data. Photon and neutron curves differed principally in their relative position along the dose axis. Comparison of the dose-response curves was made by determination of RBE. At 320 cGy, the lowest neutron dose at which taste measurements were made, RBE = 5.7. If this RBE is correct, then ...

Fluorescence quenching in dodecylammonium propionate reversed micelles

Depending on the kind of quencher (capacity and concentration), multi-exponential decay curves are obtained which can be analysed by a four-parameter equation, on the assumption that ion exchange between micelles is possible. The occurrence of probe exchange by intermicellar collisions is also suggested.

Equations of state of polymer melts and temperature dependence of the number of external degrees of freedom in a macromolecule

The theory of the liquid state developed by Flory is reviewed and its assumptions compared with those in the recent model of thermal expansion of polymer melts developed by the author. Methods of evaluation of the characteristic parameters p ∗, V ∗ and T ∗ are discussed. The number c of external degrees of freedom of a polymeric segment does depend on the temperature; consequences of disregarding this dependence are noted. It is shown that a simplification of the Flory equation of state leads to the van der Waals equation. The four-parameter equation of state proposed by the author, simplified to a three-parameter form, is consistent with the principle of corresponding states.

A Method for Determining the Biological Effect of Herbicide Mixtures

Experiments were conducted in growth chambers by growing oats (Avena sativaL. ‘Ansi’) in water culture and soil with admixed TCA (trichloroacetic acid), pyrazon [5-amino-4-chloro-2-phenyl-3-(2H)-pyridazinone], and two mixtures (pyrazon: less TCA; 1:25 and 1:50). The sigmoid log-dose relationship for dry-matter yield was fitted with a four-parameter logistic equation. Generally, the mixtures appeared to be less active than would be expected under an additive dose model. Around GR50the departure of joint action of the mixtures from the reference model was almost similar in the two growth media. At higher dose levels the mixtures approached the reference model, and the bioactivity of one mixture (1:25) in the water culture experiment even showed a higher activity than would be expected from the reference model.

Parametric correlation of formation constants in aqueous solution. 1. Ligands with small donor atoms

It is shown that four-parameter equations proposed by previous authors have poor predictive powers for data of formation constants in aqueous solution. This relates partly to the paucity of data on complexes of ligands such as ammonia which, with most metal ions, cannot exist in water because of hydrolysis. Equations previously proposed that relate the formation constants of polyamine and poly(aminbcarboxy1ate) complexes to those of the ammonia and acetate complexes are used to calculate formatio‘n constants for these hydrolysis-prone ammonia complexes. An equation of the type log K, = EAEB + CACB, where E and C are identified with the tendencies of the Lewis acid A and base B to undergo ionic and covalent bonding, was used to correlate the F-, OH-, and NH, formation constants of 27 Lewis acids to a standard deviation of 0.24 log unit. Hardness parameters l!fA and HB were defined as EA/CA and EB/CB for acids and bases, respectively, and gave reasonable orders of hardness It was found that for ligands with large donor atoms, such as C1-, and for sulfur or phosphorus donor atoms, deviations from the predictions of this equation were observed that appeared to be related to the size of the acid, so that no deviations were observed for large cations such as Ag’ and Pb2+, aith Occurrence of large deviations for smaller cations such as Cu” or Ni2+, with the very largest occurring for the proton. These deviations were attributed to steric hindrance between the large donor atom and adjacent coordinated water molecules. The important contribution of the Edwards equation’ lies in illustrating that formation constant data for complexes of unidentate ligands in aqueous solution need at least a dualbasicity scale for any kind of correlation to be obtained at all. There have been several other multiparameter equation^^-^ proposed for correlating formation constant data. The need for at least a dual-basicity equation has manifested itself in the classification of metal ions’into Schwart~enbach’s~ and Ahrland and Chatt’s6 A- and B-type metal ions and Pearson’s more general classification7 of Lewis acids and bases into hard and soft acids and bases (HSAB). Drago and co-workers* have had considerable success with a more generalized form of a four-parameter equation similar to eq 1 for the correlation of enthalpy changes on adduct formation in solvents of low dielectric constant. In eq 1, which resembles Drago’s expression, except that -AHo has been replaced with log K1, C and E are identified with the tendency of each Lewis acid A or base B to undergo covalent or ionic bonding. This interpretation of the significance of the C and E parameters seems most reasonable in the light of the work of Klopman9 on the origin of hardness and softness in acids and bases. Klopman9 found softness to be associated Rith “frontier-controlled’’ (covalent) and hardness with “charge-controlled’’ (ionic) bonding in calculations based upon polyelectronic pertubation theory. We have therefore adopted the E and C formalism of Drago and co-workers8 in this paper as probably being the best interpretation of the parameters in a four-parameter equation such as (1).

Prediction of core electron binding energies with a four-parameter equation

Chemical shifts in core electron binding energies can be predicted by the equation ..delta..E/sub B/ = aF + bR, where the parameters a and b are characteristic of the class of molecule and atom to which the binding energies pertain, and the parameters F and R are characteristic of substituent groups. The F and R parameters are analogous to the Swain and Lupton ''F'' and ''R'' parameters; i.e., they measure the sigma and ..pi.. electronegativities, respectively, of substituents. However, the F and R values are appropriate only for processes in which a localized positive charge develops on an atom, whereas the ''F'' and ''R'' values are appropriate for ordinary chemical reactions (including both electrophilic and nucleophilic substitutions). Thus lone-pair ionization potentials and proton affinities can be correlated with F and R values more satisfactorily than with ''F'' and ''R'' values.

The thermodynamic and the kinetic limits of superheat of a liquid

Thermodynamic, mechanical stability considerations place an upper limit on the domain of a metastable, superheated liquid. This spinodal temperature or thermodynamic limit of superheat (TLS) exceeds the kinetic limit of superheat (KLS) determined by droplet-heating and pulse-heating experiments or predicted by homogeneous nucleation theory. The two-parameter equations of state of van der Waals, scaled particle theory, Redlich-Kwong, and Berthelot, all fail to exceed the maximum observed KLS. These low estimates are expected because all of these equations predict a tangent to the vapor pressure curve, dP s dT , which is too small at the critical point. A four-parameter equation of state is explored with the form ( P + a T mV n ) (V - b) = RT . The parameters are adjusted so that T, P, V, and dP s dT all have the correct value at the critical point. This equation yields TLS estimates which not only exceed the maximum KLS, but also provide a reasonable extrapolation of the KLS data to the maximum possible bubble nucleation flux.

Ethyl fluoride vapor pressure

The vapor pressure of ethyl fluoride was measured from 227.65 to 322.48K (0.6893-16.77 atm). The data were In agreement wlth llterature vapor-pressure values that had been obtalned over both a lowerand a higher-pressure range than the present work. A four-parameter vaporpressure equation for ethyl fluorlde was obtalned by regressing the comblned experimental data. The vapor pressure of ethyl fluoride Is glven by In P = 66.4460 3939.51/T+ 0.0159614T9.79050 In T. In this equation, 7 is in K and P Is In atmospheres. This equation fit the comblned experimental data (0.01316-49.62 alm) to within an average error of 0.35 % and a maxlmum error of 0.89 %. The data are In agreement wlth the tabular values reported by the Thermodynamlcs Research Center Data Project but lndlcate an average vapor-pressure devlatlon greater than 30% for the values reported In the “Chemical Engineers’ Handbook.”

Additions and Corrections: A Four-Parameter Equation for Predicting Enthalpies of Adduct Formation

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTAdditions and Corrections: A Four-Parameter Equation for Predicting Enthalpies of Adduct FormationRussell Drago, Glenn Vogel, and Terence NeedhamCite this: J. Am. Chem. Soc. 1972, 94, 7, 2557Publication Date (Print):April 1, 1972Publication History Published online8 July 2003Published inissue 1 April 1972https://doi.org/10.1021/ja00762a602RIGHTS & PERMISSIONSArticle Views38Altmetric-Citations-LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (121 KB) Get e-Alerts Get e-Alerts

Prediction of gas-phase Lewis acid-base dissociation heats with an empirical four-parameter equation

Prediction of gas-phase Lewis acid-base dissociation heats with an empirical four-parameter equation

Four-parameter equation for predicting enthalpies of adduct formation

Four-parameter equation for predicting enthalpies of adduct formation

Polymer rheology: Influence of molecular weight and polydispersity

AbstractLiterature data on the non‐Newtonian flow of bulk polymer and of polymer solutions are correlated on the basis of a four‐parameter equation, η = η∞ + (η0 − η∞)/[1 + (τD)m], η being the viscosity at shear rate D, and η0 and η∞ limiting values at D = 0 and D = ∞, respectively. The parameters η0, η∞, and τ all show dependence on molecular weight, and in general there is good correlation between τ and η0. There is evidence that τ is related to a molecular weight higher than the weight‐average. The exponent m shows dependence on molecular weight distribution and approaches an upper limit of unity for a monodisperse linear polymer. For linear unblended polymers it may be expressed empirically by m = (M̄n/M̄w)1/5.

Rate correlations involving the linear combination of substituent parameters. Part IV. Base-catalysed hydrolysis of triethylphenoxysilanes

The base-catalysed hydrolysis of meta- and para-substituted triethylphenoxysilanes in 60% aqueous dioxan at 30·2° has been followed spectrophotometrically. The rate constant for spontaneous hydrolysis, ksp, and the catalytic rate constants kB and kOH have been evaluated. kB and ksp are shown to correlate better with a four-parameter equation involving σ and σ– than with the standard Hammett equation. The precision of the kOH values obtained was much less than either of the other rate constants and only an approximately linear dependence of kOH on σ could be discerned.

Induced Rotation—Translation Spectra of Solutions of H2, HD, and D2 in Liquid Argon

Dilute solutions of H2, D2, and HD in liquid argon are studied by examining the 1000–140-cm−1 region of their induced absorption spectra. Since the absorption features overlap extensively, an empirical profile analysis is performed. For H2 and D2 several relatively sharp absorptions (Δν1/2≈30 cm−1), which have a Lorentzian line shape and closely correspond to the expected transitions of the freely rotating solute molecules, are observed. The width of these lines appears to be due to uncertainty broadening which depends on the time the absorbing solute molecule spends in a particular solvent cavity. Other features, with a 150-cm−1 half-width, are fitted with a four-parameter exponential equation and are believed to involve quantized translational energy levels of the solute. The range of translational energy levels, which is determined by the distribution of solvent cavity sizes, is responsible for their increased bandwidths. The HD spectrum is unusual in that the rotational selection rules are relaxed and transitions corresponding to ΔJ = +2, +3, and +4 are observed. Furthermore, all of the rotational absorption features are anomalously broad (Δν1/2≈50 cm−1) and their maxima are shifted from the gas-phase frequencies in an irregular pattern. Because of its asymmetrical mass distribution, rotational motion of HD is accompanied by a translation of the molecule in its solvent cavity. This rotation—translation interaction has been given quantitative form in a perturbation treatment by Friedmann and Kimel, and the results are consistent with the observed selection rules, frequency shifts, and linewidths in the spectrum of HD in liquid argon.