Thermodynamic Analysis of the Hammett Equation, the Temperature Dependence of ρ, and the Isoequilibrium (Isokinetic) Relationship

1971 ◽  
Vol 49 (17) ◽  
pp. 2803-2807 ◽  
Author(s):  
Loren G. Hepler
Keyword(s):  
Differential Equations ◽  
Temperature Dependence ◽  
Thermodynamic Analysis ◽  
Activation Parameters ◽  
Isokinetic Temperature ◽  
Hammett Equation ◽  
Isokinetic Relationship ◽  
Solvent Interactions ◽  
Independent Constant ◽  
The Relationship

Exact thermodynamic analysis of the Hammett equation has led to four differential equations relating δΔH0, δΔS0, δΔCp0, dρ/dT, and d2ρ/dT2. Similar equations can be obtained in terms of activation parameters ΔH≠, etc. For temperature independent δΔH0 and δΔS0 and therefore δΔCp0 = 0, two of these differential equations lead to ρ = ρ∞ [1–(β1/T)] and the familiar isoequilibrium (isokinetic) equation δΔH0 = β1δΔS0. The "isoequilibrium (isokinetic) temperature" represented here by β1 is a temperature independent constant of integration. For constant non-zero δΔCp0 we similarly obtain more complicated expressions for ρ and the "isoequilibrium (isokinetic) temperature." These findings are considered in relation to a model in which environmental contributions (due to solute–solvent interactions) to δΔH0 and δΔS0 are related by a parameter βc. The relationship between β1, and βc is established, and it is shown that in general β1 ≠ βc.

Concerning the Problem of the Isokinetic Relationship. IV. The Evaluation of the Isosubstituent Hammett Parameter by Means of Isokinetic and Isoequilibrium Temperatures

1986 ◽  
Vol 39 (2) ◽  
pp. 199 ◽  
Author(s):  
W Linert
Keyword(s):  
Temperature Dependence ◽  
Activation Parameters ◽  
Common Point ◽  
Isokinetic Temperature ◽  
Benzoic Acids ◽  
Reaction Series ◽  
Isokinetic Relationship ◽  
The Common ◽  
Point Of Intersection ◽  
Hammett Parameter

The interrelation between the isosubstituent and the isokinetic relationship is developed and tested for several Hammett reaction series. Two methods of approach to relate the temperature dependence of Arrhenius and Hammett plots are given, one utilizing the isoequilibrium temperature of the ionization of benzoic acids and the other the isokinetic temperature of the respective Hammett reaction series. The efficiency of the approaches for the evaluation of the coordinates of the common point of intersection in the Hammett plot, i.e. the characteristics of the isosubstituent relationship, are compared with each other and with experimental results. With the former approach, by using the activation parameters of only one (commonly the unsubstituted ) member of the series, the temperature dependence of a Hammett line can be predicted provided that the isokinetic temperature of the Hammett series does not approach the experimental temperature range. Otherwise the latter approach must be used which, however, needs the temperature dependence of at least two (or better more) members of the reaction series.

Concerning the Problem of the Isokinetic Relationship. III. The Temperature-Dependance of the hammett Equation

1985 ◽  
Vol 38 (5) ◽  
pp. 677 ◽  
Author(s):  
W Linert ◽  
R Schmid ◽  
AB Kudrjawtsev
Keyword(s):  
Benzoic Acid ◽  
Hammett Equation ◽  
Reaction Series ◽  
Temperature Dependent ◽  
Isokinetic Relationship ◽  
A Value ◽  
The Common ◽  
Point Of Intersection ◽  
Temperature Dependance ◽  
The Relationship

It is shown that the temperature-dependence of the Hammett equation is, in contrast to tradition, both physically and experimentally better described by means of temperature-dependent σ and temperature- independent ρ (termed ρo). The relationship between ρo and the customary (temperature dependent) ρ is ρT = ρo(1/T-1/Tbiso)/(1/T-1/Tbiso) where Tbiso , is the isoequilibrium temperature of the benzoic acid ionization, for which the present analysis suggests a value of -255 K, and T is 298 K. In these terms, the temperature variation of the Hammett equation can be evaluated by supplying merely E(u)a (the activation energy for the reaction of the unsubstituted reactant) and ρo, in that the σ value for the isokinetic substituent , i.e., the abscissa of the common point of intersection in the Hammett plot, is σiso = (1/T-1/Tbiso)E(u)a/(2.303Rρo) = E(u)a/(2630po) Further, ρo I related to energies ρo = E(u)a/(ΔH°u-ΔH°s(iso))where ΔH°u and ΔH°s(iso) are the ionization enthalpies of the parent benzoic acid and that bearing the isokinetic substituent , respectively. Analogous equations apply to thermodynamic reaction series when substituting E(u)a for ΔH°u(series). Along these lines the interpretation of the customary Hammett plot is advanced.

The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 170-184
Author(s):  
Richard Atkins
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shortest Paths ◽  
Equivalence Problem ◽  
Second Order ◽  
System Of Differential Equations ◽  
Lagrange Equations ◽  
Invariant Functions ◽  
Underlying Geometry ◽  
The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

scholarly journals Mechanistic aspects of oxidation on L-tyrosine by diperiodatocuprate(III) complex in alkali media: a kinetic model

2009 ◽  
Vol 7 (4) ◽  
pp. 929-937 ◽  
Author(s):  
Nagaraj Shetti ◽  
Rajesh Hegde ◽  
Sharanappa. Nandibewoor
Keyword(s):  
Alkaline Medium ◽  
Activation Parameters ◽  
Spectral Studies ◽  
Slow Step ◽  
Isokinetic Temperature ◽  
Thermodynamic Quantities ◽  
Radical Intermediate ◽  
Different Temperatures ◽  
Reactive Oxidant ◽  
Reaction Constants

AbstractOxidation of an amino acid, L-tyrosine (L-Tyr) by diperiodatocuprate(III) (DPC) in alkaline medium at a constant ionic strength of 0.1 mol dm−3 was studied spectrophotometrically at different temperatures (288.1–313.1 K). The reaction between DPC and L-Tyr in alkaline medium exhibits 1:4 stoichiometry (L-Tyr:DPC). Intervention of free radicals was observed in the reaction. Based on the observed orders and experimental evidence, a mechanism involving monoperiodatocuprate(III) (MPC) as the reactive oxidant species has been proposed. A suitable mechanism is proposed through the formation of a complex and free radical intermediate. The products were identified by spot test and characterized by spectral studies. The reaction constants involved in the different steps of the mechanism were calculated. The activation parameters with respect to slow step of the mechanism were computed and are discussed. The thermodynamic quantities were determined for different equilibrium steps. Isokinetic temperature was also calculated and found to be 252.3 K.

scholarly journals The role of enzyme dynamics and tunnelling in catalysing hydride transfer: studies of distal mutants of dihydrofolate reductase

2006 ◽  
Vol 361 (1472) ◽  
pp. 1307-1315 ◽  
Author(s):  
Lin Wang ◽  
Nina M Goodey ◽  
Stephen J Benkovic ◽  
Amnon Kohen
Keyword(s):  
Temperature Dependence ◽  
Dihydrofolate Reductase ◽  
Isotope Effects ◽  
Hydride Transfer ◽  
Activation Parameters ◽  
Physical Nature ◽  
Enzyme Dynamics ◽  
Temperature Dependences ◽  
Donor Acceptor ◽  
Significant Temperature

Residues M42 and G121 of Escherichia coli dihydrofolate reductase ( ec DHFR) are on opposite sides of the catalytic centre (15 and 19 Å away from it, respectively). Theoretical studies have suggested that these distal residues might be part of a dynamics network coupled to the reaction catalysed at the active site. The ec DHFR mutant G121V has been extensively studied and appeared to have a significant effect on rate, but only a mild effect on the nature of H-transfer. The present work examines the effect of M42W on the physical nature of the catalysed hydride transfer step. Intrinsic kinetic isotope effects (KIEs), their temperature dependence and activation parameters were studied. The findings presented here are in accordance with the environmentally coupled hydrogen tunnelling. In contrast to the wild-type (WT), fluctuations of the donor–acceptor distance were required, leading to a significant temperature dependence of KIEs and deflated intercepts. A comparison of M42W and G121V to the WT enzyme revealed that the reduced rates, the inflated primary KIEs and their temperature dependences resulted from an imperfect potential surface pre-arrangement relative to the WT enzyme. Apparently, the coupling of the enzyme's dynamics to the reaction coordinate was altered by the mutation, supporting the models in which dynamics of the whole protein is coupled to its catalysed chemistry.

scholarly journals Boundedness and oscillation of third order neutral differential equations

2009 ◽  
Vol 43 (1) ◽  
pp. 137-144 ◽  
Author(s):  
Božena Mihalíková ◽  
Eva Kostiková
Keyword(s):  
Differential Equations ◽  
Third Order ◽  
Neutral Differential Equations ◽  
The Third ◽  
The Relationship ◽  
Oscillation Of Solutions

Abstract The relationship between boundedness and oscillation of solutions of the third order neutral differential equations are presented.

scholarly journals A Study on the Kinetics and Mechanism of the One-Pot Formation of 3,4,5-Substituted Furan-2(5H)-Ones in the Presence of Lactic Acid: Effect of Different Substituents

2018 ◽  
Vol 43 (3-4) ◽  
pp. 286-299 ◽  
Author(s):  
Osman Asheri ◽  
Sayyed Mostafa Habibi-Khorassani ◽  
Mehdi Shahraki
Keyword(s):  
Lactic Acid ◽  
Activation Parameters ◽  
General Mechanism ◽  
Common Mechanism ◽  
Substituted Anilines ◽  
One Pot ◽  
Isokinetic Relationship ◽  
Zero Order Kinetics ◽  
Different Temperatures ◽  
The One

The kinetics of the reaction between para-substituted anilines and dimethyl acetylenedicarboxylate (DMAD) with derivatives of benzaldehyde for the one-pot formation of 3,4,5-substituted furan-2(5 H)-ones in the presence of lactic acid as a catalyst have been studied spectrophotometrically at different temperatures. A mechanism involving four steps was proposed for the reactions, all of which followed second-order kinetics. The partial orders with respect to substituted aniline and DMAD were one and one and the reactions revealed zero-order kinetics for benzaldehyde and its derivatives. Changing of substituents on benzaldehyde left rates of reaction unaffected. However, various substituents on aniline showed that para electron-withdrawing groups decreased the rate of reaction. According to investigation of an isokinetic relationship, a common mechanism exists for all studied substituents and a general mechanism can be formulated. Kinetic values ( k and Ea) and associated activation parameters (Δ G‡, Δ S‡ and Δ H‡) of the reactions were determined.

scholarly journals Pendulum with aerodynamic and viscous damping

2020 ◽  
Vol 3 (2) ◽  
pp. 43-47
Author(s):  
Herlin Soraya
Keyword(s):  
Differential Equations ◽  
Stable State ◽  
The State ◽  
Linear Differential Equations ◽  
Viscous Damping ◽  
Viscous Damper ◽  
Aerodynamic Damping ◽  
Non Linear ◽  
Linear Differential ◽  
The Relationship

In this paper we discuss about how the relationship between non-linear differential equations on aerodynamic damping with linearly viscous damping equations. And it turns out after analyzing that the changes that occur pendulum that changes from the start of the maximum state to a stable state takes time so that changes that occur until the state is stable, this change can be reduced with the use of viscous damper

EQUIVALENCE OF THE TWO METHODS IN CALCULATING THE EQUILIBRIUM SHAPES OF CYLINDRICAL VESICLES

1999 ◽  
Vol 13 (16) ◽  
pp. 547-553
Author(s):  
SHAOGUANG ZHANG ◽  
ZHONGCAN OUYANG ◽  
JIXING LIU
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
The Other ◽  
General Shape ◽  
Calculus Of Variation ◽  
And Topology ◽  
Shape Equation ◽  
Equilibrium Shapes ◽  
Comparison Of The Results ◽  
The Relationship

So far, two methods are often used in solving the equilibrium shapes of vesicles. One method is by starting with the general shape equation and restricting it to the shapes with particular symmetry. The other method is by assuming the symmetry and topology of the vesicle first and treating it with the calculus of variation to get a set of ordinary differential equations. The relationship between these two methods in the case of cylindrical vesicles, and a comparison of the results are given.

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