Rate constant, relationship

Borosy, A.P., Balogh, B. and Matyus, P. (2005) Alignment-free descriptors for quantitative structure-rate constant relationships of [4 + 2] cycloadditions./. Mol, Struct. (Theochem), 729, 169-176. 

Hydrolysis of methyl acetate in a dilute aqueous solution at pH < 7 follows the first reaction order. Table 4.1 contains experimental reagent concentrations at different moments of time for seven experiments. Each series was measured under constant temperature. Determine the rate constant for each experiment and, based on the temperature-rate constant relationship, find the activation energy and pre-exponential factor. 

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

The relative basicities of aromatic hydrocarbons, as represented by the equilibrium constants for their protonation in mixtures of hydrogen fluoride and boron trifluoride, have been measured. The effects of substituents upon these basicities resemble their effects upon the rates of electrophilic substitutions a linear relationship exists between the logarithms of the relative basicities and the logarithms of the relative rate constants for various substitutions, such as chlorination and... 

Remembering that the observed second-order rate constant is merely the rate divided by the product of the stoichiometric concentrations of aromatic compound and nitric acid, the following relationship can be... 

Curve-Fitting Methods In the direct-computation methods discussed earlier, the analyte s concentration is determined by solving the appropriate rate equation at one or two discrete times. The relationship between the analyte s concentration and the measured response is a function of the rate constant, which must be measured in a separate experiment. This may be accomplished using a single external standard (as in Example 13.2) or with a calibration curve (as in Example 13.4). 

An important application of photochemical initiation is in the determination of the rate constants which appear in the overall analysis of the chain-growth mechanism. Although we shall take up the details of this method in Sec. 6.6, it is worthwhile to develop Eq. (6.7) somewhat further at this point. It is not possible to give a detailed treatment of light absorption here. Instead, we summarize some pertinent relationships and refer the reader who desires more information to textbooks of physical or analytical chemistry. The following results will be useful ... 

Figure 6.4 Schematic relationship between various experimental quantities (Rp, n j, and r) and the rate constants (k j,kp, and k ) derived therefrom.

Although there are a total of four different rate constants for propagation, Eq. (7.12) shows that the relationship between the relative amounts of the two monomers incorporated into the polymer and the composition of the monomer feedstock involves only two ratios of different pairs of these constants. Accordingly, we simplify the notation by defining... 

Condensation occurs most readily at a pH value equal to the piC of the participating silanol group. This representation becomes less vaUd at pH values above 10, where the rate constant of the depolymerization reaction k 2 ) becomes significant and at very low pH values where acids exert a catalytic influence on polymerization. The piC of monosilicic acid is 9.91 0.04 (51). The piC value of Si—OH decreases to 6.5 in higher order sihcate polymers (52), which is consistent with piC values of 6.8 0.2 reported for the surface silanol groups of sihca gel (53). Thus, the acidity of silanol functionahties increases as the degree of polymerization of the anion increases. However, the exact relationship between the connectivity of the silanol sihcon and SiOH acidity is not known. 

Although the correlation function formalism provides formally exact expressions for the rate constant, only the parabolic barrier has proven to be analytically tractable in this way. It is difficult to consistently follow up the relationship between the flux-flux correlation function expression and the semiclassical Im F formulae atoo. So far, the correlation function approach has mostly been used for fairly high temperatures in order to accurately study the quantum corrections to CLST, while the behavior of the functions Cf, Cf, and C, far below has not been studied. A number of papers have appeared (see, e.g., Tromp and Miller [1986], Makri [1991]) implementing the correlation function formalism for two-dimensional PES. 

The goal of a kinetic study is to establish the quantitative relationship between the concentration of reactants and catalysts and the rate of the reaction. Typically, such a study involves rate measurements at enough different concentrations of each reactant so that the kinetic order with respect to each reactant can be assessed. A complete investigation allows the reaction to be described by a rate law, which is an algebraic expression containing one or more rate constants as well as the concentrations of all reactants that are involved in the rate-determining step and steps prior to the rate-determining step. Each concentration has an exponent, which is the order of the reaction with respect to that component. The overall kinetic order of the reaction is the sum of all the exponents in the... 

The rates of hydrolysis of the ester group in compounds A and B have been compared. The effect of an added metal ion (Np+) on the rate of hydrolysis has been studied, and the observed rate constants for attack by OH are tabulated. Suggest the most favorable transition-state stmcture for the addition step of the hydrolysis reaction for each substrate under each set of conditions. Discuss the relationship between the stmctures of these transition states and the relative rates of attack by hydroxide ion. 

Here, we shall examine a series of processes from the viewpoint of their kinetics and develop model reactions for the appropriate rate equations. The equations are used to andve at an expression that relates measurable parameters of the reactions to constants and to concentration terms. The rate constant or other parameters can then be determined by graphical or numerical solutions from this relationship. If the kinetics of a process are found to fit closely with the model equation that is derived, then the model can be used as a basis for the description of the process. Kinetics is concerned about the quantities of the reactants and the products and their rates of change. Since reactants disappear in reactions, their rate expressions are given a... 

Common unit for Relationship between rate constant rate constants... 

The equations for the Zf ro order reaction ore true for any stoichiometry, hut the relationship between the rate constants depends on the stoichhnetry. 

A plot of In Pa/Pao against time t (Figure 3-23) gives a straight line with the slope equal to the rate constant kj. Therefore, the assumed first order for the reaction is coiTect. The relationship between die rado of Pa/Pao versus time t is represented by die model equadon Y = Ae . ... 

Figure 3-24 shows the relationship between 1/C as a function of time t. The graph is a straight line, therefore, the assumed order of the reaction is correct. The slope of the line from the regression analysis is the rate constant k. 

Show that the reactor temperature must be maintained at 34°C given that tlie reaction is first order with respect to anhydride concentration and the rate constant k can be determined from the relationship... 

In addition to the elimination rate constant, the half-life (T/i) another important parameter that characterizes the time-course of chemical compounds in the body. The elimination half-life (t-1/2) is the time to reduce the concentration of a chemical in plasma to half of its original level. The relationship of half-life to the elimination rate constant is ti/2 = 0.693/ki,i and, therefore, the half-life of a chemical compound can be determined after the determination of k j from the slope of the line. The half-life can also be determined through visual inspection from the log C versus time plot (Fig. 5.40). For compounds that are eliminated through first-order kinetics, the time required for the plasma concentration to be decreased by one half is constant. It is impottant to understand that the half-life of chemicals that are eliminated by first-order kinetics is independent of dose. ... 

Equation (5-69) describes rate-equilibrium relationships in terms of a single parameter, the intrinsic barrier AGo, which therefore assumes great importance in interpretations of such data. It is usually assumed that AGo is essentially constant within the reaction series then it can be estimated from a plot of AG vs. AG° as the value of AG when AG = 0. Another method is to fit the data to a quadratic in AG and to find AGq from the coefficient of the quadratic term. ... 

Chemical reaction rates increase with an increase in temperature because at a higher temperature, a larger fraction of reactant molecules possesses energy in excess of the reaction energy barrier. Chapter 5 describes the theoretical development of this idea. As noted in Section 5.1, the relationship between the rate constant k of an elementary reaction and the absolute temperature T is the Arrhenius equation ... 

If k is much larger than k", Eq. (6-64) takes the form of Eq. (6-61) for the fraction Fhs thus we may expect the experimental rate constant to be a sigmoid function of pH. If k" is larger than k, the / -pH plot should resemble the Fs-pH plot. Equation (6-64) is a very important relationship for the description of pH effects on reaction rates. Most sigmoid pH-rate profiles can be quantitatively accounted for with its use. Relatively minor modifications [such as the addition of rate terms first-order in H or OH to Eq. (6-63)] can often extend the description over the entire pH range. 

The most common manifestation of extrathermodynamic relationships is a linear correlation between the logarithms of rate or equilibrium constants for one reaction series and the logarithms of rate or equilibrium constants of a second reaction series, both sets being subjected to the same variation, usually of structure. For illustration, suppose the logarithm of the rate constants for a reaction series B is linearly correlated with the logarithm of the equilibrium constants for a reaction series A, with substituent changes being made in both series. The empirical correlation is... 

It was not until the 1970s that the statistics of the isokinetic relationship was satisfactorily worked out.Exner first took this approach Let k, and 2 be the rate constants for a member of a reaction series at temperatures T, and T2, with T2 > T, and let k° and k° be the corresponding values for the reference member of the series. Then Eqs. (7-76) and (7-77) are easily derived for the reaction series. 

These rate constants are for the cinnamoylation of hydroxy compounds by cinnamic anhydride catalyzed by A-methylimidazole. The reaction is first-order in each reactant. The kinetics were followed spectrophotometrically in acetonitrile solution. Analyze the data that is, attempt to account for the relationship between structure and reactivity. 

There is a third experimental design often used for studies in electrolyte solutions, particularly aqueous solutions. In this design the reaction rate is studied as a function of ionic strength, and a rate variation is called a salt effect. In Chapter 5 we derived this relationship between the observed rate constant k and the activity coefficients of reactants l YA, yB) and transition state (y ) ... 

The quantity on the left side of the equation is referred to as the logarithm of the rate constant corrected for protonation often the correction term is negligible. If the activity coefficient term on the right side is negligible, Eq. (8-104) predicts a linear relationship between the corrected log kobs and —//o, the slope being unity. A similar treatment of the A-Se2 mechanism also predicts a linear plot of log fcobs against —Ho-... 

In systems such as the 2- and 6-hydroxypteridines, sudden addition of an alkaline solution to a neutral buffer, or of a neutral solution to an alkaline buffer, displaces the equilibrium between hydrated and anhydrous species (because the anions are less hydrated than the neutral molecules). By measuring the time-dependent change of optical density at a selected wavelength, a first-order rate constant, obs5 can be obtained. This constant is a composite one, and to see its relationship to other quantities some discussion is necessary. 

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