Second order rate constants dimensions

First-order and second-order rate constants have different dimensions and cannot be directly compared, so the following interpretation is made. The ratio intra/ inter has the units mole per liter and is the molar concentration of reagent Y in Eq. (7-72) that would be required for the intermolecular reaction to proceed (under pseudo-first-order conditions) as fast as the intramolecular reaction. This ratio is called the effective molarity (EM) thus EM = An example is the nu-... 

Analysis of the variation of the overall rate constant of reaction with [surfactant] was discussed in Section 3 (p. 222) and the treatment allows calculation of the second-order rate constants of reaction in the micellar pseudophase. These rate constants can be compared with second-order rate constants in water provided that both constants are expressed in the same dimensions and typically the units are M-1 s-1. Inevitably the comparison... 

Since the collision radius for two particles of equal size is two times the particle radius, the effective volume swept out will be four times that given by Eq. 9-33. Since both particles are diffusing, the effective diffusion constant will be twice that used in obtaining Eq. 9-28. Thus, the effective volume swept out by the particle in a second will be eight times that given by Eq. 9-33. The volume swept out by one mole of particles is equal to /cD (recall that the second-order rate constant has dimensions of liter mol-1 s 1). Thus, when converted to a moles per liter basis and multiplied by 8, Eq. 9-33 should (and does) become identical with the Smolu-chowski equation (Eq. 9-30). 

Significance of the Michaelis Constant, Km. The Michae-lis constant Km has the dimensions of a concentration (molarity), because k x and k2, the two rate constants in the numerator of equation (23), are first-order rate constants with units expressed per second (s 1), whereas the denominator fc is a second-order rate constant with units of m-is-1. To appreciate the meaning of Km, suppose that [S] = Km. The denominator in equation (25) then is equal to 2[S], which makes the velocity v = VmaJ2. Thus, the Km is the substrate concentration at which the velocity is half maximal (fig. 7.6). 

For the reaction A + A — P, we can write —dAJdt = k(A)(A) = k A2. In this case, the rate involves a higher power of the concentration (n = 1 + 1 = 2) and the reaction is second-order. For the reaction A -I- B —> P, the rate is proportional to the first power of each reactant and —d(A)(B)/dt = k(A)(B). The reaction is first-order with respect to A or B, but the overall reaction is second-order, because the right-hand side of the equation contains the product of two concentrations. The value of tV2 will depend on the initial reactant concentration(s), and the second-order rate constants have the dimensions of reciprocal concentration times time. Reactions of higher order, such as third order, are relatively rare, and their rates are proportional to the product of three concentration terms. 

Encounter-controlled rate — A -> reaction rate corresponding to the rate of encounter of the reacting -> molecular entities. This rate is also known as diffusion-controlled rate since rates of encounter are themselves controlled by -> diffusion rates (which in turn depend on the - viscosity of the medium and the dimensions of the reactant molecular entities). For a bi-molecular reaction between solutes in water at 25 °C an encounter-controlled rate is calculated to have a second-order rate constant of about 1010 dm3 mol-1 s 1. 

The term k is, again, the rate constant for the reaction, but in a second-order process k has dimensions of concentration-1 time-1. The relationship between the half-life and the second-order rate constant, k, for initial equal concentrations of reactant can be found by substituting t = h into equation (9.5) as follows ... 

The absolute values of the four maximum rate constants, A max depend on the theory applied here. The rate constant k is a second-order rate constant with a dimension of cm s, provided that the concentration of the carrier density and of the redox system are given in units of cm, and k is related to the local rate constant k y [s" ] by... 

Equations 2 and 5 assume that reactions are sufficiently slow for equilibrium to be maintained between aqueous and micellar pseudophases, but this condition is easily satisfied for most thermal reactions. The form of equation 5 is such that rate-surfactant profiles can be fit relatively easily, but the second-order rate constants, kw and kM, have different dimensions and cannot be compared directly. Comparison can be made by converting concentration in the micellar pseudophase from a mole ratio (equation 4) into a... 

The second order rate constants, A and k, for reactions in the micellar and aqueous pseudophases have the same dimensions, and can now be compared directly, and within all the uncertainties of the treatment it seems that A and k are of similar magnitudes for most reactions, and in some systems A > A . This generalization is strongly supported by evidence for reactions of relatively hydro-phobic nucleophilic anions such as oximate, imidazolide, thiolate and aryloxide, typically with carboxylate or phosphate esters [61,82-85]. These similarities of second-order rate constants in the aqueous and micellar pseudophases are consistent with both reactants being located near the micellar surface in a water-rich region. Therefore the micellar rate enhancements of bimolecular reactions are due largely to concentration of the reactants in the small volume of the micelles. Some examples are in Table 3 for reactions of or hydrophilic nucleophilic anions and in Table 4 for reactions of more hydrophobic nucleophiles. 

However, as for reactions in non-functional micelles, sec , cannot be compared directly with second-order rate constants in water, whose dimensions are, conventionally, M /sec. But this comparison can be made provided that one specifies the volume element of reaction, which can be taken to be the molar volume of the micelles, or the assumed molar volume of the micellar Stern layer. This choice is an arbitrary one, but the volumes differ by factors of ca. 2 [107,108], so it does not materially affect the conclusions. The rate constants in the micelle for dephosphorylation, deacylation and nucleophilic substitution by a functional hydroxyethyl surfactant are similar to those in water [99], and similar results have been observed for dephosphorylation using functional surfactants with imidazole and oximate [106,107]. Similar results have been obtained by Fornasier and Tonalleto [108] for deacylation of carboxylic esters by a variety of functional comicelles. 

What is a pseudo-first-order rate constant How do its dimensions differ from those of a second-order rate constant ... 

In reactions between dipolar molecules, Laidler and Eyring have used the Kirkwood equation to derive an expression for log k which suggests a linear relationship with (sr — l)l(2sr + 1) where k is the second order rate constant. The slope of a plot of log k against sr — l)l(2sr + 1) depends on the dipole moments and radii of the reactants and the transition state. Although linear relationships have been found in many cases the results can hardly be classed as successful in that, again, there is no reasonable basis for estimating the dimensions or dipole moment of the transition state. 

In chemistry, the quantity of matter is usually expressed in moles (mole) and the concentration of matter is usually expressed in mole/liter (M). Reaction rates are expressed in mole/Uter/second (Ms 0- The first-order rate constants have the dimension oftime (s" ) and the second-order rate constants have the dimension of concentration" x time" (M s" ) zero-order rate constants have the dimension of concentration" x time" (M" s" ). 

Second-order rate constants have dimensions of concentration" time. Since molecule/cm is a valid concentration unit, the units given are appropriate. 

All of the quantities required in steady-state kinetics are either concentrations, normally measured in mol litre" or a submultiple, rates, measured in mol litre" s", or rate constants, with units that vary according to the type of rate constant a first-order rate constant has dimensions of reciprocal time, and is typically measured therefore in s", and a second-order rate constant has dimensions of reciprocal time multiplied by reciprocal concentration, and is typically measured therefore in mol" litre s" . It is obvious from elementary dimensional considerations that a pseudo-first-order rate constant has the same dimensions and units as a first-order rate constant, and that constants of different order caimot be meaningfully compared. 

Here —d[A]lcIt is the rate of decrease in concentration of [A] and —d[B]ldt is that of [B] they are equal, the reaction being A -f B >- products. The factor k is the second-order rate constant. Its dimensions are seen to be liter mole , the reciprocal of concentration, times s-. ... 

Units must be treated carefully in (9.43). The left-hand side has dimensions of a second-order rate constant while the right-hand side appears to have dimension A choice of standard state is impiicit in the definition of the free energy of activation, zlG,. . For details see A. C. Norris, J. Chem. Ed. 48, 797 (1971). 

The multiplication of the second order rate constant A (M s ) by the fixed concentration Ca(0) results in a pseudo first order constant with dimension s. For example, if the rate constant for the association of A with R is 10 M s and Ca(0)= 1 pM, then the observed rate constant is 100 s-. ... 

This describes a first order reaction with the observed rate constant cat/ M- Remembering that has the dimensions of a dissociation constant, we can compare the above equation to that derived for two step ligand binding, when the second step is preceded by a rapid pre-equilibrium (see p. 66). The constant k JK can be determined from the analysis of the record as a first order reaction. Although, even at such low initial substrate concentrations, the effects of reversibility or product inhibition may perturb the later parts of the reaction. The more usual procedure found in the literature is to plot v/ce(0) against Cs(0) from a set of initial rate measurements and to take k JK as the initial slope. As pointed out in section 3.2, one obtains an apparent second order rate constant. 

The catalytic efficiency of the enzyme (kj. in M s , with the same dimension as the second-order rate constant) can be obtained from Equation 15.7 when the concentrations of the substrate and the enzyme are known. This parameter reveals the effect of reverse micelles on the enzyme catalysis when compared with k,., in an aqueous solution. 

For a first-order reaction, therefore, a plot of In Ca (or log Ca) vs. / is linear, and the first-order rate constant can be obtained from the slope. A first-order rate constant has the dimension time , the usual unit being second. ... 

The Arrhenius equation relates the rate constant k of an elementary reaction to the absolute temperature T R is the gas constant. The parameter is the activation energy, with dimensions of energy per mole, and A is the preexponential factor, which has the units of k. If A is a first-order rate constant, A has the units seconds, so it is sometimes called the frequency factor. 

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