Rate constant, units

Figure A3.12.3. Hannonic RRKM imimolecular rate constants for C2Hj H+C2H4 dissociation classical state counting (solid curve), quantal state counting (dashed curve). Rate constant is in units of s and energy in kcal moK (Adapted from [17].)...

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. 

We have tacitly assumed that the rate constants depend only on the last unit of the chain. In such a situation, the copolymerization is called a Markov copolymerization of first order. The special case (i), r r- = 1, is a Markov copolymerization of order zero. If reactivity also depends on the penultimate unit of the chain, the polymerization is a Markov copolymerization of second order. 

The equation is sometimes written, = SRTI3000 t . The form preferred here is more correct because it makes no presupposition about the units used. Arithmetical adjustments will normally be necessary to give rate constants in the required units. 

Km for an enzymatic reaction are of significant interest in the study of cellular chemistry. From equation 13.19 we see that Vmax provides a means for determining the rate constant 2- For enzymes that follow the mechanism shown in reaction 13.15, 2 is equivalent to the enzyme s turnover number, kcat- The turnover number is the maximum number of substrate molecules converted to product by a single active site on the enzyme, per unit time. Thus, the turnover number provides a direct indication of the catalytic efficiency of an enzyme s active site. The Michaelis constant, Km, is significant because it provides an estimate of the substrate s intracellular concentration. 

In writing Eqs. (7.1)-(7.4) we make the customary assumption that the kinetic constants are independent of the size of the radical and we indicate the concentration of all radicals, whatever their chain length, ending with the Mj repeat unit by the notation [Mj ], This formalism therefore assumes that only the nature of the radical chain end influences the rate constant for propagation. We refer to this as the terminal control mechanism. If we wished to consider the effect of the next-to-last repeat unit in the radical, each of these reactions and the associated rate laws would be replaced by two alternatives. Thus reaction (7. A) becomes... 

The overall requirement is 1.0—2.0 s for low energy waste compared to typical design standards of 2.0 s for RCRA ha2ardous waste units. The most important, ie, rate limiting steps are droplet evaporation and chemical reaction. The calculated time requirements for these steps are only approximations and subject to error. For example, formation of a skin on the evaporating droplet may inhibit evaporation compared to the theory, whereas secondary atomization may accelerate it. Errors in estimates of the activation energy can significantly alter the chemical reaction rate constant, and the pre-exponential factor from equation 36 is only approximate. Also, interactions with free-radical species may accelerate the rate of chemical reaction over that estimated solely as a result of thermal excitation therefore, measurements of the time requirements are desirable. 

R is rate of reaction per unit area, a is interfacial area per unit volume, S is solubiHty of solute in continuous phase, D is diffusivity of solute, k is rate constant, kj is mass-transfer coefficient, is concentration of reactive species, and Z is stoichiometric coefficient. When Dk is considerably greater (10 times) than Ra = aS Dk. 

Fig. 2. Steps in advanced oxidation process (AOPs) involving o2one, hydrogen peroxide, and uv light of 254 nm. ( D) represents the doublet state ( ) represents quantum yield, and the other numbers associated with the reaction arrows are rate constants in units of (Af-s). Dashed arrows indicate...

Enzyme and substrate first reversibly combine to give an enzyme-substrate (ES) complex. Chemical processes then occur in a second step with a rate constant called kcat, or the turnover number, which is the maximum number of substrate molecules converted to product per active site of the enzyme per unit time. The kcat is, therefore, a rate constant that refers to the properties and reactions of the ES complex. For simple reactions kcat is the rate constant for the chemical conversion of the ES complex to free enzyme and products. 

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

The units of the rate constant depend upon the overall reaction order. 

It is important to realize that a rate and a rate constant are different quantities. However, for a simple rate equation, this interpretation can be given to the rate constant k is the number of moles per liter reacting per unit time when all concentrations are one molar. This interpretation is the basis of the synonym specific rate for the rate constant. 

Thus, a zero-order reaction yields a linear plot of c vs. t, the slope being equal to — k. It is evident that a zero-order rate constant has the units of a rate, for example, moles per liter-second (M s ). 

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. ... 

Therefore, for this type of second-order reaction, a plot of 1/ca vs. t is linear, with the slope equal to k. The usual units of a second-order rate constant are liters per mole-second (M s" ). 

We can reach two useful conclusions from the forms of these equations First, the plots of these integrated equations can be made with data on concentration ratios rather than absolute concentrations second, a first-order (or pseudo-first-order) rate constant can be evaluated without knowing any absolute concentration, whereas zero-order and second-order rate constants require for their evaluation knowledge of an absolute concentration at some point in the data treatment process. This second conclusion is obviously related to the units of the rate constants of the several orders. 

We have seen that 10" M s is about the fastest second-order rate constant that we might expect to measure this corresponds to a lifetime of about 10 " s at unit reactant concentration. Yet there is evidence, discussed by Grunwald, that certain proton transfers have lifetimes of the order 10 s. These ultrafast reactions are believed to take place via quantum mechanical tunneling through the energy barrier. This phenomenon will only be significant for very small particles, such as protons and electrons. 

As a final example we consider noncovalent molecular complex formation with the macrocyclic ligand a-cyclodextrin, a natural product consisting of six a-D-glucose units linked 1-4 to form a torus whose cavity is capable of including molecules the size of an aromatic ring. Table 4-3 gives some rate constants for this reaction, where L represents the cyclodextrin and S is the substrate ... 

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. 

To analyze the rate constant problem we start with Eq. (5-43), k = (kT/h)K. The term (kT/h) has the unit second", so consistency is achieved if the concentration units of k and are identical. As before, we pass to pure numbers, writing (for a second-order rate constant)... 

A first-order rate constant has the dimension time, but all other rate constants include a concentration unit. It follows that a change of concentration scale results in a change in the magnitude of such a rate constant. From the equilibrium assumption of transition state theory we developed these equations in Chapter 5 ... 

It is conventional to take as the activation volume the value of AV when P = 0, namely —bRT. (This is essentially equal to the value at atmospheric pressure.) Pressure has usually been measured in kilobars (kbar), or 10 dyn cm 1 kbar = 986.92 atm. The currently preferred unit is the pascal (Pa), which is 1 N m 1 kbar = 0.1 GPa. Measurements of AV usually require pressures in the range 0-10 kbar. The units of AV are cubic centimeters per mole most AV values are in the range —30 to +30 cm moP, and the typical uncertainty is 1 cm moP. Rate constant measurements should be in pressure-independent units (mole fraction or molality), not molarity. ... 

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-... 

From this expression, it is obvious that the rate is proportional to the concentration of A, and k is the proportionality constant, or rate constant, k has the units of (time) usually sec is a function of [A] to the first power, or, in the terminology of kinetics, v is first-order with respect to A. For an elementary reaction, the order for any reactant is given by its exponent in the rate equation. The number of molecules that must simultaneously interact is defined as the molecularity of the reaction. Thus, the simple elementary reaction of A P is a first-order reaction. Figure 14.4 portrays the course of a first-order reaction as a function of time. The rate of decay of a radioactive isotope, like or is a first-order reaction, as is an intramolecular rearrangement, such as A P. Both are unimolecular reactions (the molecularity equals 1). 

The rate is proportional to the concentrations of both A and B. Because it is proportional to the product of two concentration terms, the reaction is second-order overall, first-order with respect to A and first-order with respect to B. (Were the elementary reaction 2A P + Q, the rate law would be = A[A] second-order overall and second-order with respect to A.) Second-order rate constants have the units of (concentration) time) as in M sec. ... 

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