Rate constants first-order

The effective rate law correctly describes the pressure dependence of unimolecular reaction rates at least qualitatively. This is illustrated in figure A3,4,9. In the lunit of high pressures, i.e. large [M], becomes independent of [M] yielding the high-pressure rate constant of an effective first-order rate law. At very low pressures, product fonnation becomes much faster than deactivation. A j now depends linearly on [M]. This corresponds to an effective second-order rate law with the pseudo first-order rate constant Aq ... 

The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... 

Figure B2.3.13. Model 2-level system describing molecular optical excitation, with first-order excitation rate constant W 2 proportional to the laser power, and spontaneous (first-order rate constant 21) stimulated (first-order rate constant 1 2 proportional to the laser power) emission pathways.

A different kind of shape selectivity is restricted transition state shape selectivity. It is related not to transport restrictions but instead to size restrictions of the catalyst pores, which hinder the fonnation of transition states that are too large to fit thus reactions proceeding tiirough smaller transition states are favoured. The catalytic activities for the cracking of hexanes to give smaller hydrocarbons, measured as first-order rate constants at 811 K and atmospheric pressure, were found to be the following for the reactions catalysed by crystallites of HZSM-5 14 n-... 

Kinetic measurements were performed employii UV-vis spectroscopy (Perkin Elmer "K2, X5 or 12 spectrophotometer) using quartz cuvettes of 1 cm pathlength at 25 0.1 C. Second-order rate constants of the reaction of methyl vinyl ketone (4.8) with cyclopentadiene (4.6) were determined from the pseudo-first-order rate constants obtained by followirg the absorption of 4.6 at 253-260 nm in the presence of an excess of 4.8. Typical concentrations were [4.8] = 18 mM and [4.6] = 0.1 mM. In order to ensure rapid dissolution of 4.6, this compound was added from a stock solution of 5.0 )j1 in 2.00 g of 1-propanol. In order to prevent evaporation of the extremely volatile 4.6, the cuvettes were filled almost completely and sealed carefully. The water used for the experiments with MeReOj was degassed by purging with argon for 0.5 hours prior to the measurements. All rate constants were reproducible to within 3%. 

Herein k js is the observed pseudo-first-order rate constant. In the presence of micelles, analogous treatment of the experimental data will only provide an apparent second-order rate constant, which is a weighed average of the second-order rate constants in the micellar pseudophase and in the aqueous phase (Equation 5.2). 

The effect of micelles of SDS, CTAB and C12E7 on the apparent second-order rate constants of the Diels-Alder reaction between nonionic 5.1a, anionic 5.1 f and cationic 5.1g with 5.2 is reported in Table 5.1. These apparent rate constants are calculated from the observed pseudo-first-order rate constants by dividing the latter by the overall concentration of 5.2. 

Herein [5.2]i is the total number of moles of 5.2 present in the reaction mixture, divided by the total reaction volume V is the observed pseudo-first-order rate constant Vmrji,s is an estimate of the molar volume of micellised surfactant S 1 and k , are the second-order rate constants in the aqueous phase and in the micellar pseudophase, respectively (see Figure 5.2) V is the volume of the aqueous phase and Psj is the partition coefficient of 5.2 over the micellar pseudophase and water, expressed as a ratio of concentrations. From the dependence of [5.2]j/lq,fe on the concentration of surfactant, Pj... 

Assuming complete binding of the dienophile to the micelle and making use of the pseudophase model, an expression can be derived relating the observed pseudo-first-order rate constant koi . to the concentration of surfactant, [S]. Assumirg a negligible contribution of the reaction in the aqueous phase to the overall rate, the second-order rate constant in the micellar pseudophase lq is given by ... 

When the dienophile does not bind to the micelle, reaction will take place exclusively in the aqueous phase so that the second-order rate constant of the reaction in the this phase (k,) is directly related to the ratio of the observed pseudo-first-order rate constant and the concentration of diene that is left in this phase. 

The operation of the nitronium ion in these media was later proved conclusively. "- The rates of nitration of 2-phenylethanesulphonate anion ([Aromatic] < c. 0-5 mol l i), toluene-(U-sulphonate anion, p-nitrophenol, A(-methyl-2,4-dinitroaniline and A(-methyl-iV,2,4-trinitro-aniline in aqueous solutions of nitric acid depend on the first power of the concentration of the aromatic. The dependence on acidity of the rate of 0-exchange between nitric acid and water was measured, " and formal first-order rate constants for oxygen exchange were defined by dividing the rates of exchange by the concentration of water. Comparison of these constants with the corresponding results for the reactions of the aromatic compounds yielded the scale of relative reactivities sho-wn in table 2.1. 

Since the first-order rate constant for nitration is proportional to y, the equilibrium concentration of nitronium ion, the above equations show the way in which the rate constant will vary with x, the stoichiometric concentration of dinitrogen tetroxide, in the two media. An adequate fit between theory and experiment was thus obtained. A significant feature of this analysis is that the weak anticatalysis in pure nitric acid, and the substantially stronger anticatalysis in partly aqueous nitric acid, do not require separate interpretations, as have been given for the similar observations concerning nitration in organic solvents. 

The kinetics of the nitration of benzene, toluene and mesitylene in mixtures prepared from nitric acid and acetic anhydride have been studied by Hartshorn and Thompson. Under zeroth order conditions, the dependence of the rate of nitration of mesitylene on the stoichiometric concentrations of nitric acid, acetic acid and lithium nitrate were found to be as described in section 5.3.5. When the conditions were such that the rate depended upon the first power of the concentration of the aromatic substrate, the first order rate constant was found to vary with the stoichiometric concentration of nitric acid as shown on the graph below. An approximately third order dependence on this quantity was found with mesitylene and toluene, but with benzene, increasing the stoichiometric concentration of nitric acid caused a change to an approximately second order dependence. Relative reactivities, however, were found to be insensitive... 

Ratio of first order rate constant for solvolysis in indicated solvent to that for solvolysis in acetic acid at 25 C... 

The first order rate constant for ethanolysis of the allylic chloride 3 chloro 3 methyl 1 butene is over 100 times greater than that of tert butyl chloride at the same temperature... 

The value for the pseudo-first-order rate constant is determined by solving equation 13.6 for k and making appropriate substitutions thus... 

At ordinary temperatures, formaldehyde gas is readily soluble in water, alcohols, and other polar solvents. Its heat of solution in water and the lower ahphatic alcohols is approximately 63 kJ/mol (15 kcal/mol). The reaction of unhydrated formaldehyde with water is very fast the first-order rate constant... 

The conditions chosen make the reaction appear to be first-order overall, although the reaction is really not first-order overall, unlessjy and happen to be 2ero. If a simple exponential is actually observed over a reasonable extent (at least 90—95%) of decay the assumptions are considered vaUdated and is obtained with good precision. The pseudo-first-order rate constant is related to the k in the originally postulated rate law by... 

The two main termination steps for neutral solutions are HO + HO — H2O2 + 2 O3 and HO + HO3 — H2O2 + O3 + O2. An alternative mechanism has been proposed that does not involve HO and HO but has a different initiation step (26). Three ozone molecules are destroyed for each primary event. In the presence of excess HO radical scavengers, ie, bicarbonate, the pseudo-first-order rate constant at 20°C for the initiation step is 175 X. This yields an ozone half-hfe of 66 min at pH 8. In distilled water = 50 mmol/L), the half-hfe is significantly lower, ie, 7 min. 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

For those pesticides that are cometabolized, ie, not utilized as a growth substrate, the assumption of first-order kinetics is appropriate. The more accurate kinetic expression is actually pseudo-first-order kinetics, where the rate is dependent on both the pesticide concentration and the numbers of pesticide-degrading microorganisms. However, because of the difficulties in enumerating pesticide-transforming microorganisms, first-order rate constants, or half-hves, are typically reported. Based on kinetic constants, it is possible to rank the relative persistence of pesticides. Pesticides with half-hves of <10 days are considered to be relatively nonpersistent pesticides with half-hves of >100 days are considered to be relatively persistent. 

Fig. 8.4. Logarithm of the first-order rate constants for the hydrolysis of substituted benzylidene-l,l-dimethyl-ethylamines as a fiinction of pH. [Reproduced fiom J. Am. Chem. Soc. 85 2843 (1963) by permission of the American Chemical Society.]...

Fig. 8.P3I. Plot of the pseudo-first-order rate constants for hydrolysis of thioesters A (O), B ( ), C (A), D (A) as a fiinction of pH at 50°C and ionic strength 0.1 (KCI). Lines are from fits of the data to = kon(K /H+)) + (k KJK + [//+])) where koH is the hydroxide term and is the intramolecular assistance term for B and C and from linear regression for A and D. Reproduced from problem reference 31 by permission of the American Chemical Society.

The table below gives first-order rate constants for reaction of substituted benzenes with w-nitrobenzenesulfonyl peroxide. From these data, calculate the overall relative reactivity and partial rate factors. Does this reaction fit the pattern of an electrophilic aromatic substitution If so, does the active electrophile exhibit low, moderate, or high substrate and position selectivity ... 

There are also examples of electrocyclic processes involving anionic species. Since the pentadienyl anion is a six-7c-electron system, thermal cyclization to a cyclopentenyl anion should be disrotatory. Examples of this electrocyclic reaction are rare. NMR studies of pentadienyl anions indicate that they are stable and do not tend to cyclize. Cyclooctadienyllithium provides an example where cyclization of a pentadienyl anion fragment does occur, with the first-order rate constant being 8.7 x 10 min . The stereochemistry of the ring closure is consistent with the expected disrotatory nature of the reaction. 

The intermediate diphenylhydroxymethyl radical has been detected after generation by flash photolysis. Photolysis of benzophenone in benzene solution containing potential hydrogen donors results in the formation of two intermediates that are detectable, and their rates of decay have been measured. One intermediate is the PhjCOH radical. It disappears by combination with another radical in a second-order process. A much shorter-lived species disappears with first-order kinetics in the presence of excess amounts of various hydrogen donors. The pseudo-first-order rate constants vary with the structure of the donor with 2,2-diphenylethanol, for example, k = 2 x 10 s . The rate is much less with poorer hydrogen-atom donors. The rapidly reacting intermediate is the triplet excited state of benzophenone. 

Calculate die pseudo-first order rate constant kj = k2[S20g ] and, hence, k2. 

The results give the pseudo-first order rate constant, and kj = 3.312 a... 

First order rate constant k, for the rotation about the C-N bond in N, N-dimediylnicotinamide (3) measured at different temperatures by nuclear magnetic resonance (NMR) are ... 

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