Estimation of rate and equilibrium

Chiang, Y. Kresge, J. Zhu, Y. Flash photolytic generation and study of /j-quinone methide in aqueous solution. An estimate of rate and equilibrium constants for heterolysis of the carbon-bromine bond in p-hydroxybenzyl bromide. J. Am. Chem. Soc. 2002,124, 6349-6356. 

Case B Estimation of Rate and Equilibrium Constants in a Reversible Esterification Reaction Using MADONNA... 

When the uncertainty associated with AHf is 5 kcal/mol, rate and equilibrium constants can be estimated within a factor of 10 at process temperatures, i.e., 500-1,500 K. This level of accuracy may be acceptable for preliminary mechanism development work and for the identification of important reactions in a DCKM. However, it would clearly be desirable to know AHf within 1 kcal/mol, which would lead to the determination of rate and equilibrium constants that are accurate within a factor of two. Since this level of accuracy is very close to the limits of accuracy of most experimental measurements, improvements in AHf are often difficult. Consequently, computational quantum chemistry holds a great promise for the accurate determination of AHf. 

Investigation 10 was a study of fixed-bed reactor models and their application to the data of Hettinger et al. (1955) on catalytic reforming of C7 hydrocarbons. The heuristic posterior density function p 6 Y) proposed by Stewart (1987) was used to estimate the rate and equilibrium parameters of various reaction schemes, two of which were reported in the article. The data were analyzed with and without models for the intraparticle and boundary-layer transport. The detailed transport model led to a two-dimensional differential-algebraic equation system, which was solved via finite-element discretization in the reactor radial coordinate and... 

Even without taking into account side reactions changing the concentrations of the active species, a great number of rate and equilibrium constants are required in order to describe the whole course of polymerization. So far, no data are available on equilibrium (25) from which the concentrations of lactam and polymer amide anions could be estimated. Therefore, the individual rate coefficients can be obtained only from measurements of the initial rates of the isolated reactions. In this case, the participation of the reaction products in subsequent reactions can be neglected. 

Endicott s group has measured and made an extremely detailed analysis of rate and equilibrium constants for transmethylation between various cobalt N4-macrocyclic systems (see Table 7.6 and accompanying structures). The reactions are first order in each reagent. Rates can vary by 10 (even for the back reactions where AG < 0). AG is analyzed into components, intrinsic free energy barriers to transmethylation being small for cobalt corrin and large for sterically hindered neutral macrocyclic complexes. Estimates for C0-CH3 bond energies are between 33 and 48 kcal mol the bond is stabilized by unsaturation in the N4 macrocycle, but is also very sensitive to stereochemistry. 

For species 11 we will use the intrinsic barrier for hydroxide addition to trimethyl phosphate, G = 19 (calculated using rate and equilibrium data from reference 100) and assume the same value for the attack of hydroxide at sulfur on dimethyl sulfate. This (nonobservable) rate will be estimated using a Brpnsted type plot from the rate constants for diaryl sulfates (diphenyl sulfate,and bis p-nitrophenyl sulfate), estimated from the rate for phenyl dinitrophenyl sulfate,assuming equal contributions for the two nitro groups. This gives ftg = 0.95, and thus for dimethyl sulfate log k = 11.3... 

A value of kjkp = 17 000 has been determined for partitioning of the acetophenone oxocarbenium ion [12+] in water.15,16 It is not possible to estimate an equilibrium constant for the addition of water to [12+], because of the instability of the hemiketal product of this reaction. However, kinetic and thermodynamic parameters have been determined for the reaction of [12+] with methanol to form protonated acetophenone dimethyl ketal [12]-OMeH+ and for loss of a proton to form a-methoxystyrene [13] in water (Scheme 10).15,16 Substitution of these rate and equilibrium constants into equation (3) gives values of AMeoH = 6.5 kcal mol-1 and Ap = 13.8 kcal mol-1 for the intrinsic... 

Kaye, S.V. and P.B. Dunaway. 1963. Estimation of dose rate and equilibrium state from bioaccumulation of radionuclides by mammals. Pages 107-111 in V. Schultz and A.W. Klement, Jr. eds. Radioecology. Reinhold, New York. 

The rate constants kTS and kST define an equilibrium constant (ATeq) connecting the singlet and triplet carbenes. An estimate of Ktq, and hence AGSX, for BA can be obtained from the experiments described above. The time resolved spectroscopic measurements indicate that BA reacts with isopropyl alcohol with a rate constant some five times slower than the diffusion limit (Table 7). This, in conjunction with the picosecond timescale measurements, gives a value for ksr. The absence of ether formation from the sensitized irradiation, when combined with the measured rate constant for reaction of 3BA with isopropyl alcohol, gives an upper limit for k-. These values give Keq and thus AGST 2 5.2 kcal mol-1 (Table 8). 

Accumulated data to date indicates that is relatively insensitive to the details of the structure of species. Consequently, additivity rules and model compounds, in conjunction with statistical mechanics, can readily be used to estimate entropies and specific heats of molecules and radicals. These estimates generally are accurate within l.Ocal/mol-K, leading to uncertainties in rate and equilibrium constants well within 20%. 

Fromm and Cleland provide valuable discussions of the utility of Haldane relations in excluding certain kinetic reaction mechanisms based on a numerical evaluation of the constants on each side of the equal sign in the Haldane relation. If the equality is maintained, the candidate mechanism is consistent with the observed rate parameter data. Obviously, one must be concerned about the quality of experimentally derived estimates of rate parameters, because chemists have frequently observed that thermodynamic data (such as equilibrium constants) are often more accurate and precise than kinetically derived parameters. See Haldane Relations for Multisubstrate Enzymes... 

When we can predict the response of the reacting system to changes in operating conditions (how rates and equilibrium conversion change with temperature and pressure), when we are able to compare yields for alternative designs (adiabatic versus isothermal operations, single versus multiple reactor units, flow versus batch system), and when we can estimate the economics of these various alternatives, then and only then will we feel sure that we can arrive at the design well fitted for the purpose at hand. Unfortunately, real situations are rarely simple. 

In estimating these barriers Richard addresses a problem that so far has been avoided. When discussing the correlation of log h2o with pATR in Fig. 3, it was implied that the rate and equilibrium constants refer to the same reaction step. That is not strictly true, because attack of water on a carbocation yields initially a protonated alcohol which subsequently loses a proton in a rapid equilibrium step. As we are reminded in Equation (26) the equilibrium constant AR refers to the combination of these two steps. To calculate an intrinsic barrier for reaction of the carbocation with water therefore the equilibrium constant KR should be corrected for the lack of stoichiometric protonation of the alcohol. Fortunately, there have been enough measurements of pA,s of protonated alcohols240 (e.g. pAa = -2.05 for CthOHi1") for the required corrections to be made readily. 

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