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On the curvature in logarithmic plots of rate coefficients for chemical reactions.

Canepa C - Chem Cent J (2011)

Bottom Line: In terms of the reduced potential energy barrier ζ = ΔuTS/kT, the rate coefficients for chemical reactions are usually expressed as proportional to e-ζ.The coupling between vibrational modes of the medium to the reaction coordinate leads to a proportionality of the regularized gamma function of Euler Q(a,ζ) = Γ(a,ζ)/Γ(a), with a being the number of modes coupled to the reaction coordinate.The new expression affords lower deviations from the experimental points in 29 cases out of 38 and it accounts for the curvature in the logarithmic plots of rate coefficients versus inverse temperature.

View Article: PubMed Central - HTML - PubMed

Affiliation: Dipartimento di Chimica Generale e Chimica Organica, Università di Torino Corso Massimo d'Azeglio 48, 10125 Torino, Italy. carlo.canepa@unito.it.

ABSTRACT
In terms of the reduced potential energy barrier ζ = ΔuTS/kT, the rate coefficients for chemical reactions are usually expressed as proportional to e-ζ. The coupling between vibrational modes of the medium to the reaction coordinate leads to a proportionality of the regularized gamma function of Euler Q(a,ζ) = Γ(a,ζ)/Γ(a), with a being the number of modes coupled to the reaction coordinate. In this work, the experimental rate coefficients at various temperatures for several chemical reactions were fitted to the theoretical expression in terms of Q(a,ζ) to determine the extent of its validity and generality. The new expression affords lower deviations from the experimental points in 29 cases out of 38 and it accounts for the curvature in the logarithmic plots of rate coefficients versus inverse temperature. In the absence of tunneling, conventional theories predict the curvature of these plots to be identically zero.

No MeSH data available.


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Logarithmic plot of rate coefficients for the oxidation of xanthine by xanthine oxidase. The experimental points are taken from ref. [28].
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Figure 9: Logarithmic plot of rate coefficients for the oxidation of xanthine by xanthine oxidase. The experimental points are taken from ref. [28].

Mentions: The sets of data for the thermolyses of N-benzyl-N-nitrosoamides (Figures 3-4, Table 1) include only four experimentally measured rate coefficients for each reaction, and the values of σ are relatively high (from 3.87·10-2 to 1.14·10-1). With the exception of the unsubstituted pivalamide (which we consider an anomaly), the response frequencies for pivalamides are about 10-2 Hz, while for tosylamides they fall below this value. The response frequencies are higher for the hydrolysis of trans-dinitrobis(ethylenediamine) cobalt III nitrate (Figure 5, Table 2), but the coupling is less effective, as exemplified by the lower values of (0.48 ÷ 0.81). The values of the parameters for the rearrangement of bis-(4-chlorophenyl) thioncarbonate to bis-(4-chlorophenyl) thiolcarbonate (Figure 6, Table 3) fall between the two previous cases, with a response frequency of the order of 10-1 Hz and . The solvolysis of methyldiphenylsulfonium ion in water is the first case where the minimum of σ is given by a regression with a = 1, i.e. an Arrhenius plot (all the linear plots are shown in green in the Figures). The corresponding hydrolysis in ethanol exhibits a curved plot (Figure 7, Table 4) with low values of both and . Although the average curvature can in principle only be assigned with confidence in those cases where the experimental determinations exhibit sufficient accuracy, as in the neutral hydrolysis of methyl trifluoroacetate in H2O/DMSO (Figure 8), we might tentativley interpret the alternation between zero and non zero curvature as a result of the high sensitivity of the coupling between the substrate and the medium to the values of the parameters governing the dynamics of the cluster (see the effect of the parameter γ in Figure 2). This sensitivity was demonstrated for a model system of two oscillators [19] driven by an external force according to the Debye theory of solids. The oxidation of xanthine to uric acid by xanthine oxidase affords the plot that deviates the most from linearity (Figure 9, Table 6), even though the actual calculated average curvature is not high, due to the large span of ζ values. In this case we obtain a relatively large frequency and a value of near unity. Also the hydrolysis of amides by α-chymotrypsin (Figure 10, Table 7) may be interpreted along the same lines, large values of both the response frequencies and the parameter . The reactions catalyzed by Escherichia coli dihydrofolate reductase exhibits overall higher values for and lower (for an enzyme) values of (Figures 11-12, Table 8) with respect to the previous enzymes. Also, four reactions out of nine have zero curvature, albeit two of them with high values of σ. The experimental values of the deuterium kinetic isotope effects for the hydride transfer catalyzed by Escherichia coli dihydrofolate reductase in Table 9 may be an indication that the cooperative mechanism is in effect for all solvents. In fact all KIE values are in the range 2.60 ÷ 3.22, well below the estimated value of 7.15 given by classical transition state theory for a difference of 800 cm-1 between the stretching frequencies C-H and C-D. The first bimolecular reaction of our analysis is the activation step of the atom transfer radical polymerization of alkyl halides catalyzed by Cu(I)Br(PDMETA) (PMDETA = N,N,N',N',N''-pentamethyldiethylenetriamine) in acetonitrile [30] (Figures 13-14, Table 10). The measured rate constants are relevant to the cleavage of the C-X bond (X = Cl, Br) of various halides. With respect to the unimolecular and enzyme-catalyzed reactions discussed previously, this process exhibits low values of a and correspondingly low values of the activation barriers. Values of σ are relatively high (never below 3·10-2, corresponding to an average deviation of 19%) and the values of are lower than optimal. Also, three processes have plots with zero curvature. Both the hydration of CO2 by water (Figure 15, Table 11) and the solvolysis of 4-methylbenzhydryl 4-nitrobenzoate in 90% acetone (Figure 16, Table 12) are cases of nearly linear plots with low values of a. The numerical analysis however reveals the non-zero curvature. In these cases, the experimental error may decide between linear and non linear plots and we are not in a position to draw firm conclusions. Concerning the cases that exhibit a mixed behavior of zero and negative curvatures within the same class of reactions, we limit ourselves to the observation that the value of the deviation σ, averaged over the processes with zero curvature (6.04 10-2) is about twice the value averaged over the reactions of the same class with negative curvature (3.58 10-2).


On the curvature in logarithmic plots of rate coefficients for chemical reactions.

Canepa C - Chem Cent J (2011)

Logarithmic plot of rate coefficients for the oxidation of xanthine by xanthine oxidase. The experimental points are taken from ref. [28].
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3117758&req=5

Figure 9: Logarithmic plot of rate coefficients for the oxidation of xanthine by xanthine oxidase. The experimental points are taken from ref. [28].
Mentions: The sets of data for the thermolyses of N-benzyl-N-nitrosoamides (Figures 3-4, Table 1) include only four experimentally measured rate coefficients for each reaction, and the values of σ are relatively high (from 3.87·10-2 to 1.14·10-1). With the exception of the unsubstituted pivalamide (which we consider an anomaly), the response frequencies for pivalamides are about 10-2 Hz, while for tosylamides they fall below this value. The response frequencies are higher for the hydrolysis of trans-dinitrobis(ethylenediamine) cobalt III nitrate (Figure 5, Table 2), but the coupling is less effective, as exemplified by the lower values of (0.48 ÷ 0.81). The values of the parameters for the rearrangement of bis-(4-chlorophenyl) thioncarbonate to bis-(4-chlorophenyl) thiolcarbonate (Figure 6, Table 3) fall between the two previous cases, with a response frequency of the order of 10-1 Hz and . The solvolysis of methyldiphenylsulfonium ion in water is the first case where the minimum of σ is given by a regression with a = 1, i.e. an Arrhenius plot (all the linear plots are shown in green in the Figures). The corresponding hydrolysis in ethanol exhibits a curved plot (Figure 7, Table 4) with low values of both and . Although the average curvature can in principle only be assigned with confidence in those cases where the experimental determinations exhibit sufficient accuracy, as in the neutral hydrolysis of methyl trifluoroacetate in H2O/DMSO (Figure 8), we might tentativley interpret the alternation between zero and non zero curvature as a result of the high sensitivity of the coupling between the substrate and the medium to the values of the parameters governing the dynamics of the cluster (see the effect of the parameter γ in Figure 2). This sensitivity was demonstrated for a model system of two oscillators [19] driven by an external force according to the Debye theory of solids. The oxidation of xanthine to uric acid by xanthine oxidase affords the plot that deviates the most from linearity (Figure 9, Table 6), even though the actual calculated average curvature is not high, due to the large span of ζ values. In this case we obtain a relatively large frequency and a value of near unity. Also the hydrolysis of amides by α-chymotrypsin (Figure 10, Table 7) may be interpreted along the same lines, large values of both the response frequencies and the parameter . The reactions catalyzed by Escherichia coli dihydrofolate reductase exhibits overall higher values for and lower (for an enzyme) values of (Figures 11-12, Table 8) with respect to the previous enzymes. Also, four reactions out of nine have zero curvature, albeit two of them with high values of σ. The experimental values of the deuterium kinetic isotope effects for the hydride transfer catalyzed by Escherichia coli dihydrofolate reductase in Table 9 may be an indication that the cooperative mechanism is in effect for all solvents. In fact all KIE values are in the range 2.60 ÷ 3.22, well below the estimated value of 7.15 given by classical transition state theory for a difference of 800 cm-1 between the stretching frequencies C-H and C-D. The first bimolecular reaction of our analysis is the activation step of the atom transfer radical polymerization of alkyl halides catalyzed by Cu(I)Br(PDMETA) (PMDETA = N,N,N',N',N''-pentamethyldiethylenetriamine) in acetonitrile [30] (Figures 13-14, Table 10). The measured rate constants are relevant to the cleavage of the C-X bond (X = Cl, Br) of various halides. With respect to the unimolecular and enzyme-catalyzed reactions discussed previously, this process exhibits low values of a and correspondingly low values of the activation barriers. Values of σ are relatively high (never below 3·10-2, corresponding to an average deviation of 19%) and the values of are lower than optimal. Also, three processes have plots with zero curvature. Both the hydration of CO2 by water (Figure 15, Table 11) and the solvolysis of 4-methylbenzhydryl 4-nitrobenzoate in 90% acetone (Figure 16, Table 12) are cases of nearly linear plots with low values of a. The numerical analysis however reveals the non-zero curvature. In these cases, the experimental error may decide between linear and non linear plots and we are not in a position to draw firm conclusions. Concerning the cases that exhibit a mixed behavior of zero and negative curvatures within the same class of reactions, we limit ourselves to the observation that the value of the deviation σ, averaged over the processes with zero curvature (6.04 10-2) is about twice the value averaged over the reactions of the same class with negative curvature (3.58 10-2).

Bottom Line: In terms of the reduced potential energy barrier ζ = ΔuTS/kT, the rate coefficients for chemical reactions are usually expressed as proportional to e-ζ.The coupling between vibrational modes of the medium to the reaction coordinate leads to a proportionality of the regularized gamma function of Euler Q(a,ζ) = Γ(a,ζ)/Γ(a), with a being the number of modes coupled to the reaction coordinate.The new expression affords lower deviations from the experimental points in 29 cases out of 38 and it accounts for the curvature in the logarithmic plots of rate coefficients versus inverse temperature.

View Article: PubMed Central - HTML - PubMed

Affiliation: Dipartimento di Chimica Generale e Chimica Organica, Università di Torino Corso Massimo d'Azeglio 48, 10125 Torino, Italy. carlo.canepa@unito.it.

ABSTRACT
In terms of the reduced potential energy barrier ζ = ΔuTS/kT, the rate coefficients for chemical reactions are usually expressed as proportional to e-ζ. The coupling between vibrational modes of the medium to the reaction coordinate leads to a proportionality of the regularized gamma function of Euler Q(a,ζ) = Γ(a,ζ)/Γ(a), with a being the number of modes coupled to the reaction coordinate. In this work, the experimental rate coefficients at various temperatures for several chemical reactions were fitted to the theoretical expression in terms of Q(a,ζ) to determine the extent of its validity and generality. The new expression affords lower deviations from the experimental points in 29 cases out of 38 and it accounts for the curvature in the logarithmic plots of rate coefficients versus inverse temperature. In the absence of tunneling, conventional theories predict the curvature of these plots to be identically zero.

No MeSH data available.


Related in: MedlinePlus