Rate constant vs. temperature

The activation energy (Ea) of the migration copolymerization of (n-C4H9)2SnH2 with (I) calculated from the plot of the rate constant vs. temperature is 12.2 kcal/mol. 

Figure 16.8 Dependence of the rate constant on temperature. A, In the hydrolysis of the ester ethyl acetate, CH3COOCH2CH3 + H2O CH3COOH + CH3CH2OH, when reactant concentrations are held constant and temperature increases, the rate and rate constant increase. Note the near doubling of k with each rise of 10 K (10°C). B, A plot of rate constant vs. temperature for this reaction shows an exponentially Increasing curve.

The variation of a rate constant with temperature is described by the Arrhenius equation. According to its logarithmic form (Equation ), a plot of in k vs. 1 / Z, with temperature expressed in keIvins, shou id be a straight line. 

Figure 7. Variation of unimolecular dissociation rate constant vs. reciprocal temperature for the proton-bound methoxide ion, ((CH30)2H]. Arrhenius parameters derived...

Figure 7.10 Arrhenius plots of In apparent rate constants vs. reciprocal absolute temperature for potato tissue treated in water.

When several temperature-dependent rate constants have been determined or at least estimated, the adherence of the decay in the system to Arrhenius behavior can be easily determined. If a plot of these rate constants vs. reciprocal temperature (1/7) produces a linear correlation, the system is adhering to the well-studied Arrhenius kinetic model and some prediction of the rate of decay at any temperature can be made. As detailed in Figure 17, Carstensen s adaptation of data, originally described by Tardif (99), demonstrates the pseudo-first-order decay behavior of the decomposition of ascorbic acid in solid dosage forms at temperatures of 50° C, 60°C, and 70°C (100). Further analysis of the data confirmed that the system adhered closely to Arrhenius behavior as the plot of the rate constants with respect to reciprocal temperature (1/7) showed linearity (Fig. 18). Carsten-sen suggests that it is not always necessary to determine the mechanism of decay if some relevant property of the degradation can be explained as a function of time, and therefore logically quantified and rationally predicted. 

The current evidence concerning the details of the temperature dependence of e.t. processes, cannot be considered as decisive one way or the other it may be surmised that this will become an area of very active research in the future. The reported observations of a bell-shaped temperature dependence of some e.t. processes at least is, however, not in favour of the radiationless transition model, because this does not predict the existence of any normal region in the rate constant vs energy function. On the other hand, the observation of temperature independent e.t. processes of widely different exergonicities (Ref. 101) would certainly speak against the Marcus model, which according to Eq. 3 predicts that the exponential factor which contains the AG /RT term becomes constant only in the special condition AG° X. 

An Arrhenius plot of log rate constant vs. reciprocal temperature (Figure 6) indicates an activation energy of 61.9 1.3 kcal/mol for dodecene cracking. This is similar to the activation energies measured for n-paraffin cracking and is somewhat higher than previously measured values for alpha-olefins (6). 

Fig. 19. Logarithm of the true forward rate constant vs. the reciprocal temperature T and the corrected standard Gibbs energy of transfer (Bronsted correlation projected on the plane of the page) for (O) Pi , ( ) Me3PrP +, (A)Me3EtP, (V) Et4N+, and ( ) M04N+ ion transfer from the solution of 0.05 M LiCl in water to the solution of 0.05 M Bu4NPh4B in nitrobenzene. The numbers on the lines indicate the temperature (K). (After [115]).

Fig. 2. (A) Flash-induced AA in the SDS-fractionated PS-1 core complex (CPI) at 5 K [ with and o without DCIP] (B) Flash-induced AA in TSF-I particles containing dithionite and neutral red at pH 10 and frozen while being illuminated (C) left AA induced by 300-ns, dye laser flashes [710 nm for the blue and green region 590 nm for the red region] insets show individual AA transients at 696 and 480 nm (C) right The difference between the difference spectrum in the left panel and that of P700. (D) Plot of the rate constant vs. reciprocal temperature. Figure source (A) Mathis, Sauer and Remy (1978) Rapidly reversible flash-induced electron transfer on a P-700 chlorophyll-protein complex isolated with SDS. FEBS Lett 88 277 (8) Sauer, Mathis, Acker and van Best (1979) Absorption changes of P-700 reversible in milliseconds at low temperature in Triion-solubilized photosystem I particles. Biochim Biophys Acta 545 469 (C and D) Shuvalov, Dolan and Ke (1979) Spectral and kinetic evidence for two eariy electron acceptors in phoiosystem I. Proc Nat Acad Sci, USA 76 771,773.

Degradation rate constants were obtained by linear regression least squares analysis of plots of log % EDB remaining vs time. Pseudo-first order rate constants were used to generate Arrhenius plots (log rate constant vs 1/T °K) to estimate activation energies (E ) and to make extrapolated estimates of rate constants and half-life values at ambient temperature. 

The residence time of the individual tests was constant. Because the reaction rate constant is temperature dependent, an Arrhenius plot of In (In [1 — /] ) vs. the reciprocal of the observed mixing-zone temperature should yield a straight line. This is indeed the case, as shown in Figure 3. The temperature effect on the rate of methane decomposition is quite pronounced, corresponding to an activation energy of 30 kca 1/ gram-mole. 

Figure 7.13 Arrhenius plots of 02-reduction rate constant vs the reciprocal of temperature at two pH levels. Reprinted...

Generalized charts are appHcable to a wide range of industrially important chemicals. Properties for which charts are available include all thermodynamic properties, eg, enthalpy, entropy, Gibbs energy and PVT data, compressibiUty factors, Hquid densities, fugacity coefficients, surface tensions, diffusivities, transport properties, and rate constants for chemical reactions. Charts and tables of compressibiHty factors vs reduced pressure and reduced temperature have been produced. Data is available in both tabular and graphical form (61—72). 

A simpler phenomenological form of Eq. 13 or 12 is useful. This may be approached by using Eq. 4 or its equivalent, Eq. 9, with the rate constants determined for Na+ transport. Solving for the AG using Eqn. (3) and taking AG to equal AHf, that is the AS = 0, the temperature dependence of ix can be calculated as shown in Fig. 16A. In spite of the complex series of barriers and states of the channel, a plot of log ix vs the inverse temperature (°K) is linear. Accordingly, the series of barriers can be expressed as a simple rate process with a mean enthalpy of activation AH even though the transport requires ten rate constants to describe it mechanistically. This... 

The model GASPP was used to correlate yield vs. time for the 20 C boost to 100 C reaction temperature. With the first run, a value of kg = 0.00198 cm/sec was required to achieve the low yield reported. His second run had a yield of 13750 at 4.68 hr. Model GASPP requires kg = 0.00294 cm/sec to give this result at 100 C. This rate constant is only 2% greater than the kg reported in Table I here for the lowest activity BASF TiCi s. On this basis,... 

We can determine the activation energy from a series of measurements by plotting the logarithm of the rate constant against the reciprocal temperature, as rearrangement of Eq. (45) sho vs ... 

Figure 10.11 Arrhenius plots of the ORR rate constants obtained at various electrodes. The symbols are the same as those in Fig. 10.10. Each solid line is the least squares fit of all the data at the constant applied potential. Since the standard potential E° and [RHE(r)] shift to less positive values in a different maimer, the corrected potential E is applied so as to keep a constant overpotential for the ORR at each temperature. The applied potentials of -0.485, -0.525, and -0.585 V vs. E° correspond to 0.80, 0.76, and 0.70 V vs. RHE, respectively, at 30 °C. (From Yano et al. [2006b], reproduced by permission of the PCCP Owner Societies.)...

The quantum yield for the formation of the cycloaddition product has been found to be temperature dependent, increasing by a factor of approximately three as the temperature is lowered from 65 ( = 0.24) to 5°C ( = 0.69). Photolysis of mixtures of the olefin and f/my-stilbene in the presence of sensitizers yielded no cycloaddition product (42) but rather only m-stilbene. This suggests that the cycloadduct is produced via a singlet reaction. This conclusion is supported by the fact that tetramethylethylene quenches fluorescence from the /rans-stilbene singlet. A plot of l/ (42) vs. 1/[TME] (TME = tetramethylethylene) is linear. The slope of this plot yields rate constants for cycloadduct formation which show a negative temperature dependence. To account for this fact, a reversibly formed exciplex leading to (42) was proposed in the following mechanism<82) ... 

In the meantime temperature-dependent stopped-flow measurements were conducted on the latter complex in order to determine the activation parameters of the N-N cleavage reaction (24). Plots of the absorption intensity at 418 nm vs. time at T — —35 to +15°C indicate biphasic kinetics with two rate constants 0bs(p and obs(2)> in analogy to our measurements of the tungsten complex. This time, however, both rates depended upon the acid concentration. Interestingly much smaller rate constants 0bs(i) and 0bs(2)> were found for all acid concentrations than given by Henderson et al. for his (single) rate constant kobs (up to 1 order of magnitude). Furthermore plots of 0bs(i) and kohs(2) vs. the acid concentration showed no saturation behavior but linear dependencies with slopes k and k and intercepts k und k, respectively (s — acid dependent and i — acid independent), Eq. (2) ... 

The plot of In (1 + [anti-2 /[syn-2]) vs. t shows good linearity in the temperature range of 200-220 °C, and the rate constants are listed in Table II. The thermodynamic parameters determined by use of the following equations are also listed in Table II. 

Finally, to conclude our discussion on coupling with chemistry, we should note that in principle fairly complex reaction schemes can be used to define the reaction source terms. However, as in single-phase flows, adding many fast chemical reactions can lead to slow convergence in CFD simulations, and the user is advised to attempt to eliminate instantaneous reaction steps whenever possible. The question of determining the rate constants (and their dependence on temperature) is also an important consideration. Ideally, this should be done under laboratory conditions for which the mass/heat-transfer rates are all faster than those likely to occur in the production-scale reactor. Note that it is not necessary to completely eliminate mass/heat-transfer limitations to determine usable rate parameters. Indeed, as long as the rate parameters found in the lab are reliable under well-mixed (vs. perfect-mixed) conditions, the actual mass/ heat-transfer rates in the reactor will be lower, leading to accurate predictions of chemical species under mass/heat-transfer-limited conditions. 

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