Rate constants at different pressures

Values of AV can be determined from experimental measurements of rates, and thereby rate constants, at different pressures, by plotting In k versus P. The slope at any pressure is equal to —AV /RT. If the plot is linear, the volume of activation is independent of pressure, meaning that... 

Effect of Pressure on Reaction Rate. The variation of Klason lignin contents of red spruce residue with extraction pressure Is shown In Figure 3. Based on these data points, the rate constants at different pressures have been estimated and are given In Table III. 

Some typical experimental values of the rate constants at different pressures and 603°K are [20] ... 

Measuring the reaction rate constant at different pressures p, one can find AVg p from the dependence... 

Representation of the low-pressure rate constant in the form pkg, where kg is given in terms of the strong-collision mechanism and p by Eq. (17.15), provides in principle the possibility of obtaining the mean transferred energy by comparing the theoretical and experimental rate constants. It has been found that p is relatively frequently substantially lower than unity, i.e. (AE) is appreciably smaller than kT [487]. Strictly speaking, this means that the calculation of the rate constant at intermediate pressures is to be based on the diffusion equation, rather than on equation (17.4). However, the expected difference in the fall-off curves is small [487]. 

Information concerning unimolecular potential energy surfaces can be acquired from several sources. Thermochemical measurements provide bond dissociation energies Dq and heats of reaction AHq. Analyses of the vibrational spectra of a unimolecular reaction s reactants and products yields their quadratic force constants, and if the data is sufficiently complete, also, their cubic and quartic force constants. From kinetic measurements of the unimolecular rate constant at high pressure the phenomenological Arrhenius A factor Aee and activation energy can be derived. If one can show that that the activated complex theory is valid for a specific unimolecular reaction its threshold energy Eq and the entropy difference between the activated complex and molecule can be found from... 

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-... 

To answer this question one has to look at the way the rate equation is derived. A rate equation based on a certain reaction mechanism may have been derived after the introduction of some approximations valid at atmospheric pressure. If, at higher pressure these approximations are no longer valid, a continuous use of the rate equation may lead to erroneous results. As approximations usually are introduced to reduce the number of parameters, it should be evident that equations with differing numbers of parameters most probably have different algebraic forms. The omission of a critical, initially small, but increasingly more important rate constant with increasing pressure will unavoidably lead to suspect interpretations. 

Experimental determination of Ay for a reaction requires the rate constant k to be determined at different pressures, k is obtained as a fit parameter by the reproduction of the experimental kinetic data with a suitable model. The data are the concentration of the reactants or of the products, or any other coordinate representing their concentration, as a function of time. The choice of a kinetic model for a solid-state chemical reaction is not trivial because many steps, having comparable rates, may be involved in making the kinetic law the superposition of the kinetics of all the different, and often unknown, processes. The evolution of the reaction should be analyzed considering all the fundamental aspects of condensed phase reactions and, in particular, beside the strictly chemical transformations, also the diffusion (transport of matter to and from the reaction center) and the nucleation processes. 

Figure 26. Dimerization of butadiene in the crystalline phase. Lower panel Logarithmic plots of the room-temperature evolution of the integrated absorption of characteristic vinylcyclohexene absorption bands at different pressures. The linear evolution unambiguously demonstrates the first-order kinetics of the reaction. Upper panel Evolution of the natural logarithm of the dimerization rate constant as a function of pressure (full squares, left axis the dotted line is intended as a guide for the eye) and evolution of the intensity ratio between selected polymer and dimer (vinylcyclohexene) bands (empty dots, right axis).

The volume of activation, AV, is defined as the difference between the partial molar volumes of the transition state for the reaction and the reactants and is related to the pressure variation of the rate constant at a constant temperature T by Eq. (5) (19,20) ... 

Fig. 17. Propylene polymerization rate at constant temperature (70°) and at different pressures as function of polymerization time (a-TiCU sample A 3.60 g./l. [AUCjH,),] 5.88 X 10- mol./l.).

In Fig. 11.4, burning rates are plotted as a function of 7 at different pressures. The burning rate increases with increasing Tfin the range g 1 constant... 

Because of the results obtained for the kinetics of sorption of methanol on both acetylated and unacetylated coals, the mechanism for sorption of methanol on coal must explain the following experimental observations sorption follows a second-order rate equation the experimental rate constants vary with pressure, and two different behaviors are noted the rate constants decrease upon partial acetylation at equilibrium one molecule of methanol is sorbed on one site. In addition, the mechanism must also account for the observations that the reverse desorption experimental rates are independent of the original pressure of sorption, and increased acetylation substantially decreases the rate of desorption. The following mechanism is postulated ... 

In order to determine the activation volume, the rate constants from tests at different pressures are plotted on a logarithmic scale versus the pressure (Fig. 3.3-8). As outlined in Chapter 3.2.3, from the slope of the resulting straight line an activation volume of +7 ml/mol is obtained. 

Effect of Hydrogen Pressure. Reactions carried out at different pressures of hydrogen resulted in essentially the same second-order rate constant, k = 6.2 x 104M 1 min-1 (first order in rhodium complex and first order in hydrogen), as shown in Table III. The solubility of hydrogen in toluene was calculated from data available in the International Critical Tables. All three reactions resulted in linear... 

Although the measured decomposition rates are not rate-limited by diffusion of gas out of the crystal, it is clear that in the case of readily reversible decompositions, such as those of the carbonates, the measured rate will depend on the difference between the equilibrium pressure pe and the actual pressure p at the reaction interface. Thus, although the functional dependence of a on / may be independent of pe - p, the actual rates are not. It is largely for this reason that it is not possible to tabulate standard reference data rate constants, since the pressure p is not usually known. Instead, one finds decompositions carried out in streams of various sweep gases, which may or may not remove product gases from the reaction interface, or in a vacuum. In the latter case, the question as to whether the manometrically measured pressure equals the pressure at the interface needs to be answered. Small sample size and a porous oxide favor this condition. 

From the rate of polymerization rp0j measured at different pressures (Figure 3) an activation volume for the polymerization of ethylene catalyzed by metallocenes can be determined. For this purpose first an overall rate constant kpo] was evaluated from the relation... 

Within the limits of experimental error, the effective diffusity increases with temperature and decreases with pressure as it is observed for molecular diffusivity. In general, the tortuosity factor was found to be constant at different flow rates and for different particle sizes. Figure 3 illustrates a comparison of predicted and experimental extraction curves for both small particles... 

There are several studies that have been successful in determining the dissolution rate at conditions near seawater saturation. Acker et al. (1987) was able to employ very precise determinations of pH to measure the rate of dissolution of a single pteropod shell at different pressures from 15 atm to 300 atm. Because his measurements were at different pressures and is a function of pressure, he was able to determine whether the rate constant is indeed a function of K p. He found that Equation (9) fit his data better than (10), suggesting that the constant is not pressure dependent and the former is a more accurate universal rate law. An exponent oin= 1.9 was obtained for this surface-controlled dissolution reaction and a partial molal volume. Ay, of —39 cm mol (very close to the mean of the values determined in laboratory experiments for calcite) best fit the data. 

Using the above type of experimental setup for propylene polymerization with TiCls-AlEts in n-heptane, the rate of polymerization was measured [5] at different speeds of stirring and constant propylene pressure. The results obtained indicated that there were two different steady-state rate curves for the stirring speeds of 400 and 600 rpm. In each case, a steady bulk monomer concentration was reached in about 3-4 hours. Show how the overall process at steady state can be modelled to show dependence of the polymerization rate on stirring speed and to enable determination of both the mass transfer rate constant and polymerization rate constant from rate measurements at different stirring speeds. 

Figure 7-37 shows the relationship between burning rate and (N02) at different pressures 281. The burning rate decreases linearly as (N02) increases at a constant pressure in a (N02) versus In rplot. The burning rate is represented by... 

Figures 41 (a) and (b) show the predicted and experimental rate coefficients at different temperatures as a function of N2 pressure. In these figures, the solid 1 ines represent the calculated results using the predicted AH°o = -15.8 kcal/mol the dotted and dashed lines represent the results using experimentally estimated AH°o = -14.8 [130] and -11.1 [131] kcal/mol, respectively the dash-dot-dotted line is the results based AH°o = -17.7 kcal/mol which is the upper-limit reported by Hayman and Cox [130]. Symbols are experimental results [130, 131, 141]. Apparently, the pressure-dependent rate constants are strongly sensitive to AH°o for the low-dissociation energy process our predicted values based on Affo = -15.8 kcal/mol are in good agreement with experimental data.

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