Arrhenius rate expression, diffusion

From this molecular theory, we see that the diffusion coefficient depends on the frequency of cooperative main-chain motions of the polymer, v, which cause chain separations equal to or greater than the penetrant diameter. Pace and Datyner were able to estimate v by adopting an Arrhenius rate expression in which the pre-exponential factor, A, is a function of both AE and T. The diffusion coefficient is given by,... 

The ability to model the detailed chemistry of ignition and combustion of energetic materials requires the simultaneous treatment of the chemical kinetics behavior of large chemical reaction systems combined with convective and diffusive transport of mass, momentum, and energy. Such models require the evaluation of equations of state, thermodynamic properties, chemical rate expressions, and transport properties. The computer software used to evaluate these quantities is referred to as the Chemkin package.33-35 includes an interpreter for the chemical reactions, a thermochemical data base, a linking file, and gas-phase subroutine libraries. The interpreter reads in the list of elementary chemical reactions. The forward reaction rates are given in the form of the Arrhenius rate expression... 

Pk temperature exponent in Arrhenius rate expression (—) r generalized diffusion coefficient (variable units)... 

Increasing temperature permits greater thermal motion of diffusant and elastomer chains, thereby easing the passage of diffusant, and increasing rates Arrhenius-type expressions apply to the diffusion coefficient applying at each temperature," so that plots of the logarithm of D versus reciprocal temperature (K) are linear. A similar linear relationship also exists for solubUity coefficient s at different temperatures because Q = Ds, the same approach applies to permeation coefficient Q as well. 

We have thus far written unimolecular surface reaction rates as r" = kCAs assuming that rates are simply first order in the reactant concentration. This is the simplest form, and we used it to introduce the complexities of external mass transfer and pore diffusion on surface reactions. In fact there are many situations where surface reactions do not obey simple rate expressions, and they frequently give rate expressions that do not obey simple power-law dependences on concentrations or simple Arrhenius temperatures dependences. 

Self diffusion coefficients can be obtained from the rate of diffusion of isotopically labeled solvent molecules as well as from nuclear magnetic resonance band widths. The self-diffusion coefficient of water at 25°C is D= 2.27 x 10-5 cm2 s 1, and that of heavy water, D20, is 1.87 x 10-5 cm2 s 1. Values for many solvents at 25 °C, in 10-5 cm2 s 1, are shown in Table 3.9. The diffusion coefficient for all solvents depends strongly on the temperature, similarly to the viscosity, following an Arrhenius-type expression D=Ad exp( AEq/RT). In fact, for solvents that can be described as being globular (see above), the Stokes-Einstein expression holds ... 

In a diffusion-free enzyme reaction the reaction rate increases up to a certain critical value exponentially and is described by the Arrhenius equation [82]. In diffusion-controlled reactions the reaction rate is a matter of the efficiency factor ri [see Eqs. (3 - 5)]. In more detail, the maximum reaction rate is expressed within the root of Eq. (4). Conclusively, the temperature dependence is a function of the square root of the enzyme activity. In practice, immobilized enzymes are much less temperature dependent when their reaction rate is diffusion controlled. 

Finally, another area of overlap is the potential presence of a false activation energy. This occurs when film diffusion is coupled with, say, a hrst-order reaction at the catalyst surface. The mass transfer rate r = k a ic - Cj). In contrast, the inherent reaction rate is r = These two can be combined to give an expression for the snrface concentration c, = kf cl(k + k, which is difficult to measure. This expression can be snbstitnted in the original reaction rate expression to give r = with an effective rate constant of = ll(llk, + k. Thns, if the inherent rate obeys the Arrhenius dependence on temperatnre, k, = A exp(-A yR7), and if kf is constant, the observed Arrhenius activation energy, AEq = RT d In k(/d(l/T), would be deceptively low. 

There is a large amount of data on nonhomogeneous track chemistry of energetic electrons at room temperature, and the track structure and diffusion-limited kinetics are well parameterized. This wealth of knowledge contrasts with the limited information about the effects of radiation on aqueous solutions at elevated temperatures. The majority of the studies at elevated temperatures have been performed at AECL, Canada [93], or at the Cookridge Radiation Laboratory, University of Leeds, UK [94]. These two groups have focused on measuring the rate coefficients of the reactions of the radiation-induced radicals and ions of water. The majority of the temperature dependencies can be fitted with an empirical Arrhenius-type expression, k = A This type of parameterization provides a... 

K is the reactivity rate constant, D is a constant, and is the critical extent of curing at which the glassy state is attained. Because switching from a reactivity-controlled reaction takes place gradually, the overall rate constant k (x, T) can be expressed using the Rabinovitch model [13] in terms of the Arrhenius rate constant (k) or reactivity rate constant (k ) and diffusion rate constant (k ) as follows [19, 20] ... 

In practice it is usual to use Arrhenius type expressions for the temperature dependence of the rate constants. These expressions lead to small but finite values of the A , at the cold boundary temperature Tq. In this case, solutions of the type mentioned above do not exist and it is necessary to invoke the concept of a flameholder. The flameholder acts primarily as a heat sink and as a semipermeable membrane which passes only the fuel molecules and prevents the back diffusion of the product molecules. 

The rate of diffusion is governed by the diffusion coefficient in conjunction with Pick s laws of diffusion. Because diffusion is thermally activated, the diffusion coefficients follow the Arrhenius behavior and it is a common practice to express their thermal dependence in terms of a pre-exponential coefficient Do and an activation energy Q,... 

The presence of diffusion limitations has a strong effect on the apparent activation energy one measures. We can express both the rate constant, k, and the diffusion constant, Defr, in the Arrhenius form ... 

The increase of the lateral diffusion rate with increasing temperature was used to estimate the activation energy for diffusion in the LC and GI phases. The temperature dependence of the correlation-time for molecular diffusion, Xd, can be formulated in terms of the activation energy E ) for the motion affecting Xd in an Arrhenius expression (t > = exp( a/R7 ))- Since D = a ldx ... 

In what I regard as the world of change (essentially chemical kinetics and dynamics), there are three central equations. One is the form of a rate law, v = /[A],[B]...), and all its implications for the prediction of the outcome of reactions, their mechanisms, and, increasingly, nonlinear phenomena, and the other closely related, augmenting expression, is the Arrhenius relation, k = Aexp(-EJRT), and its implications for the temperature-dependence of reaction rates. Lurking behind discussions of this kind is the diffusion equation, in its various flavors starting from the vanilla dP/dt = -d2P/dl2 (which elsewhere I have referred to as summarizing the fact that Nature abhors a wrinkle ). 

A pure gas is absorbed into a liquid with which it reacts. The concentration in the liquid is sufficiently low for the mass transfer to be governed by Fick s law and the reaction is first order with respect to the solute gas. It may be assumed that the film theory may be applied to the liquid and that the concentration of solute gas falls from the saturation value to zero across the film. Obtain an expression for the mass transfer rate across the gas-liquid interface in terms of the molecular diffusivity, D, the first-order reaction rate constant ft, the film thickness L and the concentration Cas of solute in a saturated solution. The reaction is initially carried out at 293 K. By what factor will the mass transfer rate across the interface change, if the temperature is raised to 313 K Reaction rate constant at 293 K = 2.5 x 10 6 s 1. Energy of activation for reaction (in Arrhenius equation) = 26430 kJ/kmol. Universal gas constant R = 8.314 kJ/kmol K. Molecular diffusivity D = 10-9 m2/s. Film thickness, L = 10 mm. Solubility of gas at 313 K is 80% of solubility at 293 K. 

Table 6.8 presents the details of calculations for spherical particles with an equivalent diameter of 2.4mm. It may be observed that the pore diffusion considerably affects the process rate, particularly at higher temperatures. The external mass transfer plays a minor role. Their combination leads to a global effectiveness that drops from 75% to 35% when the temperature varies from 160 to 220°C. Based on the above elements the apparent reaction constant may be expressed by the following Arrhenius law ... 

This expression has the Arrhenius form and E is the maximum value of the potential energy, an activation energy for deposition. This is expected because the potential profile of fig. 2 resembles the plot of the energy against reaction coordinate used in the theory of rate processes. The factor /(//m) accounts for the dependence of the diffusion coefficient on the distance and evaluations show that it can decrease the frequency factor in eqn (16) by two orders of magnitude. 

The temperature dependence of the rate constant k is normally expressed by an Arrhenius law with the intrinsic activation energy E. In contrast, the temperature dependence of the effective diffusivity De is much weaker. Normally, De is obtained from... 

Transport through a dense polymer may be considered as an activated process, which can be represented by an Arrhenius type of equation. This implies that temperature may have a large effect on the transport rate. Equations 4.7 and 4.8 express the temperature dependence of the diffusion coefficient and solubility coefficient in Equation 4.5 ... 

Operative. For the non isothermal case, effectiveness factors greater than unity are possible. Weisz and Hicks have considered this problem in some detail and constructed a number of graphs for various heats of reaction and activation energies. When a reaction is limited by pore diffusion, the reaction rate is proportional to yjky. If the temperature effects can be expressed as a simple Arrhenius relationship = A txp —E/RT), then the measured activation energy E will be about half the true activation energy. Very low values of the activation energy, i.e, 1-2 kcal. mole are only observed when mass transfer to the external catalyst surface is limiting the rate. 

Temperature and Transitions. Diffusion in solids and liquids is extremely affected by temperature. Like reaction rates, diffusion may be thought of as an activated process following an expression of the Arrhenius form... 

---