Temperature dependence of the diffusion constant

the integrand in eqn (14.22) can be replaced with its value for small x. Using eqn (14.18) for the typical values for k, this condition translates into 

If we now replace k with its mean value ( k 2)1/2 and perform the integration in eqn (14.22) using the small-.x expansion, we obtain 

for T T0, we find rac 1/T3/2 which using eqn (14.12) for the diffusion constant leads to 

This relationship for Frenkel excitons was derived in (14) it can be seen from its derivation that it is independent of the model and, therefore, is valid also for ground state large-radius excitons as well as for electrons and holes in semiconductors. 

To end this section, we make a few remarks. First it should be noted that the applicability of the above D(T) relationships is limited by the condition of applicability of the Boltzmann equation. This condition reads A where = (v)t and A is the thermal de Broglie wavelength of the excitons. Therefore, we should expect that this condition may be satisfied only for sufficiently low 

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Applying this prediction to the cooling rate dependence of a break points in the specific volume curves, one obtains a Vogel-Fulcher temperature of To = 0.35 that agrees well with that determined from the temperature dependence of the diffusion constant in this model, which is T = 0.32. 

Figure 3. Arhenius plot of the temperature dependence of the diffusion constant for HP Xe diffusing into a cylinder of porous Vycor glass.

From these figures he computed the energy of activation for diffusion normal to the 201 and 001 faces respectively to be 5400 and 9140 cal./mol. The temperature dependence of the diffusion constants he found did not depend appreciably upon the amount of water in the lattice, although we have seen that their absolute magnitudes do. 

The temperature dependence of the permeability arises from the temperature dependencies of the diffusion coefficient and the solubility coefficient. Equations 13 and 14 express these dependencies where and are constants, is the activation energy for diffusion, and is the heat of solution... 

In addition, the temperature dependence of the diffusion potentials and the temperature dependence of the reference electrode potential itself must be considered. Also, the temperature dependence of the solubility of metal salts is important in Eq. (2-29). For these reasons reference electrodes with constant salt concentration are sometimes preferred to those with saturated solutions. For practical reasons, reference electrodes are often situated outside the system under investigation at room temperature and connected with the medium via a salt bridge in which pressure and temperature differences can be neglected. This is the case for all data on potentials given in this handbook unless otherwise stated. 

On the other hand, the low temperature dependance of the rate constants with activation energies around 5 kcal/mole indicates a diffusion limited reaction rate which could refer to diffusion of oxygene into the fibers of the board, i.e. into the fiberwalls. The corresponding negative activation energy for the groundwood based hardboard and the effect of fire retardants there upon are difficult to understand. 

The layer-growth kinetics were found to be parabolic for both compounds (Fig. 2.18), indicative of diffusion control. This is an expectable result since the layer thickness varied from about 10 pm to 300 pm for the Al12Mg17 intermetallic compound and from about 80 pm to more than 900 pm for the Al3Mg2 intermetallic compound. Diffusional constants were calculated using parabolic equations of the type x2 = 2k t. The temperature dependence of the diffusional constants was found to obey the Arrhenius relation ... 

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... 

The experimental a versus x dependence for these samples, together with the fitting curves, are shown in Fig. 53. Note that in contrast to the previous example, these data are obtained at a constant sample composition. Now, Variations of the parameters a and x are induced by temperature variation. As mentioned above, the exponents a as well as the relaxation time x are functions of different experimentally controlled parameters. The same parameters can affect the structure or the diffusion simultaneously. In particular, both a and x are functions of temperature. Thus, the temperature dependence of the diffusion coefficient in (144) should be considered. Let us consider the temperature dependence of the diffusion coefficient D ... 

The numerical value of each of these constants depends on temperature due to the temperature dependence of the diffusion coefficients, chemical reaction rate constant, and equilibrium constant. 

A theory of exciton-phonon coupling is presented and the consequences of this coupling for spectral line shapes and exciton transport are discussed. The theory is valid for arbitrary phonon and exciton bandwidths and for arbitrary exciton phonon coupling strengths. The dependence of the diffusion constant on temperature and the other parameters is analyzed. 

Fig. 7.9 shows the temperature dependence of the dielectric constant and dielectric loss at 1 kHz for the PMN-PT ceramics obtained by sintering the calcined powders from a soft-mechanochemical route at 1200°C for 2 h. A diffuse phase transition, being typical for a relaxor, is observed for each ceramics. As x increases from 0 to 0.2, the maximum dielectric constant, K, , increases from 13000 to 27000. The temperature correspondent to K ,... 

The temperature dependence of the rate constant is illustrated in Fig. 19. Thble 2 summarizes the apparent activation energies of the ion transfer F fr = -Z 8In(l/T), and of diffusion F a = -Rd nD/Q /T). Ion diffusion coefficients at various temperatures were evaluated from voltammetric data [115]. Provided that the temperature dependencies of the friction coefficient in the bulk of the solution, and at the location a, have equal slopes, a relationship can be derived from Eq. (48) [115] ... 

By way of example, Figure 6.2 shows a first order rate plot for the stirred contact of sodium form styrenesulfonate cation exchange resin (12% DVB) with 0.001 M hydrochloric acid solution at 25 C. The system data and calculated rate constant are given in Table 6.1. The activation energy may be found from the temperature dependence of the rate constant and was found to equal 16.7 kj eq. This same data is redeployed later according to more rigorous diffusion theory. 

To describe the kinetics of sand-grain dissolution in the system Si02 —CaO—Na20, Sasek in the study mentioned above applied the Hixson-Crowell (1931) solution which takes into consideration the variable solvent concentration during mass transfer by steady-state diffusion. His results indicate that solution of this type describes well isothermal dissolution kinetics however, the temperature dependence of the rate constant does not have the expected simple behaviour. According to the author s explanation, this is due to the change in the character of the rate--controlling step which is surface reaction up to about 1 ISO and diffusion only above 1400 °C. 

The temperature dependence of the diffusion coefficient or diffusion constant is generally expressed as ... 

The diffusion of oxygen in polymers is unaffected by any complication due to the concentration dependence of the diffusion constant [583]. The effect of temperature on the diffusion is given by the relationship... 

The proportionality constant D is called the diffusion coefficient and quantifies the chaotic translation motion of the molecules in solution. Its basic evaluation is given by the Stokes formula (Eq. 33). The diffusion coefficient decreases as the size of the molecule increases. For typical biomolecules in aqueous medium, D is usually between 10 cm s and 10 cm s Temperature dependence of the diffusion coefficient follows T/rj, where T is absolute temperature and t] viscosity of the solvent, unless the temperature change does not alter the molecular shape. 

Since the data were obtained in the transition region where intrinsic and extrinsic defects are contributing to the total defect concentration, the calculation of an enthalpy of motion cannot be made in a simple way because the temperature dependence of the Frenkel constant is not known. However, Ail. probably increases with temperature while the extrinsic defect concentration decreases with temperature. If, to a first approximation, these two trends cancel, then the enthalpy of motion is just the experimentally determined activation energy. Using this value from (16) and the defect concentration shown in Table VII, the preexponential constant Dq and hence the diffusion coefficient can be determined. 

The parameters of the model were estimated from the experimental data using a non linear multivariate curve fitting technique. In this process the temperature dependence of the diffusion coefficient for glucose was assumed to be small in the range of temperatures studied. The equilibrium constant was assumed to be given by ... 

These predictions were made for 200 um membranes. Industrial application of this technology will require the use of membranes which are two orders of magnitude thinner. In order to use the model to predict facilitation factors for thinner membranes, it is necessary to determine whether the reaction equilibrium assumption still applies. The parameter (tanh )/ has a value of 0 if the system is diffusion limited and 1 if the facilitated transport system is reaction rate limited. At a thickness of Ipm, the value of (tanh X)/X is of the order 10 , which implies that the system is diffusion limited and that the simplified analytical model can be used to predict facilitation factors. If the solubility of HjS, the pressure and temperature dependence of the equilibrium constant and the diffusion coefficients are known, then F could be estimated at industrial conditions. 

The diffusion laws which apply to gases are not valid for the condensed phase, where diffusion is governed by migration processes. These require activation energy E, and therefore the temperature dependency of the diffusion coefficient can be described by the Anhenius-type Eq. 5, where R is the universal gas constant. 

Acetyl transfer between aspirin and sulfadiazine is a bimolecular reaction in which the translational diffusion of reactant molecules becomes rate determining when molecular mobility is limited in the solid state [33]. Therefore, it can offer a useful reaction model for understanding the ways in which chemical degradation rates in lyophilized formulations are affected by molecular mobility. Figure 17A shows the temperature dependence of the rate constant of acetyl transfer in lyophilized formulations containing dextran. Figure 17B shows the pseudo rate constant of aspirin hydrolysis that occurs in parallel with acetyl transfer in the presence of water. The rate constant of acetyl transfer ( t) and the pseudo rate constant of hydrolysis ( H> pseudo) are described by following equations ... 

Dq being a constant characterising the polymer-diffusing couple, E the energy of activation, expressing the temperature dependency of the diffusivity, R is the ideal gas constant, and T the local temperature which varies with time and space. 

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