Arrhenius-type relationship

An Arrhenius-type relationship is obtained, with a slope determined by the energy of formation of the defects. 

The polymer rheology is modeled by extending the usual power-law equation to include second-order shear-rate effects and temperature dependence assuming Arrhenius type relationship. 

The parameter values were then plotted versus the inverse temperature and were found to follow an Arrhenius type relationship... 

Let us reconsider the hydrogenation of 3-hydroxypropanal (HPA) to 1,3-propanediol (PD) over Ni/Si02/Al203 catalyst powder that used as an example earlier. For the same mathematical model of the system you are asked to regress simultaneously the data provided in Table 16.23 as well as the additional data given here in Table 16.28 for experiments performed at 60°C (333 K) and 80°C (353 K). Obviously an Arrhenius type relationship must be used in this case. Zhu et al. (1997) reported parameters for the above conditions and they are shown in Table 16.28. 

Examine whether any of the estimated parameters follow an Arrhenius-type relationship. If they do, re-estimate these parameters simultaneously. A better way to numerically evaluate Arrhenius type constants is through the use of a reference value. For example, if we consider the death rate, kd as a function of temperature we have... 

The effect of temperature on the rate and degree of polymerization is of prime importance in determining the manner of performing a polymerization. Increasing the reaction temperature usually increases the polymerization rate and decreases the polymer molecular weight. Figure 3-13 shows this effect for the thermal, self-initiated polymerization of styrene. However, the quantitative effect of temperature is complex since Rp and X depend on a combination of three rate constants—kd, kp, and kt. Each of the rate constants for initiation, propagation, and termination can be expressed by an Arrhenius-type relationship... 

SocJ 30, 151-58(1960) (Recent advances in condensed media detonation) 37b) Dunkle s Syllabus (1960-1961), pp 4a 4b (Initiation of shock waves) lOa-lOg (Initiation of deflgrn and deton) p 12a (Frank-Kamenet-skii formulation) p 13b (Initiation by electric discharge) p 13f (Thermal Decomposition and Initiation of Explosives, as discussed by B. Reitzner) pp 17a to 17e (Mechanism of initiation and propagation of detonation in solid explosives) pp 17e 17f [Marlow Skidmore (Ref 31) concluded from their investigations that the problem of shock initiation is somehow related to the temperature distribution in the shock pulse and its effect on the chemical reaction rate. They used an Arrhenius type relationship for the rate increase in the frac-... 

Temperature Dependence of Pure Metal Viscosity. Practically speaking, empirical and semiempirical relationships do a much better job of correlating viscosity with nsefnl parameters such as temperature than do equations like (4.7). There are nnmerons models and their resnlting equations that can be used for this purpose, and the interested student is referred to the many excellent references listed at the end of this chapter. A useful empirical relationship that we have already studied, and that is applicable to viscosity, is an Arrhenius-type relationship. For viscosity, this is... 

Experiments snch as the one illnstrated in Fignre 4.38 not only give us self-diffusion coefficients for certain snbstances, bnt as the temperatnre of the experiment is varied, they give us the temperature dependence of the process and a measurement of the activation energy barrier to diffnsion. Diffusion in solid systems, then, can be modeled as an activated process that is, an Arrhenius-type relationship can be written in which an activation energy, Ea, and temperatnre dependence are incorporated, along with a preexponential factor. Do, sometimes called ht frequency factor ... 

The variation of creep with time as a function of both load and temperature is illustrated in Figure 5.44. Arrhenius-type relationships have been developed for steady-state creep as a function of both variables such as... 

Although this simple relationship holds for some gases, for other gases and most vapours it does not and, as noted above, the permeability constant is then not a constant. It depends on the solubility and diffusion characteristics but these may vary with different conditions. The permeability constant varies with temperature and, although simple theory predicts that the change will follow an Arrhenius type relationship, this also is not true for many vapours. 

The aqueous geochemistry data obtained from the Nickel Rim site indicates that the rate of sulfate reduction is dependent on temperature, with sulfate reduction more rapid during the warm summer months than during the cooler winter months. The relationship between temperature and reaction rate could be explained using an Arrhenius-type relationship (Benner et al., 2002). 

Similar to solutions of other random coil polysaccharides, the viscosity of pectin dispersion decreases with increasing temperature but increases with increasing concentration, and the effect of temperature is stronger at higher concentrations (Kar and Arslan, 1999b). The effect of temperature on apparent viscosity is usually analyzed by an Arrhenius-type relationship to calculate activation energy of flow... 

Instead of following the WLF equation, values of the shift factor follow an Arrhenius-type relationship with temperature, indicating-that the chain segment mobility is restricted even in the rubbery state. 

In conclusion, the temperature dependence of shift factors for the networks studied here do not follow the WLF equation, but rather an Arrhenius-type relationship. The apparent activation energies are independent of stoichiometric variation [as they are when is varied by changing prepolymer molecular weight (13)]. ... 

Effect of Temperature on Viscosity. The viscosity of mobility control polymers decreases with increasing temperature and an Arrhenius type relationship is obeyed ... 

The above equations are based on the assumption that the diffusion coefficient D is independent of the gas concentration. However, that may not be necessarily true. Also, the diffusion coefficient is usually dependent on temperature following an Arrhenius type relationship. For a sphere, the Equation 2 can be represented in terms of concentration gradient in radial direction (3c/3r) as follows ... 

By substituting Equations 2 and 3 into Equation 1, the permeability can also be expressed in terms of an Arrhenius type relationship. 

It is clear from Eq. (6.24) that the polymerization rate Rp depends on the combination of three rate constants kd,kp, and kt, which makes the quantitative effect of temperature on Rp rather complex. Expressing the rate constants kp,kd, and k by an Arrhenius type relationship (Young and Lovell, 1990) ... 

Other equations have been developed to describe the shear thinning behavior of polymer melts, for instance, the Yasuda-Carreau equation, which is written here as Equation 22.19 [41]. In this equation, as in the power-law model, the effect of temperature on viscosity of the system can be taken into account by means of an Arrhenius-type relationship ... 

It is also possible to divide the biomass into living and dead biomass and to include equations for temperature-related death. Arrhenius-type relationships can be used to describe the effect of temperature on both the specific growth rate and the specific death rate, giving a net growth rate [99] ... 

Considerable differences in the resistivity between that of ordinary proteins and that of electron transfer proteins have been reportedand also between ferrocytochrome-c and ferricytochrome-c. This is attributed to the electronic state of the central metal atom. The anhydrous cytochromes exhibit resistivities many orders of magnitude higher thus anhydrous ferricytochrome has a room temperature of 4.1 x 10 Q cm and the ferrocytochrome Cj a value of 1.6 x 10 these data are even more remarkable because the ferri compound is said not to follow an Arrhenius type relationship in its temperature dependence while the ferro compound is reported to yield a negative value for its thermal activation energy. Cytochromes will be discussed in more detail in the next section. 

The viscosity of low molecular weight polymers is related to their temperature by an Arrhenius-type relationship [19, 22, 25] ... 

To study TSA systems with the solute movement analysis we must determine the effect of tenperature changes on the solute waves, the rate at which a tenperature wave moves in the column, and the effect of temperature changes on concentration. The first of these is easy. As tenperature increases the equilibrium constants, and K, both decrease, often following an Arrhenius type relationship as shown in Eq. fl8-7). If the effect of temperature on the equilibrium constants is known, new values of the equilibrium constants can be calculated and new solute velocities can be determined. 

---