Equation, Arrhenius

Virtually all models currently used for describing the aging of plastics are based on the Arrhenius equation. 

Exposure tests at increased temperatures based on the Arrhenius equation are state-of-the-art for evaluating the service life of plastic materials and plastics components. The Arrhenius equation has general validity and describes the temperature dependence of simple reactions and physical processes. The reaction rate constants of these simple chemical reactions increase exponentially with temperature. Thus, a temperature increase will cause an accelerated reaction. 

However, if the Arrhenius equation is applied to complex reactions and processes with many individual reactions, it can lead to a distortion of the results, because the total activation energy of the combined reaction is influenced by the activation energies of the individual reactions so that the rate constant of the total reaction does not progress exponentially. 

To apply the Arrhenius equation is to presume that the rate of plastics aging is determined by one (dominant) chemical reaction or one (dominant) physical process with corresponding dependency of aging rate on temperature. The relation found by Arrhenius for gas reactions between the reaction rate constant and absolute temperature T states that  

The limitation of the Arrhenius equation is caused by the fact that it is applied to an unsuitable problem. The Arrhenius equation describes a simple chemical reaction, whereas the reactions are more complex in practice. In addition, it is presumed that the reactions at service temperature are the same as at test temperature, that activation energy is independent of temperature, and that chemical transformations relate directly to physically measured properties. If one of these assumptions is not valid, the Arrhenius equation is of limited use. Moreover, geometrical and physical effects are largely ignored. 

In order to explain the observed kinetic behavior, Arrhenius expressed the view that, in a reacting system, there is only a certain number of active molecules that can undergo reaction. Arrhenius claimed that the nonreactive molecules could be activated by absorption of the heat of normal molecules. He found that the effect of temperature on the reaction rate could be represented by the equation  

Arrhenius believed that there was an equilibrium between active and inert molecules at any given temperature and that this equilibrium is shifted in favor of active molecules with temperature rise. It has been found for many chemical reactions that a plot of In k against 1/r gives a straight line. 

The next most important idea was that the temperature dependence of the reaction rate constant must be a function similar to the temperature dependence of the reaction equilibrium constant, because the equilibrium constant is equal to the ratio of forward and reverse rate constants (K = k lk ). Since van t Hoff s differential equation  

Arrhenius could not define the physico-chemical meaning of his pre-exponential factor A. Nevertheless, the Arrhenius equation contributed substantially to the development of the theory of reaction rates. 

Discrepancies between the results predicted by the collision theory (equation 1.3.7) and the experimental values were corrected by introducing the so-called steric factor P hence the equation (1.3.12) assumed the form (1.3.13)  

The rate of a reaction r is dependent on the reactant concentrations. For example, a bimolecular reaction between the reactants B and C could have a rate 

The simplest expression for the temperature dependence of the rate constant 

Although the Arrhenius equation does not predict rate constants without parameters obtained from another source, it does predict the temperatme dependence of reaction rates. The Arrhenius parameters are often obtained from experimental kinetics results since these are an easy way to compare reaction kinetics. The Arrhenius equation is also often used to describe chemical kinetics in computational fluid dynamics programs for the purposes of designing chemical manufacturing equipment, such as flow reactors. Many computational predictions are based on computing the Arrhenius parameters. 

Some reactions, such as ion-molecule association reactions, have no energy barrier. These reactions cannot be described well by the Arrhenius equation or 

Rates of reactions change as a function of the temperature of the reactants. Frequently, reactions go faster 

The energy of the reactants is higher than that of the products, so heat energy is released when the reaction goes in the forward direction. 

Arrhenius published this equation in 1889 and suggested that molecules must be given enough energy to become activated before they could react. Collision and transition-state theories, which enlarged on this concept, were developed later (1920s and 1930s, respectively). 

Rate constants for most chemical reactions closely follow an equation of the form 

The mathematical equation k = Ae °, which expresses the dependence of the rate constant on temperature, is called the Arrhenius equation, after its formulator, the Swedish chemist Svante Arrhenius. Here e is the base of natural logarithms, 2.718. .. is the activation energy R is the gas constant, 8.31 J/(K mol) and T is the absolute temperature. The symbol A in the Arrhenius equation, which is assumed to be a constant, is called the frequency factor. The frequency factor is related to the frequency of collisions with proper orientation (pZ). (The frequency factor does have a slight dependence on temperature, as you see from collision theory, but usually it can be ignored.) 

It is useful to recast Arrhenius s equation in logarithmic form. Taking the natural logarithm of both sides of the Arrhenius equation gives 

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Arrhenius equation The variation in the rate of a chemical reaction with temperature can be represented quantitatively by the Arrhenius equation... 

In praetiee, one of the most important aspeets of interpreting experimental kinetie data in tenns of model parameters eoneems the temperature dependenee of rate eonstants. It ean often be deseribed phenomenologieally by the Arrhenius equation [39, 40 and 41]... 

This expression corresponds to the Arrhenius equation with an exponential dependence on the tlireshold energy and the temperature T. The factor in front of the exponential function contains the collision cross section and implicitly also the mean velocity of the electrons. 

There are a few cases where the rate of one reaction relative to another is needed, but the absolute rate is not required. One such example is predicting the regioselectivity of reactions. Relative rates can be predicted from a ratio of Arrhenius equations if the relative activation energies are known. Reasonably accurate relative activation energies can often be computed with HF wave functions using moderate-size basis sets. 

Note that Eqs. (6.5) and (6.12) are both first-order rate laws, although the physical significance of the proportionality factors is quite different in the two cases. The rate constants shown in Eqs. (6.5) and (6.6) show a temperature dependence described by the Arrhenius equation ... 

The activation energies for the decomposition (subscript d) reaction of several different initiators in various solvents are shown in Table 6.2. Also listed are values of k for these systems at the temperature shown. The Arrhenius equation can be used in the form ln(k j/k j) (E /R)(l/Ti - I/T2) to evaluate k j values for these systems at temperatures different from those given in Table 6.2. 

Combination and disproportionation are competitive processes and do not occur to the same extent for all polymers. For example, at 60°C termination is virtually 100% by combination for polyacrylonitrile and 100% by disproportionation for poly (vinyl acetate). For polystyrene and poly (methyl methacrylate), both reactions contribute to termination, although each in different proportions. Each of the rate constants for termination individually follows the Arrhenius equation, so the relative amounts of termination by the two modes is given by... 

The apparent rate constant in Eq. (6.26) follows the Arrhenius equation and yields an apparent activation energy ... 

The mechanistic analysis of the rate of polymerization and the fact that the separate constants individually follow the Arrhenius equation means that... 

Applying the Arrhenius equation to Eq. (6.116) shows that the apparent activation energy for the overall rate of polymerization is given by... 

The temperature dependence of the reactivity ratio rj also involves the Ell Ej2 difference through the Arrhenius equation hence... 

The Arrhenius equation enables us to expand on this still further ... 

E. R. Bixon and D. Robertson, "Lifetime Predictions for Single Base PropeUant Based on the Arrhenius Equation," in Fifth International Gun Propellant and Propulsion Symposium, ARDEC, Dover, N.J., Nov. 1991. 

The experimentally measured dependence of the rates of chemical reactions on thermodynamic conditions is accounted for by assigning temperature and pressure dependence to rate constants. The temperature variation is well described by the Arrhenius equation. 

Rheology of LLDPE. AH LLDPE processiag technologies iavolve resia melting viscosities of typical LLDPE melts are between 5000 and 70, 000 Pa-s (50,000—700,000 P). The main factor that affects melt viscosity is the resia molecular weight the other factor is temperature. Its effect is described by the Arrhenius equation with an activation energy of 29—32 kj/mol (7—7.5 kcal/mol) (58). 

A common expression relating viscosity to temperature is the Arrhenius equation, rj = otrj = A-10, where M and B are constants... 

The Arrhenius equation may also be expressed in logarithmic form (eq. 6) ... 

The Arrhenius equation holds for many solutions and for polymer melts well above their glass-transition temperatures. For polymers closer to their T and for concentrated polymer and oligomer solutions, the WiUiams-Landel-Ferry (WLF) equation (24) works better (25,26). With a proper choice of reference temperature T, the ratio of the viscosity to the viscosity at the reference temperature can be expressed as a single universal equation (eq. 8) ... 

Effects of Rate Conditions. It is essential for commercial a-quartz crystals to have usable perfection growth at a high rate and at pressure and temperature conditions that allow economical equipment design. The dependence of rate on the process parameters has been studied (8,14) and may be summarized as follows. Growth rate depends on crystallographic direction the (0001) is one of the fastest directions. Because AS is approximately linear with AT, the growth rate is linear with AT. Growth rate has an Arrhenius equation dependence on the temperature in the crystallization zone ... 

FIG. 7-1 Constants of the power law and Arrhenius equations hy linearization (a) integrated equation, (h) integrated fimt order, (c) differential equation, (d) half-time method, (e) Arrhenius equation, (f) variahle aotivation energy, and (g) ehange of meohanism with temperature (T in K),... 

The most common ways of evaluating the constants are from linear rearrangements of the rate equations or their integrals. Figure 7-1 examines power law and Arrhenius equations, and Fig. 7-2 has some more complex cases. 

The two basic laws of kinetics are the law of mass action for the rate of a reac tion and the Arrhenius equation for its dependence on temperature. Both of these are strictly empirical. They depend on the structures of the molecules, but at present the constants of the equations cannot be derived from the structures of reac ting molecules. For a reaction, aA + hE Products, the combined law is... 

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