The Arrhenius Equation

To see if the removal is a first-order process, we draw up the following table  

The graph of the data is shown in Fig. 6.8. The plot is straight, confirming a first-order process. Its least-squares best-fit slope is -7.6 x 10 so = 7.6 X 10 min and ti/2 = 91 min at 310 K, body temperature. 

The rates of most chemical reactions increase as the temperatare is raised. Many organic reactions in solution He somewhere in the range spaimed by the hydrolysis of methyl ethanoate (for which the rate constant at 35 C is 1.8 times that at 25°C) and the hydrolysis of sucrose (for which the factor is 4.1). Reactions in the gas phase typically have rates that are only weakly sensitive to the temperature. Enzyme-catalyzed reactions may show a more complex temperature dependence because raising the temperature may provoke conformational changes and even denaturation and degradation, which lower the effectiveness of the enzyme. We saw in the discussion of the hydrophobic effect (Section 2.7) that lowering the temperature can also result in denaturation, so an enzyme may lose its effectiveness at low temperatures too. 

The balance of reactions in organisms depends strongly on the temperature that is one function of a fever, which modifies reaction rates in the infecting organism and hence destroys it. To discuss the effect quantitatively, we need to know the factors that make a reaction rate more or less sensitive to temperature. 

As data on reaction rates were accumulated toward the end of the nineteenth century, the Swedish chemist Svante Arrhenius noted that almost all of them showed a similar dependence on temperature. In particular, he noted that a graph of In k where k, is the rate constant for the reaction, against 1/T, where T is the (absolute) temperature at which k, is measured, gives a straight line with a slope that is characteristic of the reaction (Fig. 6.15). The mathematical expression of this conclusion is that the rate constant varies with temperature as 

In this equation k is the rate constant, is the activation energy, R is the gas constant (8.314 J/mol-K), and T is the absolute temperature. The frequency factor. A, is constant, or nearly so, as temperature is varied. This factor is related to the frequency of collisions and the probability that the collisions are favorably oriented for reaction. As the magnitude of increases, k decrea.ses because the fraction of molecules that possess the required energy is smaller. Thus, reaction rates decrease as E increases. 

Relating Energy Profiles to Activation Energies and Speeds of Reaction 

Consider a series of reactions having these energy profiles  

Rank the reactions from slowest to fastest assuming that they have nearly the same value for the frequency factor A. 

The lower the activation energy, the faster the reaction. The value of AB does not affect the rate. Hence, the order from slowest reaction to fastest is 2 3 1. 

Rank the reverse reactions from slowest to fastest. 

Early in the development of classical chemical kinetics Arrhenius proposed a formula for describing the effect of temperature on a chemical reaction. It may be expressed as follows  

The energy of activation E can be determined readily from the temperature coefficient of the reaction rate either by direct substitution into equation (15) of the value of k at two different temper- 

The differential equation on which this relation is based is  

This calculation is illustrated in Fig. 5 using data on the decomposition of nitrogen pentoxide (page 64). The data of Table II are obtained by interpolation and extrapolation with this formula. 

The more rigorous derivation of equation (16) by statistical mechanics is given by Tolman.4 The energy of activation E is defined as the average energy 

The dependence of the rate constant of a reaction on temperature can be expressed by the Arrhenius equation, 

a plot of In k versus HT gives a straight line whose slope is equal to -EJR and whose intercept b) is equal to In A. 

Sample Problem 14.8 demonstrates a graphical method for determining the activation energy of a reaction. 

Strategy Plot In k versus HT, and determine the slope of the resulting line. According to Equation 14.10, slope = —EJR. 

Setup R = 8.314 J/K mol. Taking the natural log of each value of k and the inverse of each value 

A large portion of the field of chemical kinetics can be described by, or discussed in terms of, Eq. (5-1), the Arrhenius equation. 

The Arrhenius equation relates the rate constant k of an elementary reaction to the absolute temperature T R is the gas constant. The parameter is the activation energy, with dimensions of energy per mole, and A is the preexponential factor, which has the units of k. If A is a first-order rate constant, A has the units seconds, so it is sometimes called the frequency factor. 

The Arrhenius equation is best viewed as an empirical relationship that describes kinetic data very well. It is commonly applied in the linearized form 

Van t Hoff s Contribution A close examination of the history of the appearance and subsequent development of the Arrhenius equation [9], which forms the basis for chemical kinetics, suggests that its formulation was initiated to a large extent by the fundamental treatise of van t Hoff [10] published in 1884. Van t Hoff showed that the equilibrium constant, K, of a reaction is related to temperature, T, and heat of the reaction, AHt, through the relationship that is known as the van t Hoff equation 

Prom here, it follows, as pointed out by van t Hoff, that because the equilibrium constant K is the ratio of the rate constants ki and k i in the direct and reverse directions, these constants should obey a similar equation, i.e., 

Van t Hoff was thus the first to formulate and substantiate thermodynamically the exponential dependence of the reaction rate on temperature and Eqs. (3.2) and (3.3) could, justifiably, have been called the van t Hoff equations. 

Validating the Arrhenius Equation This equation did not receive immediate recognition. In 1899, Bodenstein [13] published a series of papers bearing a common title Gasreaktionen in der chemischen Kinetik . In a comprehensive study of the decomposition and formation reactions of HI, H2S, H2Se, and H2O conducted at different temperatures, he showed convincingly that they obeyed the van t Hoff-Arrhenius theory. 

As for the solid-state reactions, the Arrhenius concept on active particles was first used apparently only in 1921 in a paper by Hinshelwood and Bowen 

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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. 

Although the Arrhenius equation does not predict rate constants without parameters obtained from another source, it does predict the temperature 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... 

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) ... 

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... 

Various Langmiiir-Hinshelwood mechanisms were assumed. GO and GO2 were assumed to adsorb on one kind of active site, si, and H2 and H2O on another kind, s2. The H2 adsorbed with dissociation and all participants were assumed to be in adsorptive equilibrium. Some 48 possible controlling mechanisms were examined, each with 7 empirical constants. Variance analysis of the experimental data reduced the number to three possibilities. The rate equations of the three reactions are stated for the mechanisms finally adopted, with the constants correlated by the Arrhenius equation. 

As predicted by the Arrhenius equation (Sec. 4), a plot of microbial death rate versus the reciprocal or the temperature is usually linear with a slope that is a measure of the susceptibility of microorganisms to heat. Correlations other than the Arrhenius equation are used, particularly in the food processing industry. A common temperature relationship of the thermal resistance is decimal reduction time (DRT), defined as the time required to reduce the microbial population by one-tenth. Over short temperature internals (e.g., 5.5°C) DRT is useful, but extrapolation over a wide temperature internal gives serious errors. 

Later, in the 1890s, Arrhenius moved to quite different concerns, but it is intriguing that materials scientists today do not think of him in terms of the concept of ions (which are so familiar that few are concerned about who first thought up the concept), but rather venerate him for the Arrhenius equation for the rate of a chemical reaction (Arrhenius 1889), with its universally familiar exponential temperature dependence. That equation was in fact first proposed by van t HofT, but Arrhenius claimed that van t Hoff s derivation was not watertight and so it is now called after Arrhenius rather than van t Hoff" (who was in any case an almost pathologically modest and retiring man). 

The temperature dependence of reactions can also be expressed in terms of the Arrhenius equation ... 

Comparing the form of Eq. (4.9) with Eq. (4.5) indicates that 4 in the Arrhenius equation corresponds to (KkT/h)e l. The Arrhenius equation shows that a plot of In it, versus 1 /T will have the slope —EJR. For reactions in solution at a constant pressure, A/f and... 

We ean evaluate the effeet of temperature on the reaetion rate from the Arrhenius equation, k = k e , as ... 

The aetivation energy that is expeeted aeeording to a relevant ehain reaetion meehanism ean be determined if eaeh elementary rate eon-stant is expressed with a temperature dependenee aeeording to the Arrhenius equation,... 

The reaetion rate usually rises exponentially with temperature as shown in Figure 3-1. The Arrhenius equation as expressed in Chapter 1 is a good approximation to die temperature dependeney. The temperature dependent term fits if plotted as In (rates) versus 1/T at fixed eoneentration C, Cg (Figure 3-2). 

The aetivation energy E is a measure of the temperature sensitivity of the rate eonstant. A high E eorresponds to a rate eonstant that inereases rapidly with temperature. Erom the Arrhenius equation... 

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