The temperature dependence of reaction rates

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. 

The temperature dependence of reaction rates permits evaluation of the enthalpy and entropy components of the free energy of activation. The terms in Eq. (4.4) corresponding to can be expressed as... 

Among other contributions of Arrhenius, the most important were probably in chemical kinetics (Chapter 11). In 1889 he derived the relation for the temperature dependence of reaction rate. In quite a different area in 1896 Arrhenius published an article, "On the Influence of Carbon Dioxide in the Air on the Temperature of the Ground." He presented the basic idea of the greenhouse effect, discussed in Chapter 17. 

Although the mean relative speed of the molecules increases with temperature, and the collision frequency therefore increases as well, Eq. 16 shows that the mean relative speed increases only as the square root of the temperature. This dependence is far too weak to account for observation. If we used Eq. 16 to predict the temperature dependence of reaction rates, we would conclude that an increase in temperature of 10°C at about room temperature (from 273 K to 283 K) increases the collision frequency by a factor of only 1.02, whereas experiments show that many reaction rates double over that range. Another factor must be affecting the rate. 

Section 5.1 shows how nonlinear regression analysis is used to model the temperature dependence of reaction rate constants. The functional form of the reaction rate was assumed e.g., St = kab for an irreversible, second-order reaction. The rate constant k was measured at several temperatures and was fit to an Arrhenius form, k = ko exp —Tact/T). This section expands the use of nonlinear regression to fit the compositional and temperature dependence of reaction rates. The general reaction is... 

Since data are almost invariably taken under isothermal conditions to eliminate the temperature dependence of reaction rate constants, one is primarily concerned with determining the concentration dependence of the rate expression [0(Ct)] and the rate constant at the temperature in question. We will now consider two differential methods that can be used in data analysis. 

These equations must be solved simultaneously using a knowledge of the temperature dependence of reaction rate expression. 

Chemists exploit the temperature dependence of reaction rates by carrying out chemical reactions at elevated temperatures to speed them up. In organic chemistry, especially, reactions are commonly performed under reflux that is, while boiling the reactants. To prevent reactants and products from escaping as gases, a water-cooled condenser tube is fitted to the reaction vessel. The tube condenses the vapours to liquids and returns them to the reaction vessel. Figure 6.15 shows an experiment performed under reflux. 

These complications show wli we emphasize simple and qualitative problems in this course. In reactor engineering the third decimal place is almost always meaningless, and even the second decimal place is fiequently suspect. Our answers may be in error by several orders of magnitude through no fault of our own, as in our example of the temperature dependence of reaction rates. We must be suspicious of our calculations and make estimates with several approximations to place bounds on what may happen. Whenever a chemical process goes badly wrong, we are blamed. This is why chemical reaction engineers must be clever people. The chemical reactor is the least understood and the most complex unif of any chemical process, and its operation usually dominates the overall operation and controls the economics of most chemical processes. 

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 ). 

The other problem we will discuss is the most likely distribution of a fixed amount of energy between a large number of molecules (the Boltzmann distribution). This distribution leads directly to the ideal gas law, predicts the temperature dependence of reaction rates, and ultimately provides the connection between molecular structure and thermodynamics. In fact, the Boltzmann distribution will appear again in every later chapter of this book. 

Temperature effects on the rates of chemical reactions cannot be calculated in lerms of the temperature dependence of reaction rate coefficients as easily as demonstrated in Eq. 1.42 because of the model-specific nature of overall i( action rale laws (Section 1.3). One empirical approach that is in widespread... 

The hydrogen-iodine reaction is a classic in chemical kinetics. The work of Bodenstein on this reaction is one of the first systematic studies of the temperature dependence of reaction rates. For many years the formation of HI from H2 and I2 was regarded as the textbook example of a bimolecular four-center reaction as was its reverse. Recently, however, experimental results inconsistent with this interpretation have been obtained. ... 

Because of experimental limitations, quantitative studies of the temperature dependence of reaction rates have not been possible. Riley and coworkers, using a continuous-flow reactor, have obtained qualitative temperature-dependent data. In this area significant improvement in experimental capabilities is needed. 

Arrhenius proposed a widely used and fundamental equation for the temperature dependence of reaction rates in the year 1889, which was based entirely on experimental measurements ... 

Both the Arrhenius and the Eyring equation describe the temperature dependence of reaction rate. Strictly speaking, the Arrhenius equation can be applied only to gas reactions. The Eyring equation is used in the smdy of gas, condensed and mixed phase reactions - aU places where the simple collision model is not very helpful. The Arrhenius equation is founded on the empirical observation that conducting a reaction at a higher temperature increases the reaction rate. The Eyring equation is a theoretical construct, based on transition state model. 

On a modest level of detail, kinetic studies aim at determining overall phenomenological rate laws. These may serve to discriminate between different mechanistic models. However, to it prove a compound reaction mechanism, it is necessary to determine the rate constant of each elementary step individually. Many kinetic experiments are devoted to the investigations of the temperature dependence of reaction rates. In addition to the obvious practical aspects, the temperature dependence of rate constants is also of great theoretical importance. Many statistical theories of chemical reactions are based on thermal equilibrium assumptions. Non-equilibrium effects are not only important for theories going beyond the classical transition-state picture. Eventually they might even be exploited to control chemical reactions [24]. This has led to the increased importance of energy or even quantum-state-resolved kinetic studies, which can be directly compared with detailed quantum-mechanical models of chemical reaction dynamics [25,26]. 

In general reaction rates increase with increasing temperature. Kooij and van t Hoff (1893) proposed an equation for the temperature dependence of reaction rates... 

Ever since the rates of chemical reactions were first systematically measured, chemical kineticists have sought to measure, characterise, and understand the effect of temperature on these rates. One of the earliest and most celebrated efforts in this direction was the proposal by Svante Arrhenius [1] that the temperature-dependence of reaction rate constants can be expressed in terms of what we now know as the Arrhenius equation ... 

The temperature dependence of reaction rates is the reason that cold-blooded animals become more sluggish at lower temperatures. The reactions required for them to think and move become slower, resulting in tiie sluggish behavior. 

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