The rate constant normally depends on the absolute temperature, and the functional form of this relationship was first proposed by Arrhenius in 1889 (see Rule III in Chapter 1) to be:
where the activation energy, E, and the pre-exponential factor, A, both do not depend on the absolute temperature. The Arrhenius form of the reaction rate constant is an empirical relationship. However, transition-state theory provides a justification for the Arrhenius formulation, as will be shown below. Note that the Arrhenius law (Equation 2.2.1) gives a linear relationship between In k and T~ 1
It is easy to see from Equation (2.2.5) that if (E2 – E1) > 0 then the reaction is exothermic and likewise if (E2 – E1) < 0 it is endothermic (refer to Appendix A for temperature dependence ofKc). In a typical situation, the highest yields of products are desired. That is, the ratio (CSCW/CACB)eq will be as large as possible. If the reaction is endothermic, Equation (2.2.5) suggests that in order to maximize the product yield, the reaction should be accomplished at the highest possible temperature. To do so, it is necessary to make exp[(E2 – E1)/(RgT) ] as large as possible by maximizing (RgT), since (E2 – E1) is negative. Note that as the temperature increases, so do both the rates (forward and reverse). Thus, for endothermic reactions, the rate and yield must both increase with temperature. For exothermic reactions, there is always a trade-off between the equilibrium yield of products and the reaction rate. Therefore, a balance between rate and yield is used and the T chosen is dependent upon the situation.