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Unformatted text preview: M. S. Shell 2010 1/13 last modified 12/3/2010 Reaction pathways and rates ChE210A Kinetics from statistical thermodynamics For most of this course, we have discussed equilibrium properties, for which time is not a relevant variable. Indeed, thermodynamics is not concerned with time. The very notion of equilibrium invokes, in part, the concept of time-independence. The current topic will now take a slight departure from that perspective. Here, we will consider how one might extract rates of some molecular process. In order to extract these kinetic quantities, we will have to make certain assumptions about the equilibrium behavior and molecular trajectories that take us slightly beyond the mere province of thermodynamics and into the realm of kinetics. We will find, however, that thermodynamic quantities like the free energy are strong determinants of kinetic rate coefficients. Consider this classical picture of a chemical reaction that you may have seen in your early chemistry courses: Here, we envision some coordinate that takes us along a reaction pathway, from start to finish. Along this coordinate we track a free energy, which in some simple cases can be just the potential energy. There is a barrier in free energy the system must overcome, g , and that barrier determines how fast the reaction can proceed. The barrier is often called the activation energy. The state of the system when it is at the height of this barrier is called the transition state , or the rate-limiting state . After reacting, if the reaction is favorable, there will be a net lowering in free energy at the product state. free energy reaction pathway g reactant product g rxn M. S. Shell 2010 2/13 last modified 12/3/2010 In this lecture we will understand the origins of such diagrams in terms of microscopic proper- ties. We will find that the emergence of this kind of picture really stems from many simplifica- tions about microscopic interactions, but in many cases, it is able to give a quantitative descrip- tion of the kinetic rates. This kind of analysis is called classical transition state theory. Macroscopic considerations For chemical reactions, reaction rates are typically written as the product of a rate coefficient times a concentration of a reactant raised to its stoichiometric coefficient. Consider the reac- tion g G 2 . The rate of change of the components is usually written in the following form: g g G 2 g G 2 g Here, and are the forward and reverse rate coefficients , and the brackets indicate concentration (such as moles/molecules per volume). Each component changes concentration due to forward and reverse reactions. At equilibrium, the change of concentrations with time is zero. Setting the time derivatives equal to zero, we find: g From the previous lecture we know that this ratio of concentrations must give the equilibrium...
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