2.11: Thermodynamic Probability and Equilibrium in an Isomerization Reaction
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To relate these ideas to a change in a more specific macroscopic system, let us consider isomeric substances \(A\) and \(B\). (We consider this example further in Chapter 21.) In principle, we can solve the Schrödinger equation for a molecule of isomer \(A\) and for a molecule of isomer \(B\). We obtain all possible energy levels for a molecule of each isomer. \({}^{1}\) If we list these energy levels in order, beginning with the lowest, some of these levels belong to isomer \(A\) and the others belong to isomer \(B\).
Now let us consider a mixture of \(N_A\) molecules of \(A\) and \(N_B\) molecules of \(B\). We suppose that individual molecules are distinguishable and that intermolecular interactions can be ignored. Since a group of atoms that can form an \(A\) molecule can also form a \(B\) molecule, every energy level is accessible to this group of atoms; that is, we can view both sets of energy levels as being available to the atoms that make up the molecules. For a given system energy, there will be many population sets in which only the energy levels belonging to isomer \(A\) are occupied. For each of these population sets, there is a corresponding thermodynamic probability, \(W\). Let \(W^{max}_A\) be the largest of these thermodynamic probabilities. Similarly, there will be many population sets in which only the energy levels corresponding to isomer \(B\) are occupied. Let \(W^{max}_B\) be the largest of the thermodynamic probabilities associated with these population sets. Finally, there will be many population sets in which the occupied energy levels belong to both isomer \(A\) and isomer \(B\). Let \(W^{max}_{A,B}\) be the largest of the thermodynamic probabilities associated with this group of population sets.
Now, \(W^{max}_A\) is a good approximation to the number of ways that the atoms of the system can come together to form isomer \(A\). \(W^{max}_B\) is a good approximation to the the number of ways that the atoms of the system can come together to form isomer \(B\). At equilibrium, therefore, we expect
\[K=\frac{N_B}{N_A}=\frac{W^{max}_B}{W^{max}_A} \nonumber \]
If we consider the illustrative—if somewhat unrealistic—case of isomeric molecules whose energy levels all have the same degeneracy (\(g_i=g\) for all \(i\)), we can readily see that the equilibrium system must contain some amount of each isomer. For a system containing \(N\) molecules, \(N!g^N\) is the numerator in each of the thermodynamic probabilities \(W^{max}_A\), \(W^{max}_B\), and \(W^{max}_{A,B}\). The denominators are different. The denominator of \(W^{max}_{A,B}\) must contain terms, \(N_i!\), for essentially all of the levels represented in the denominator of \(W^{max}_A\). Likewise, it must contain terms, \(N_j!\), for essentially all of the energy levels represented in the denominator of \(W^{max}_B\). Then the denominator of \(W^{max}_{A,B}\) is a product of \(N_k!\) terms that are generally smaller than the corresponding factorial terms in the denominators of \(W^{max}_A\) and \(W^{max}_B\). As a result, the denominators of \(W^{max}_A\) and \(W^{max}_B\) are larger than the denominator of \(W^{max}_{A,B}\). In consequence, \(W^{max}_{A,B}>W^{max}_A\) and \(W^{max}_{A,B}>W^{max}_B\). (See problems 5 and 6.)
If we create the system as a collection of \(A\) molecules, or as a collection of \(B\) molecules, redistribution of the sets of atoms among all of the available energy levels must eventually produce a mixture of \(A\) molecules and \(B\) molecules. Viewed as a consequence of the principle of equal a priori probabilities, this occurs because there are necessarily more microstates of the same energy available to some mixture of \(A\) and \(B\) molecules than there are microstates available to either \(A\) molecules alone or \(B\) molecules alone. Viewed as a consequence of the tendency of the isolated system to attain the state of maximum entropy, this occurs because \(k{ \ln W^{max}_{A,B}>k{ \ln W^{max}_A\ }\ }\) and \(k{ \ln W^{max}_{A,B}>k{ \ln W^{max}_B\ }\ }\).