# 2.11: Thermodynamic Probability and Equilibrium in an Isomerization Reaction

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

$K=\frac{N_B}{N_A}=\frac{W^{max}_B}{W^{max}_A}$

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 $$k{ \ln W^{max}_{A,B}>k{ \ln W^{max}_B\ }\ }$$.