20.1: The Independent-Molecule Approximation Because the energies that characterize the interactions of atoms within a molecule are much greater than the energies that characterize the interaction of one molecule with another, the energy of the system can be viewed as the sum of two terms. One term is a sum of the energies that the component molecules would have if they were all infinitely far apart. The other term is a sum of the energies of all of the intermolecular interactions. 20.2: The Probability of An Energy Level at Constant N, V, and T 20.3: The Population Sets of a System at Equilibrium at Constant N, V, and T In developing Boltzmann statistics, we assume that we can tell different molecules of the same substance apart. We say that the molecules are distinguishable. This assumption is valid for molecules that occupy lattice sites in a crystal. In a crystal, we can specify a particular molecule by specifying its position in the lattice. In other systems, we may be unable to distinguish between different molecules of the same substance. 20.4: How can Infinitely Many Probabilities Sum to Unity? 20.5: The Total Probability Sum at Constant N, V, and T 20.6: The Most Probable Population Set at Constant N, V, and T 20.7: The Microstates of a Given Population Set Thus far, we have considered only the probabilities associated with the assignments of distinguishable molecules to the allowed energy levels. 20.8: The Probabilities of Microstates that Have the Same Energy Our development of statistical thermodynamics relies on the principle of equal a priori probabilities. The equal-probability idea is useful only if it leads us to theoretical models that successfully mirror the behavior of real macroscopic systems. This it does. Accordingly, we recognize that the equal-probability idea is really a fundamental postulate about the behavior of quantum-mechanical systems. 20.9: The Probabilities of the Population Sets of an Isolated System In principle, the energy of an equilibrium system that is in contact with a constant-temperature heat reservoir can vary slightly with time. In contrast, the energy of an isolated system is constant. A more traditional and less general statement of the equal a priori probability principle focuses on isolated systems, for which all possible microstates necessarily have the same energy: All microstates of an isolated (constant energy) system occur with equal probability. 20.10: Entropy and Equilibrium in an Isolated System If an isolated system can undergo change, and we re-examine it at after a few molecules have moved to different energy levels, we expect to find it in one of the microstates of a more-probable population set. Evidently, the largest- W population set characterizes the equilibrium state. Either the system can undergo change until W reaches a maximum. Thereafter, it is at equilibrium and can undergo no further macroscopically observable change. 20.11: 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. 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. 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. 20.12: The Degeneracy of an Isolated System and Its Entropy The sum of the thermodynamic probabilities over all allowed population sets is just the number of microstates that have energy E . This sum is just the degeneracy of the system energy, E . 20.13: The Degeneracy of an Isolated System and its Entropy 20.14: Effective Equivalence of the Isothermal and Constant-energy Conditions In principle, an isolated system is different from a system with identical macroscopic properties that is in equilibrium with its surroundings. We emphasize this point, because this distinction is important in the logic of our development. However, our development also depends on the assumption that, when N is a number that approximates the number molecules in a macroscopic system, the constant-temperature and constant-energy systems are functionally equivalent. 20.15: Problems