Pedagogical Approach to Microcanonical Statistical Mechanics via Consistency with the Combined First and Second Law of Thermodynamics for a Nonideal Fluid

Thermodynamics forms an important part of the science and engineering curriculum at the undergraduate and graduate levels. Over the years, the importance of statistical mechanics and molecular simulations in the curriculum has increased. In this work, we present a pedagogical approach to the microcanonical formulation of statistical mechanics via its consistency with the combined first and second law of thermodynamics. We start with Boltzmann's entropy formula and use differential calculus to establish that dE = TdS - PdV for an isolated, nonideal fluid in an arbitrary number of dimensions, with a constant number of particles (N), volume (V), and energy (E) and with temperature T, pressure P, and entropy S. To this end, we write the partition function for an isolated monatomic fluid. Furthermore, we derive the average of the inverse kinetic energy, which appears in the microcanonical ensemble, and show that it is equal to the inverse of the average kinetic energy, thus introducing the system's temperature. Subsequently, we obtain an expression for the pressure of a system involving many-body interactions and introduce it in the combined first and second law via Clausius's virial theorem. Overall, we show that the statistical mechanics of an isolated (microcanonical) nonideal fluid is consistent with the fundamental thermodynamic relationship dE = TdS - PdV, thereby providing deeper insight into equilibrium statistical thermodynamics. We also demonstrate that this material resulted in favorable learning outcomes when taught as a 1.5 h lecture; therefore, it may be incorporated into graduate-level courses on statistical mechanics and/or molecular simulations.