Chemical Engineering Seminar Series: Prof. Ananth Govind Rajan

Microcanonical Statistical Mechanics of a Non-Ideal Fluid: A Pedagogical Approach to Obtain the Combined First and Second Law of Thermodynamics

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, especially at the graduate level. Given this shift, in this lecture, I will elucidate the statistical-mechanical roots of the combined first and second law of thermodynamics. To accomplish this, I will employ the microcanonical ensemble, which is typically considered to be intractable, and show that it is possible to use it as a pedagogical tool. Accordingly, I will start with Boltzmann’s entropy formula and use differential calculus to establish that dE=TdS – PdV, for an isolated, non-ideal 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, I will develop the partition function for an isolated, monoatomic fluid. Further, I will derive the average of the inverse kinetic energy, which appears in the formulation of the combined first and second law, and show that it is equal to the inverse of the average kinetic energy. Subsequently, I will obtain an expression for the system pressure involving many-body interactions and introduce the system pressure in the combined first and second law via Clausius’s virial theorem. I will discuss how intermolecular interactions affect the entropy and pressure of a thermodynamic system. Overall, the lecture will inform the derivation of the fundamental thermodynamic relationship dE=TdS-PdV using an isolated (microcanonical) system, thereby providing deeper insight into equilibrium statistical thermodynamics. The material presented here may be incorporated into graduate-level courses on statistical mechanics and/or molecular simulations in various disciplines, including chemistry and physics, as well as chemical, materials, and mechanical engineering. As part of the lecture, responses to questions will be collected before and after the lecture to assess the pedagogical utility of the material.