A Pedagogical Approach to Obtain the Combined First and Second Law of Thermodynamics from Classical Statistical Mechanics

The combined first and second law of thermodynamics for a closed system is written as dE=TdS - PdV, where E is the internal energy, S is the entropy, V is the volume, T is the temperature, and P is the pressure of the system. This equation forms the basis for understanding physical phenomena ranging from heat engines to chemical reactors to biological systems. In this work, we present a pedagogical approach to obtain the combined first and second law of thermodynamics beginning with the principles of classical statistical mechanics, thereby establishing a fundamental link between energy conservation, heat, work, and entropy. We start with Boltzmann's entropy formula and use differential calculus to establish this link. Some new aspects of this work include the use of the microcanonical ensemble, which is typically considered to be intractable, to write the partition function for a general system of matter; deriving the average of the inverse kinetic energy, which appears in the microcanonical formulation of the combined first and second law, and showing that it is equal to the inverse of the average kinetic energy; obtaining an expression for the pressure of a system involving many-body interactions; and introducing the system pressure in the combined first and second law via Clausius's virial theorem. Overall, this work informs the derivation of fundamental thermodynamic relations from an understanding of classical statistical mechanics. The material presented herein could be incorporated into senior undergraduate/graduate-level courses in statistical thermodynamics and/or molecular simulations.