From the Boltzmann Entropy Equation to Equilibrium Constant: A Microscopic Approach to Equilibrium

The equilibrium constant and Gibbs energy are essential ideas for understanding and controlling chemical processes. However, a major pedagogical disjoint exists at the undergraduate level in connecting these concepts to the Boltzmann entropy equation, S=kB ln⁡(Ω), the fundamental equation of statistical thermodynamics. This gap can lead to many students and practitioners misunderstanding entropy and the connections between equilibrium, entropy, and Gibbs energy. To address these issues, we propose an approach to derive the equilibrium constant from the Boltzmann entropy equation. Gibbs energy and the Boltzmann distribution also naturally emerge within this framework, eliminating the need for an ad hoc introduction. We emphasize the conceptual and pedagogical advantages of this approach and provide examples to demonstrate its applicability and generality for solving many equilibrium problems. These ideas are tailored toward advanced undergraduate students.