Teaching Entropy from Phase Space Perspective: Connecting the Statistical and Thermodynamic Views Using a Simple One-Dimensional Model

Abstract

Connecting the thermodynamic definition of entropy, dS = dQ/T (Clausius’s equation), with the statistical definition, S = kB ln Ω (Boltzmann’s equation), has been a persistent challenge in chemical education at the undergraduate level. Not meeting this challenge results in students taking away the meaning of entropy in a vague and subjective way as a measure of “disorder” or increase in number of configurations without any meaningful way of connecting it to heat. To address this challenge, we present a simple model that connects these two definitions. This approach relies centrally on emphasizing that the number of configurations, Ω, includes configurations in both real space and momentum space, collectively known as the phase space. Without including momentum configurations (i.e., how fast the particles move), connecting heat to entropy change is not possible. We construct the phase space for an ensemble of simple one-dimensional systems at equilibrium and show that delivery of heat dQ to the system results in an increase in the number of momentum configurations and consequently an expansion of the phase space area by dΩ. Relating dQ to dΩ is the linchpin between the two views and, when integrated, leads to Boltzmann’s equation. We further show that understanding entropy in terms of volume of phase space removes common ambiguities in teaching the subject. Among other examples, we show that understanding adiabatic compression, if treated using the usual approach, results in contradictions that are resolved if a phase space view is adopted. We propose this approach at the undergraduate physical chemistry and physics level.