Chapter 40: Principles of Statistical Mechanics

Chapter 40 delves into the foundational concepts of classical statistical mechanics, successfully linking large-scale thermal phenomena with the average positions and movements of vast numbers of interacting molecules. A central assertion of the theory is that the mean kinetic energy associated with every independent direction of motion (degree of freedom) is directly proportional to the absolute temperature multiplied by the Boltzmann constant. The discussion begins with the exponential atmosphere, demonstrating how the concentration of gas molecules decreases in a uniform gravitational field as the altitude increases, following an exponential decay. This idea is generalized into the Boltzmann law, a crucial principle stating that the likelihood or concentration of finding molecules in a spatial region is exponentially related to the negative of the potential energy divided by the thermal energy. Applying this framework, the chapter examines the challenging physics of evaporation and liquids, showing that the spatial arrangement of attracting molecules is also governed by an exponential function involving their interaction potential energies. The sources note that higher temperatures reduce the influence of these potential interactions, making them less significant. Furthermore, the text rigorously derives the distribution of molecular speeds, establishing that the probability of a molecule possessing a particular velocity is proportional to an exponential factor involving the molecule's kinetic energy and the thermal energy. The final and historically significant section addresses the failure of classical physics to correctly predict the specific heats of gases; classical models failed dramatically when tested against experimental data for gases like hydrogen at low temperatures, where certain internal molecular motions appeared to stop contributing to the specific heat. This profound experimental contradiction necessitated the incorporation of quantum mechanics into statistical mechanics. In the quantum view, where energy exists only in discrete, quantized levels, the relative probability of molecules residing in a higher energy state compared to a lower state is determined by an exponential factor related to the difference in energy between the states divided by the thermal energy. This quantum modification successfully explains the phenomenon of motions "freezing out"—such as rotational and vibrational states—at low temperatures.