Gas Kinetic Theory of Entropy

Abstract

This chapter will attempt to provide a physical understanding of the concept of entropy based on the kinetic theory of gases. Entropy in classical thermodynamics is a mathematical concept that is derived from a closed cycle on a reversible Carnot heat engine. For many students it lacks physical meaning. Most students have a physical understanding of variables like volume, temperature, and pressure. Internal energy and enthalpy are easy to understand, if not intuitive. However, entropy is a bit more difficult. The discussion that follows is an attempt to provide physical insight into the concept of entropy at the introductory level. This discussion closely follows the excellent text “Elements of Statistical Thermodynamics” by L. K. Nash, Dover 2006 [1–4].

Problems

Problem 9.1: Calculate the number of macrostates and their frequency that will be observed in a solid with distinguishable atoms if 3 quanta are distributed among six atoms.

Problem 9.2: Consider a micro-dot of silicon that can be represented as a hemisphere 1 micron in diameter. What is the probability of observing a deviation from the most likely macrostate of one part in a trillion (10⁻¹²)? The density of silicon is 2.33 gm/cc and its atomic weight is 28.

Problem 9.3: What is the ratio of microstates contributing to the most likely macrostate for iron at 300 K compared to the number of microstates contributing to the most likely macrostate at 290 K? Use a constant β at 295 K.

Problem 9.4: Estimate the level of degeneracy for a hydrogen molecule moving in a 1 l box with an average energy of 0.0253 eV. The mass of a hydrogen molecule is 2 amu.

Problem 9.5: Calculate the average velocity, rms velocity, and most probable velocity for a carbon dioxide molecule in a container at 3 MPa and 500 K.