Modern Engineering Thermodynamics

CHAPTER 18
Introduction to Statistical Thermodynamics
CONTENTS
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727
18.2 Why Use a Statistical Approach? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728
18.3 Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728
18.4 Intermolecular Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732
18.5 Molecular Velocity Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
18.6 Equipartition of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738
18.7 Introduction to Mathematical Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741
18.8 Quantum Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747
18.9 Three Classical Quantum Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749
18.10 Maxwell-Boltzmann Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750
18.11 Monatomic Maxwell-Boltzmann Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751
18.12 Diatomic Maxwell-Boltzmann Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
18.13 Polyatomic Maxwell-Boltzmann Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
18.1 INTRODUCTION
In this chapter, we explore some of the basic concepts of statistical thermodynamics that lead to useful engineering
results. In Chapter 2, we discuss the difference between microscopic and macroscopic systems and note that classical
thermodynamics is based on a continuum macroscopic system approach. Recognition of the existence of atoms
and molecules was not necessary for the development of classical thermodynamics, the results of which are valid
for all processes in which the continuum hypothesis holds. Statistical thermodynamics, on the other hand, is
based on the use of standard statistical methods in the analysis of molecular behavior and, therefore, corresponds
to a microscopic system approach.
There are four basic attributes of statistical thermodynamics. First, it can be used to explain certain apparent
discontinuities in physical behavior, such as superconductivity. Second, it can be used to extend classical thermodynamic
results into regions where the continuum hypothesis is no longer valid, as in the case of rarefied
gases. Third, it can often provide a molecular interpretation of physical phenomena that are observed at the
macroscopic level but originate at the molecular level (such as fluid viscosity). Fourth and perhaps most important,
it can function as a tool to provide accurate equations of state that describe the behavior of nonmeasurable
thermodynamic properties, such as internal energy, enthalpy, and entropy, as a function of measurable properties,
such as pressure, temperature, and density, without resorting to experimental measurements. These
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