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Thermodynamics and Kinetics in Materials ScienceA Short Course$
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Boris S. Bokstein, Mikhail I. Mendelev, and David J. Srolovitz

Print publication date: 2005

Print ISBN-13: 9780198528036

Published to Oxford Scholarship Online:

DOI: 10.1093/oso/9780198528036.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 20 June 2021

Introduction to statistical thermodynamics of gases

Introduction to statistical thermodynamics of gases

(p.192) 12 Introduction to statistical thermodynamics of gases
Thermodynamics and Kinetics in Materials Science

Boris S. Bokstein

Mikhail I. Mendelev

David J. Srolovitz

Oxford University Press

As we discussed earlier in this book, thermodynamics provides very general relations between the properties of a system. On the other hand, thermodynamics is unable to predict any of the individual properties without the addition of either empirical or microscopic information. For example, we used thermodynamics to obtain Raoult’s law from Henry’s law, but we cannot derive Henry’s law from thermodynamic principles. Statistical mechanics provides an approach to determine individual thermodynamic properties from microscopic considerations. When applied in the realm of physical chemistry, we refer to this approach as statistical thermodynamics. In this chapter, we provide a simplified derivation of the Gibbs distribution, which is the basis of much of statistical thermodynamics. We then use statistical mechanics to show how the properties of an ideal gas can be obtained from a small number of properties of the molecules in the gas. This will allow us to determine such quantities as the equilibrium and rate constants of gas phase chemical reactions. As a result, we will gain new insight into the phenomena which we have already considered on the basis of phenomenological thermodynamics or formal kinetics. This approach will also show how to determine some of the parameters we previously introduced as input data in our thermodynamic considerations. As we have already seen, a finite system will eventually come into equilibrium with its surroundings. We even showed that when thermodynamic equilibrium is established, the temperatures, pressures, and chemical potentials of the system and its surroundings are equal (see Section 1.5.2). However, we never discussed what equilibrium actually is. For example, does this mean that the energy of the system is truly constant or is it only constant on average? When the system has a particular energy, does this mean that it is in a unique physical state or can it be in any one of several states that have exactly the same energy? In the latter case, can we simply talk about the probability the system is in each of these states? If the energy can fluctuate, what is the probability that the system has a particular energy? A very general approach to these types of questions was suggested by Gibbs and is now known as Gibbs statistics.

Keywords:   Born-Oppenheimer approximation, Gibbs statistics, Maxwell distribution, activated complex, collision cross-section, dividing surface, ergodic hypothesis, mean free path, partition function, reaction path

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