Study of Two Prototype Physical Models from Statistical Route: an Ideal Gas and a Collection of N Three Dimensional Classical Harmonic

P.K. Ahluwalia

1. Learning Outcomes

After studying this module, you shall be able to

• Understand the physical importance of the prototype model of statistical physics, studied in this module, called monoatomic ideal gas model in the development of equilibrium statistical mechanics.
• Apply the statistical approach to get thermodynamic properties of this prototype model in an exploratory manner without explicit calculation of the number of microstates and further derive equation of state, pressure as energy density and equation of state of an irreversible adiabatic process.
• Evaluate explicitily (a possibility only in ideal cases), the number of microstates for an ideal gas consisting of N particles in the asymptotic limit and get an expression for entropy.
• See the violation of extensive nature of entropy desired by thermodynamics derived for the ideal gas.
• Derive expression of entropy asymptotically for system of quasi particles resulting from the case of a collection of distinguishable harmonic oscillators of same frequency.and ponder on the fact why here entropy turns out to be extensive in nature unlike the case of the monoatomic ideal gas.

2. Introduction

In this module we embark upon calculation of number of microstates of an ideal monoatomic gas enclosed in a volume V, having total Energy E and number of gas particles N, so that entropy of the gas may be calculated from the statistical route. N is an enormously large number typically of the order of Avogadro’s number, making it a fit system to apply statistical methods to understand its behavior.
This gas is ideal in the sense that there is negligible interaction among the particles of the gas. In other words, particles of the gas are free to move such that compared to their kinetic energy mutual interaction potential energy between the particles can be neglected. To say that there is negligible interaction is as good as saying no interaction and is, therefore, an idealization. In real gases there is always an interaction.
There is however a dilemma, if there is no interaction the speed of the atoms of the each gas atom shall be conserved. Therefore, in this case system can not go through all possible microstates violating equal apriori probability axiom studied in module 7. So allowing a weak interaction is desirable, howsoever small it may be. One may prefer to describe such a gas as an ideal gas or a real gas in a dilute limit.

3. Classical Monoatomic Ideal Gas and Derivation of Entropy from Statistical Route

4.Thermodynamic properties

6. Entropy of N Three Dimensional Classical Harmonic Oscillators

9. Summary

In this module we have learnt

• The application of statistical physics to a monoatomic ideal gas and discussed the thermodynamic properties of the ideal gas derived from the calculated entropy using the methodology of counting the microstates in asymptotical limit.

References:-

1. Pathria R.K. and Beale P. D., Statistical Mechanics, 3rd ed. (Elsevier, 2011).
2. Landau L.D., Lifshitz E.M., “ Statistical Physics Part 1,” 3rd Edition, Oxford: Pergamon Press., 1982
3. Pal P.B., “An Introductory Course of Statistical Mechanics”, New Delhi: Narosa Publishing House Pvt. Ltd., 2008.
4. Panat P.V., “Thermodynamics and Statistical Mechanics,” New Delhi: Narosa Publishing House Pvt. Ltd., 2008
5. Yoshioka D., “Statistical Physics An Introduction,” Berlin Heidelberg: Springer-Verlag, 2007

Appendices:-