# Classical and Quantum Gases

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Classical and Quantum Gases
Fundamental Ideas Density of States Internal Energy Fermi-Dirac and Bose-Einstein Statistics Chemical potential Quantum concentration

Density of States Derived by considering the gas particles as wave-like and confined in a certain volume, V. Density of states as a function of momentum, g(p), between p and p + dp: gs = number of polarisations 2 for protons, neutrons, electrons and photons

Internal Energy The energy of a particle with momentum p is given by:
Hence the total energy is: No. of quantum states in p to p +dp Average no. of particles in state with energy Ep

Total Number of Particles
No. of quantum states in p to p +dp Average no. of particles in state with energy Ep

Fermi-Dirac Statistics
For fermions, no more than one particle can occupy a given quantum state Pauli exclusion principle Hence:

Bose-Einstein Statistics
For Bosons, any number of particles can occupy a given quantum state Hence:

F-D vs. B-E Statistics

The Maxwellian Limit Note that Fermi-Dirac and Bose-Einstein statistics coincide for large E/kT and small occupancy Maxwellian limit

Ideal Classical Gases Classical Þ occupancy of any one quantum state is small I.e., Maxwellian Equation of State: Valid for both non- and ultra-relativistic gases

Ideal Classical Gases Recall: Non-relativistic: Ultra-relativistic
Pressure = 2/3 kinetic energy density Hence average KE = 2/3 kT Ultra-relativistic Pressure = 1/3 kinetic energy density Hence average KE = 1/3 kT

Ideal Classical Gases Total number of particles N in a volume V is given by:

Ideal Classical Gases Rearranging, we obtain an expression for m, the chemical potential

Ideal Classical Gases Interpretation of m
From statistical mechanics, the change of energy of a system brought about by a change in the number of particles is:

Ideal Classical Gases Interpretation of nQ (non-relativistic)
Consider the de Broglie Wavelength Hence, since the average separation of particles in a gas of density n is ~n-1/3 If n << nQ , the average separation is greater than l and the gas is classical rather than quantum

Ideal Classical Gases A similar calculation is possible for a gas of ultra-relativistic particles:

Quantum Gases Low concentration/high temperature electron gases behave classically Quantum effects large for high electron concentration/”low” temperature Electrons obey Fermi-Dirac statistics All states occupied up to an energy Ef , the Fermi Energy with a momentum pf Described as a degenerate gas

Quantum Gases Equations of State: (See Physics of Stars secn 2.2)
Non-relativistic: Ultra-relativistic:

Quantum Gases Note: Pressure rises more slowly with density for an ultra-relativistic degenerate gas compared to non-relativistic Consequences for the upper mass of degenerate stellar cores and white dwarfs

Reminder Assignment 1 available today on unit website

Next Lecture The Saha Equation Derivation
Consequences for ionisation and absorption

Next Week Private Study Week - Suggestions Assessment Worksheet
Review Lectures 1-5 Photons in Stars (Phillips ch. 2 secn 2.3) The Photon Gas Radiation Pressure Reactions at High Temperatures (Phillips ch. 2 secn 2.6) Pair Production Photodisintegration of Nuclei