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

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Presentation on theme: "Classical and Quantum Gases n Fundamental Ideas –Density of States –Internal Energy –Fermi-Dirac and Bose-Einstein Statistics –Chemical potential –Quantum."— Presentation transcript:

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

2 Density of States n 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: –g s = number of polarisations n 2 for protons, neutrons, electrons and photons

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

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

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

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

7 F-D vs. B-E Statistics

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

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

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

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

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

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

14 Ideal Classical Gases n Interpretation of n Q (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 << n Q, the average separation is greater than and the gas is classical rather than quantum

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

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

17 Quantum Gases n Equations of State: –(See Physics of Stars sec n 2.2) –Non-relativistic: –Ultra-relativistic:

18 Quantum Gases n 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

19 Reminder n Assignment 1 available today on unit website

20 Next Lecture n The Saha Equation –Derivation –Consequences for ionisation and absorption

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


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