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Chapter 13 (III)

# Chapter 13 (III) - 13.4 Dilute gases& Maxwell-Boltzmann...

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Unformatted text preview: 13.4 Dilute gases & Maxwell-Boltzmann Distribution , 1 << j j g N , 1 1 1 / ) ( << ± = = − kT j j j j e g N f μ ε For a dilute gas, the occupation number at each energy level is much smaller than the available number of quantum states, i.e., for each j. (13.27) This gives in Fermi-Dirac or Bose-Einstein distribution. It follows that (13.28) . / ) ( kT j j j j e g N f μ ε − − ≈ = Since (13.29) Thus, Eq(13.28) can be rewritten as (13.30) It follows that for dilute gases, both Fermi-Dirac and Bose Einstein statistics reduce to the Boltzmann distribution. This classical limit is called Maxwell-Boltzmann distribution f MB . , 1 N N n i i = ∑ = . , / 1 / / ) ( 1 Z N e g N e N e g kT n i j kT kT n i j j j = = ⇒ = − = − − = ∑ ∑ ε μ μ ε . / B kT j j j f e Z N g N f j = ≈ = − ε In fact, there is another approach to get Eq(13.30). For dilute gases (N j <<g j ), we can have So that Similarly, we have . ) 1 )...( 2 )( 1 ( )! ( )! )( 1 )...( 2 )( 1 ( )! ( ! j N j j j j j j j j j j j j j j j j j j g N g g g g N g N g N g g g g N g g ≈ + − − − = − − + − − − = − ....
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Chapter 13 (III) - 13.4 Dilute gases& Maxwell-Boltzmann...

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