6. The Theory of Simple Gases

6. The Theory of Simple Gases An Ideal Gas in a Quantum Mechanical Microcanonical Ensemble An Ideal Gas in Other Quantum Mechanical Ensembles Statistics of the Occupation Numbers Kinetic Considerations Gaseous Systems Composed of Molecules with Internal Motion Chemical Equilibrium

6.1. An Ideal Gas in a Quantum Mechanical Microcanonical Ensemble N non-interacting, indistinguishable particles in V with E. ( N, V, E ) = # of distinct microstates Let  be the average energy of a group of g >> 1 unresolved levels. Let n be the # of particles in level .  Let W { n } = # of distinct microstates associated with a given set of { n }.  Let w(n) = # of distinct microstates associated with level  when it contains n particles. 

Bosons ( Bose-Einstein statistics) : See § 3.8 Fermions ( Fermi-Dirac statistics ) : w(n ) = distinct ways to divide g levels into 2 groups; n of them with 1 particle, and g  n with none.

Classical particles ( Maxwell-Boltzmann statistics ) : w(n ) = distinct ways to put n distinguishable particles into g levels. Gibbs corrected

Method of most probable value ( also see Prob 3.4 ) Lagrange multipliers n* extremize

  BE FD  Most probable occupation per level MB

BE FD   MB: 

6.2. An Ideal Gas in Other Quantum Mechanical Ensembles Canonical ensemble : Ideal gas, = 1-p’cle energy : = statistical weight factor for { n}. Actual g absorbed in  ( here is treated as non-degenerate: g = 1).

Maxwell-Boltzmann :  multinomial theorem

partition function (MB) grand partition function (MB)

Bose-Einstein / Fermi-Dirac : Difficult to evaluate (constraint on N ) 

B.E. F.D. BE FD Grand potential : q potential :

BE FD  MB : c.f. §4.4 Alternatively 

Mean Occupation Number For free particles : BE FD   see §6.1

6.3. Statistics of the Occupation Numbers BE FD Mean occupation number : BE : B.E. condensation FD :  MB : Classical : high T   must be negative & large From §4.4 :  same as §5.5

Statistical Fluctuations of n BE FD  

BE FD above normal below normal Einstein on black-body radiation : +1 ~ wave character  n 1 ~ particle character see Kittel, “Thermal Phys.” Statistical correlations in photon beams : see refs on pp.151-2

Probability Distributions of n Let p(n)= probability of having n particles in a state of energy  .  BE FD 

BE FD  BE :   FD : 

MB : Gibbs’ correction  Alternatively   Poisson distribution  “normal” behavior of un-correlated events  prob of occupying state 

“normal” behavior of un-correlated events BE : Geometric ( indep of n ) > MB for large n : Positive correlation  FD :  < MB for large n : Negative correlation

n - Representation Let n= number of particles in 1-particle state  . State of system in the n- representation : Non-interacting particles :

Mean Occupation Number Let F be an operator of the form e.g.,

6.4. Kinetic Considerations BE FD From § 6.1 Free particles :

BE FD Let p( ) be theprobability of a particle in state  . Then     s = 1 : phonons s = 2 : free p’cles All statistics

 pressure is due to particle motion (kinetics) Let n f(u) d3u= density of particles with velocity between u & u+du.  # of particles to strike wall area dA in time dt = # of particles with u dA >0within volume udA dt Each particle imparts on dA a normal impluse = Total impulse imparted on dA =  

Rate of Effusion # of particles to strike wall area dA in time dt  Rate of gas effusion per unit area through a hole in the wall is   All statistics R  u   Effused particles more energetic. u > 0  Effused particles carry net momentum (vessel recoils) Prob.6.14

6. The Theory of Simple Gases

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