6.The Theory of Simple Gases 1.An Ideal Gas in a Quantum Mechanical Microcanonical Ensemble 2.An Ideal Gas in Other Quantum Mechanical Ensembles 3.Statistics of the Occupation Numbers 4.Kinetic Considerations 5.Gaseous Systems Composed of Molecules with Internal Motion 6.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 ) n  * extremize Lagrange multipliers

BE FD

 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 : g{ n  } = statistical weight factor for { n  }.

 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. Grand potential : BE FD 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 :  MB : FD : BE :B.E. condensation 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 state of energy .  BE FD 

BE :  BE FD FD : 

MB :Gibbs’ correction   Poisson distribution  Alternatively  “normal” behavior of un-correlated events

BE : FD :  Geometric ( indep of n ) > MB for large n : Positive correlation  < MB for large n : Negative correlation

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

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

BE FD Let p(  ) be the probability 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) d 3 u = density of particles with velocity between u & u+du.  # of particles to strike wall area dA in time dt = # of particles with u  dA >0 within volume u  dA dt Total impulse imparted on dA = Each particle imparts on dA a normal impluse =  

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

6.5.Gaseous Systems Composed of Molecules with Internal Motion Assumptions ( ideal Boltzmannian gas ) : 1. Molecules are free particles ( non-interacting). 2. Non-degeneracy (MB stat) :  = quantum # for internal DoF

Internal DoF   Molecules : Homopolar molecules (A-A) :

6.5.A. Monatomic Molecules Let ( All atoms are neutral & in electronic ground state ) Nuclear spin  Hyperfine structure : T ~ 10  1 – 10 0 K.  Level-splitting treated as degeneracy : Inert gases ( He, Ne, Ar,... ) : Ground state  L = S = 0 :   = 0 denotes ground state.  0 = 0.  L = 0; S  0 : 

  L = 0, S  0 

 L  0, S  0   Ground state  0 = 0.  C V, int = 0 in both limits  C V has a maximum.

6.5.B. Diatomic Molecules Let ( All atoms are neutral & in electronic ground state ) Non-degenerate ground state ( most cases )  g e = 1& j elec (T) = 1 Degenerate ground state ( seldom ) : 1. Orbital angular momentum   0, but spin S = 0 : In the absence of B,  depends on |  z |   doublet (  z =  M ) is degenerate ( g e = 2 = j(T) )  C V = 0 2.  = 0, S  0 : g e = 2S + 1 = j(T)  C V = 0 3.   0 & S  0 : Spin-orbit coupling  B eff  fine structure

E.g., NO (  1/2, 3/2 ) ( splitting of  doublet ) :  C V has max. for some kT ~ 

Vibrational States for diatomic gases  Full contribution for T  10 4 K No contribution for T  10 2 K Harmonic oscillations (small amplitude) : From § 3.8 :  equipartition value  vib DoF frozen out

Very high T  anharmonic effects C vib  T( Prob )

Nuclear Spin & Rotational States: Heteropolar Molecules Heteropolar molecules ( AB ) : no exchange effects  interaction between nuclear spin & rotational states negligible. From § 6.5.A :  C nucl = 0 Molecule ~ rigid rotator with moment of inertia ( bond // z-axis ) = reduced mass r 0 = equilibrium bond length 

Homopolar molecules

6.6.Chemical Equilibrium