Chapter IV The Maxwell Boltzmann Gas

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Chapter IV The Maxwell Boltzmann Gas

• Preceding chapter General statistical laws for
  ideal gases both BE, FD distributions and Maxwell
  Boltzmann distribution.
• 1) MB distribution is simpler than
• 2) MB distribution is an approximate one.
• 3) But it is applicable.
• 4) until at very low temperatures, MB is a very
  good mathematical description.

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40. The Maxwell Boltzmann monoatomic Gas in the
Classical Approximation. Phase Volume of a Cell
and the Zero Point Entropy

• Discuss
• 1) a molecular ideal gas at sufficiently high
  temperature, and following MB distribution
• 2) ignoring the energy quantization
• 3) calculating the thermodynamic functions
• U?S?H or W(enthalpy, heat content)?F?G
• Only a ?(-PV) potential can be defined uniquely.

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????????????

• ????N???U?????,????MB????????????a???
• ??(denominator)???????(integral).
• ??
• ?????

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????????????

• ?????????CV?
• ???????? ??????,???????????cell???a?(?????Z1?????h
  0r )
• ?????????
• ????
• ??????????????? a ??????

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??????????

• According to
• The integral of state is expressed by
• Introducing the integration variable x ? / T
• The internal energy of a monoatomic gas is

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Thermodynamic quantities T

• Chemical potential
• From Eq.(38.15), the ?-potential is
• The enthalpy is
• The free energy and entropy is
• ??????????

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• The entropy is
• It should be noted that the formulae of entropy
  is inapplicable at low temperatures.
• Because, as T?0, S ?-?, which contradicts with
  the Nerst theorem of S(T?0)?0.
• The thermodynamic functions at low temperatures
  must take into account the degeneracy, the
  quantization of energy.

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41. The Maxwell distribution

• This section is devoted to the translational(??)
  motion ?? of a MB ideal gas. From Eq.(37.6)

• ??dN?????????(???)???????,?????? ?d? ?????dW

???dW??????????,??????????????,??????????????(Z),?
???????????????????????
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?????????

• ????????????????,???????????,the energy of a
  molecule

U(x,y,z) is the potential energy of a molecule in
a applied field. (such as Gravity). If the space
is divided into two factors momentum and
coordinates(?????). The probability is
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• The momentum and coordinate distributions are
  independent. Integrating over coordinate
• Calculating the integrating in the denominator
• The momentum distribution of molecules

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• The velocity distribution is
• ???Maxwell????,??????????
• 1)?????
• 2)????
• 3)?????

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The energy of translational motion

• From above Eq.
• The form of the distribution

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In vx?vy?vz momentum space

• 4?v2dv dvxdvydvz, therefore,

• Maxwell ???????????? ????????? ??????????

zero
??? ???Maxwell??????
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• ?Maxwell?????????,?? ??

??,?????????????????????,??????(kinetic
energy)???,???(mechanics)????????????aihpiph??????
???????
?????????????????????????????????????????????????
??????????????,?????,????,????,?????????
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• ??,???????,????????????????????????n??????,???????
  ?????,???????3n-5??????????????,???????3n-6?

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• ??,?????????????????????????????????,????????
• brsqrqs, brs?????????
• ?????????????????,?????
• ??????,??i,k??1?3n,r,s??1?3n-5?3n-6???????

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• V???????????????B??????????????pl ,
  qs,??T1/2???,??????T????l ????????????,?????
• ???????
• ??????

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????????

• ?????????????????????????????NAT/2????????????2??
  ?,?NAT ?
• ?????????????
• ?????????,????3 NAT/2????????
• ?????,????Cv7 NA/2Cp 9NA/2?
• ???? Cv5 NA/2Cp 7NA/2?
• ??????????Cv7 NA/2,????????,Cv???,????7
  NA/2?,????????????????(??)?

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• ??????????
• ????????????????kT/2. ???p??q???,?????????

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44. The Maxwell-Boltzmann Gas with Two Energy
Levels

• Example N ????????????????? ,????1/2,
  ?????????????H ???????,?????? -?H????,?????? ?H?

?????????????????????????,???????-?H ,????????H
??????????,?????????? 0,??????? ?1?
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??????????

• ???????????????????????????In a two-level system,
  g0 and g1 are the degeneracies of two energy
  levels. The occupation numbers have the
  expressions
• Here, ?is the chemical potential, N N0N1?

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????????????

• ?????????????????
• ??????????????????????

???????,???????????????????,???????,??????????,???
??????????,?????????????
??????,???????????????
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• ????????(7.4)?????

??

• ??????????

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• ?????

• ??????????? ???

????????
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????????

• T?0 K?,?????????e-? /T????,???????????,?????
• ????Tmax? ?lng1/g0gtgt1,??dC/dT0,????
• ??????
• ????
• ?????????

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???????

• ?????????,??Tmax?,
• ???????, ???????
• ???????????????,
• ???????,???????,
• ????????N1/N0 g1/g0?
• ????,????????,??????????????,???????,?????????????
  ?????????,????????????,?????