Quantum ideal gases team 1 reviewed

QUANTUM IDEAL GASES
Let
•   us consider 1 mol of quantum diatomic gases:
The internal energy of gases includes
where
• energytranslational motion
• motion
• : energy vibrational motion
• : energy associated with electron shell
• : energy associated with atomic nuclear
Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
• Heat
  capacity of gases , we get:
 At the gas temperature being much smaller than the
ionisation temperature ():

 The total heat capacity of the gas is the sum of several
terms:

Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
• For

  translational motion (with freedom degree of 3)
 Heat capacity per mol translational motion

Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
Quantum
oscillation (1)
•  
 Single particle energy (which is nondegenerate )

 Partition sum

Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
Quantum
oscillation (2)
•  
 Mean energy
Since , then
Derive the mean energy of oscillation

Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
Quantum
oscillation (3)
•  
 Heat capacity of gases for vibrational motion

Heat capacity per mol vibrational motion
• At low temperature ():
• At high temperature ():

Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
Quantum
rotator (1)
•  
 Single particle energy (with degenerate factor of )

Where :
• : the moment of inertia of the molecule
• : the rotational quantum number

Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
Quantum
rotator (2)
•  
 Partition sum

 Accoding to Quantum canonical distribution, the mean
energy:

Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
Quantum
rotator (3)
•  
 Heat capacity of gases for rotational motion
• At low temperature

• At high temperature

Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
Conclusions
•  
 If

 If temperature is decreased so that , the vibrational motion stops
contributing to heat capacity ( )

 If continuously descreasing temperature so that , the rotational
motion also stops contributing

Team 2 – Class 15SVL

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QUANTUM IDEAL GASES
Chart

Team 2 – Class 15SVL

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