The Kinetic Theory of Gases

1
Chapter-19

• The Kinetic Theory of Gases

2
Chapter-19 The Kinetic Theory of Gases

• Topics to be covered
• Ideal gas law
• Internal energy of an ideal gas
• Distribution of speeds among the atoms in a gas
• Specific heat under constant volume
• Specific heat under constant volume.
• Adiabatic expansion of an ideal gas

3
Ch 19-2, 3 Avogadro Number

• Kinetic Theory of gases Relates motion of atoms
  to the volume, pressure and temperature.
• Mole one mole is number of atoms in a 12 g
  sample of carbon-12
• Avogadro Number one mole contains Avogadro
  number NA of atoms
• NA 6.02 x 1023 atoms/mol
• n number of moles
• N number of molecules
• M Molar mass of a substance
• Msamp mass of a sample then

• M m NA N n NA Msampn M NA
• Ideal gas law
• An ideal gas obey the law
• pVnRT
• where R8.31 J/mol.K
• Boltzman constant k
• k R/NA
• pVnRTNkT

4
Ch 19-3 Ideal Gas and work done by the ideal gas

• Work done by an ideal gas at constant Temperature
  (Isothermal expansion )
• An ideal gas is allowed to expand from initial
  state pi,Vi to pf,Vf at constant T, the work W
  is
• W?VfVipdV?VfVi (nRT/V)dV
• WnRTln(Vf/Vi)
• Work done by an ideal gas at constant volume
• W?VfVipdV0 and ?EintQ
• Work done by an ideal gas at constant pressure
• Wp(Vf-Vi)p?VnR?T

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Ch 19-4,5 Pressure, Temperature, RMS Speed and
Translational Kinetic Energy

• Pressure p relation to root-mean -square speed
  vrms and temperature T
• P(nM vrms 2)/3V
• but pVnRT then
• Vrms ? (3RT)/M
• Translational Kinetic Energy K Average
  translational kinetic energy of a molecule
  Kavg(mv2/2)avgm(vrms2)/2
• Kavgm(vrms2)/2(m/2)3RT/M
• (3/2)(m/M)RT(3/2)(R/NA)T
• Kavg(3/2)(R/NA)T(3/2)kT

6
Ch 19-8 The Molar Specific Heats of an Ideal Gas

• Internal Energy Eint Ideal gas is monatomic and
  its Eint is sum of translational kinetic energies
  of its atom. For a sample containing n moles, its
  internal energy Eint
• EintnNAKavgnNA(3/2)kT (3/2)n(NAk)T
• Eint (3/2)nRT
• Molar Specific Heat at Constant Volume
• For an ideal gas process at constant volume pi,Ti
  increases to pf,Tf and heat absorbed QVnCV?T and
  W0. Then
• ?Eint (3/2)nR?T QnCV?T
• CV 3R/2
• Where CV is molar specific heat at constant volume

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Ch 19-8 The Molar Specific Heats of an Ideal Gas

• Cp Molar Specific Heat at Constant Pressure
• For an ideal gas process at constant pressure
  Vi,Ti increases to Vf,Tf and heat absorbed
  QPnCp?T and Wp?V nR ?T. Then
• ?Eint (3/2)nR?T Q-W nCp?T- nR?T.
• Cp 3R/2R5R/2
• Where Cp is molar specific heat at constant
  pressure
• Cp CV R specic heat ration ? Cp/ CV
• For monatomic gas Cp 5R/2 CV 3R/2
• and ? Cp/ CV 5/3

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Ch 19-11 The Adiabatic Expansion of an Ideal Gas

• Adiabatic process In an adiabatic expansion of
  an ideal gas no heat enters or leaves the system
  i.e. Q0
• P, V and T are related to the initial and final
  states with the following relations
• PiVi? PfVf?
• TiVi?-1 TfVf?-1
• Also T?/(? -1) V? constant then
• piTi(?-1)/? pfTf(?-1)/?

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Ch 19-11 The Adiabatic Expansion of an Ideal Gas
Free Expansion

• Free Expansion of an ideal gas-
• An ideal gas expands in an adiabatic process
  such that no work is done on or by the gas and no
  change in the internal energy of the system i.e.
  TiTf
• Also in this adiabatic process since ( pVnRT),
  piVipfVf ( not PiVi? PfVf?)