Physics 1710 Chapter 21 Kinetic Theory of Gases

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Physics 1710Chapter 21 Kinetic Theory of Gases

• Quiz
• A person emits approximately
• 100 W of heat. Show that this requires a diet of
  approximately 2000 Cal per day.

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Physics 1710Chapter 21 Kinetic theory of Gases

• 1' Lesson
• The Ideal Gas Law results from the cumulative
  action of atoms or molecules.
• The average kinetic energy of the atoms or
  molecules of an ideal gas is equal to 3/2 kT.
• Energy average distributes equally (is
  equipartitioned) into all available states.
• The distribution of particles among available
  energy states obeys the Boltzmann distribution
  law.
• nV no e E/kT

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Physics 1710Chapter 21 Kinetic Theory of Gases

• Molecular Model of Ideal Gas
• Key concept gas is ensemble of non-interacting
  atoms or molecules.
• Pressure due to a single molecule
• at wall of vessel
• P1 -F1 /A -(?px / ?t)/A
• Impulse
• ?px - mvx (mvx ) -2 mvx
• ?t 2d /vx
• Thus
• P1 - F1 /A mvx2 /(d?A)

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Physics 1710Chapter 21 Kinetic Theory of Gases

• P1 - F1 /A mvx2 /(d?A)
• Total Pressure
• P NltP1 gt Nmltvx2 gt/(d?A)
• P (N/V) mltvx2 gt
• Average vx2 ltvx2 gt
• ltv2 gtltvx2 gt ltvy2 gt ltvz2 gt
• ltv2 gt3ltvx2 gt ltvx2 gt 1/3 ltv 2gt
• P ?(N/V)(½ mltv 2gt)

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Physics 1710Chapter 21 Kinetic Theory of Gases

• P ?(N/V)(½ mltv2 gt)
• But
• P (N/V) kT
• Thus
• T 2/(3k)(½ mltv 2gt)
• ½ mltv2gt 3/2 kT
• ½ mltvx2gt 1/3 ½ mltv2gt ½ kT

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Physics 1710Chapter 21 Kinetic Theory of Gases

• Principle of Equipartition of Energy
• Each degree of freedom contributes 1/2 kT to the
  energy of a system.

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Physics 1710Chapter 21 Kinetic Theory of Gases

• Each molecule in a gas contributes 3 degrees of
  freedom to the system
• N( 1/2 mltv 2gt) 3N(½ kT) 3/2 nRT
• vltv 2gt vrms v(3 kT/m) v(3RT/M)

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Physics 1710Chapter 21 Kinetic Theory of Gases

• (Molar) Specific Heat of an Ideal Gas
• ?Q n CV ?T (constant volume)
• ?Q n CP ?T (constant pressure)
• W ? P dV at constant volume W 0.
• ?Eint ?Q n CV ?T
• Eint n CV T
• CV (1/n) d Eint /dT
• CV 3/2 R 3/2 No kT 12.5 J/mol?K

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Physics 1710Chapter 21 Kinetic Theory of Gases

• ?Eint ?Q W nCP ?T - P?V
• nCV ?T nCP ?T n R ?T
• CP - CV R
• CP 5/2 R
• ? CP / CV (5/2 R)/(3/2 R) 5/3
• ? 5/3

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Physics 1710Chapter 21 Kinetic Theory of Gases

• Adiabatic Expansion of an Ideal Gas
• For adiabatic case
• dEint n CV dT - PdV
• So that
• dT -P dV /(nCV )
• Also
• PV nRT
• PdV VdP nR dT
• PdV VdP -RP /(nCV ) dV

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Physics 1710dChapter 21 Kinetic Theory of Gases

• PdV VdP -R/(nCV ) PdV
• Rearranging
• dP/P 1 R/(nCV) dV/V
• dP/P - ? dV/V
• ln P - ? lnV ln K
• PV ? constant

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Physics 1710Chapter 21 Kinetic Theory of Gases

• Bulk Modulus of an Ideal Gas
• B -?P/ (?V/V)
• P K V - ?
• dP - ? KV - ? - 1dV
• B -dP/(dV/V)
• B (? KV 1dV)/(dV/V)
• B ? KV - ?
• B ? P

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Physics 1710Chapter 21 Kinetic Theory of Gases

• Law of Atmospheres
• dP -mg nV dy
• P nV kT
• dP kT dnV
• kT dnV -mg nV dy
• dnV/ nV -(mg/kT) dy
• nV no e (mgy/kT)

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Physics 1710Chapter 21 Kinetic Theory of Gases

• Boltzmann Distribution Function
• nV no e (mgy/kT)
• nV no e U/kT
• nV (E) no e E/kT

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Physics 1710Chapter 21 Kinetic theory of Gases

• Summary
• The Ideal Gas Law results from the cumulative
  action of atoms or molecules.
• The average kinetic energy of the atoms or
  molecules of an ideal gas is equal to 3/2 kT.
• ½ mltv2gt 3/2 kT
• Energy average distributes equally (is
  equipartitioned) into all available states.
• Each degree of freedom contributes 1/2 kT to the
  energy of a system.

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Physics 1710Chapter 21 Kinetic Theory of Gases

• Summary (contd.)
• ? CP / CV
• PV ? constant
• B ? P
• The distribution of particles among available
  energy states obeys the Boltzmann distribution
  law.
• nV no e E/kT