Thermodynamics

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Thermodynamics

• Phys 2101
• Gabriela González

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Ideal gases so far

• pV n R T
• ?Eint Q W
• Eint (3/2) n R T
• Constant volume
• W0,
• Q ?
• ?Eint (3/2) n R ?T
• Constant pressure
• W p ?V
• Q ?
• ?Eint (3/2) n R ?T
• Constant temperature
• W nRT ln (Vf/Vi),
• ?Eint 0
• Q W nRT ln (Vf/Vi)

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Molar specific heat

• We defined heat capacity C and specific heat c
  (heat capacity per mass) as
• Q C ?T c m ?T
• For water, c 1cal/goC 1 Btu/lboF 4190 J/kg
  K
• The molar specific heat is the heat capacity per
  mole.
• However, the amount of heat needed to raise the
  temperature of 1 mole by 1 kelvin depends on
  whether we perform the operation at constant
  pressure or at constant volume
• Q n CP ?T or Q n CV ?T

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Molar specific heat

• Consider a process at constant volume, raising
  the temperature by ?T . Q n
  CV ?T
• 
  Q ?Eint W
• 
  W 0 !
• ?Eint
  (3/2) nR ?T Q n CV ?T
• 
  ? CV (3/2)R 12.5 J/molK
• 
  
• 
  

The molar specific heat is the same for all
(monoatomic) ideal gases!
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Molar specific heat

• Consider a process at constant pressure, raising
  the temperature by ?T. Q n CP ?T
• ?Eint Q
  W
• 
  W p ?V n R ?T
• 
  ?Eint (3/2) nR ?T
• 
  Q ?Eint W (5/2) nR ?T
• 
  ? CP (5/2)R CV R
• 
  

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Ideal gases so far

• pV n R T
• ?Eint Q W
• Eint (3/2) n R T n CV T
• CP CV R CV(3/2)R
• Constant volume
• W0,
• Q n CV ?T
• ?Eint n CV ?T
• Constant pressure
• W p ?V nR?T
• Q n CP ?T
• ?Eint n CV ?T
• Constant temperature
• W nRT ln (Vf/Vi),
• ?Eint 0
• Q W nRT ln (Vf/Vi)

True for monoatomic gases, but not for others!?
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Polyatomic molecules

• We used Eint nNA Kavg nNA (3/2) kT
  (3/2) n R T
• Is translational kinetic energy the only kinetic
  energy in molecules? Not for polyatomic
  molecules! They can spin and rotate, in a number
  of different ways.
• Every kind of molecule has a certain number f of
  degrees of freedom in which the molecule can
  store energy. Each d.o.f. has, in average, (1/2)
  kT kinetic energy per molecule (or (1/2)R T per
  mole). Thus,
• 
  Eint nNA Kavg (f /2) nRT
• 
  and CV (f /2) R
• 
  Cp CV R
• 
  ? Cp/CV 1 2/f
  
  Cp, CV and ? can easily be measured.

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Adiabatic processes

• Adiabatic processes happen when Q 0, because
  the system is thermally insulated, or because the
  process happens very quickly. Temperature,
  pressure and volume all change, but the
  quantities
• pV? and T V?-1
  are constant,
• where ? CP/CV
  (4/31.67 for monoatomic gases)
• 
  The curve in a pV
  diagram is
  called an adiabat

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Summary
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Example

• One liter of gas with ? 1.3 is at 292 K and
  1.4 atm pressure. It is suddenly compressed
  (adiabatically) to half its original volume.
• Find its final pressure and temperature.
• The gas is now cooled back to 292 K at constant
  pressure. What is its final volume?

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Example

• One mole of an ideal diatomic gas undergoes a
  transition from a to c along the diagonal path
  shown in the figure.
• What is the change in internal energy? -5000 J
• How much heat is added to the gas? 2000J
• How much heat is added to the gasif it goes back
  to the original state a through the point b?
  -5000J