Thermodynamics

1
Thermodynamics

• Phys 2101
• Gabriela González

2
Work done by an ideal gas
pVnRT

• Constant temperature keep temperature
  constant, change the volume. The pressure will
  change, following p nRT/V
• How much work is done by the gas?
  W nRT ln (Vf/Vi)
• Constant pressure keep pressure constant,
  change the volume. The temperature will change,
  following TpV/nR. How much work is done by the
  gas? W p ?V p(Vf-Vi)
  
• Constant volume pressure and temperature may
  change, but no work is done!
• W 0

isotherms
3
Example

• A gas can be taken from the initial state i to
  the final state f in many different ways, usually
  following constant pressure curves, constant
  volume curves, and isotherms.
• If the initial pressure is 1 atm, and the
  initialvolume is 1m3, how many moles are
  therein the gas?
• If the final volume is 1.1 m3, what is the
  final pressure?
• What is the path from i to f where the gas does
  minimum work?
• What is the temperature at intermediate points
  A, B?
• If the system is taken to the final state
  through the 310 K isotherm, and then back to the
  original state through point B, what is the total
  heat added to the system?

B
A
4
Kinetic energy, internal energy

• We learned the ideal gas law pV nRT nKT.
• We know that ?Eint Q - W, and we can calculate
  the work for a process from the area in the p-V
  diagram. Can we calculate ?Eint and Q from p,V
  and/or T?
• The average translational kinetic energy of
  molecules in a gas is related only to temperature
  (not pressure, or volume). The internal energy of
  a gas is also related just to temperature (not to
  pressure, or volume). For a monoatomic, ideal
  gas,
• Eint nNA Kavg
  (3/2) nRT (3/2) N k T(The factor 3/2 can be
  derived from three directions of motion for each
  molecule, each with (1/2) kT energy).
• Thus, the internal energy of all states on an
  isotherm curve have the same internal energy.
• The work done by a gas between two states on an
  isotherm must be equal to the heatabsorbed by
  the gas.
• Whats the heat absorbed between states on
  different isotherms?

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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!?