Internal Energy

1
Internal Energy

• Physics 202
• Professor Lee Carkner
• Lecture 16

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PAL 15 Kinetic Theory

• Which process is isothermal?
• Since T is constant, nRT is constant and thus pV
  is constant
• A is isothermal
• 3 moles at 2 m3, expand isothermally from 3500 Pa
  to 2000 Pa
• Need T, Vf
• Vf nRT/pf (3)(8.31)(281)/(2000) 3.5 m3
• Since T is constant, DE 0, Q W 3917 J

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PAL 16 Internal Energy

• p,V and T for nitrogen
• Pressure increases (since lid pops off)
• Heat flow
• How do you find work?
• Assume all work is contained in lifting the lid

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Ideal Gas

• We will approximate most gases as ideal gases
  which can be represented by
• vrms (3RT/M)½

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Internal Energy

• We have looked at the work of an ideal gas, what
  about the internal energy?
• If the internal energy is the sum of the kinetic
  energies of each molecule then
• Eint (3/2) nRT
• Strictly true only for monatomic gasses

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Molar Specific Heats

• How does heat affect an ideal gas?
• The equation for specific heat is
• From the first law of thermodynamics
• Consider a gas with constant V (W0),
• But DEint/DT (3/2)nR, so
• CV 3/2 R 12.5 J/mol K

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Specific Heat and Internal Energy

• If CV (3/2)R we can find the internal energy in
  terms of CV
• Eint nCVT
• Change in internal energy depends only on the
  change in temperature

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Specific Heat at Constant Pressure

• We can also find the specific heat at constant
  pressure
• DEint Q - W
• Q nCpDT
• Solving for Cp we find
• Cp CV R
• Molar specific heat at constant pressure

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Degrees of Freedom

• Our relation CV (3/2)R 12.5 agrees with
  experiment only for monatomic gases
• Kinetic energy
• For polyatomic gasses energy can also be stored
  in modes of rotational motion

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Rotational Motions
Polyatomic 2 Rotational Degrees of Freedom
Monatomic No Rotation
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DegreesofFreedom
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Equipartition of Energy

• Equipartition of Energy
• We can now write CV as
• CV f/2 R 4.16f J/mol K

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Oscillation

• At high temperatures we also have oscillatory
  motion
• So there are 3 types of microscopic motion a
  molecule can experience
• If the gas gets too hot the molecules will
  disassociate

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Internal Energy of H2
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Adiabatic Expansion

• Lets consider a process in which no heat transfer
  takes place (adiabatic)
• It can be shown that the pressure and temperature
  are related by
• where g Cp/CV
• You can also write

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Isotherms
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Isobaric Process
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Summary Ideal Gas Processes

• Isothermal
• Constant temperature
• W nRTln(Vf/Vi)
• Isobaric
• Constant pressure
• WpDV

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Adiabatic Process
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IsochoricProcess
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Ideal Gas Processes

• Adiabatic
• No heat (pVg constant, TVg-1 constant)
• W-DEint
• Isochoric
• Constant volume
• W 0

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Four Thermodynamic Processes
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P-V Diagram
Isobaric (pconst.)
p
Isothermal (Tconst)
Adiabatic (Q0)
Isochoric (Vconst)
V
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Processes

• For each type of process you should know
• First law and ideal gas law always apply

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The Arrow of Time

• If you see a film of shards of ceramic forming
  themselves into a plate you know that the film is
  running backwards
• Examples

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Entropy

• What do irreversible processes have in common?
• The degree of randomness of system is called
  entropy

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Determining Entropy

• In any thermodynamic process that proceeds from
  an initial to a final point, the change in
  entropy depends on the heat and temperature,
  specifically

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Isothermal Reversible Process
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Isothermal Expansion

• Consider an example, isothermal expansion
• A cylinder of gas rests on a thermal reservoir
  with a piston on top
• The temperature is constant so
• DS Sf-Si(1/T)?dQ

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Closed Systems

• Consider a closed system
• The entropy change in the gas is balanced by the
  entropy change in the reservoir (DSQ/T)