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Physics 103
May 16, 2017
Lecture 32
Degrees of Freedom (Ch. 21)
Equipartition Theorem (Ch. 21)
Isochoric vs. Isobaric Heat Capacity (Ch. 21)
Adiabatic Processes (Ch. 21)
Monatomic Molecules
• Energy of a monatomic molecule (atom):
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Inertia of Continuous Body
• Moment of inertia for
continuous bodies:
mass density
volume element,
e.g.
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Diatomic Molecules
• Energy of a diatomic molecule:
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Degrees of Freedom
• Each degree of freedom (d.o.f.) of a molecule is one way
that it can store internal energy.
• A monatomic molecule can store energy in three ways:
– 3 translational d.o.f.
• A diatomic molecule can store energy in seven ways:
– 3 translational d.o.f.
– 2 rotational d.o.f.
– 2 vibrational d.o.f.
• Let
denote degrees of freedom:
– For monatomic gas:
– For cold diatomic gas:
– For hot diatomic gas:
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Equipartition Theorem
• Equipartition Theorem: In a gas at temperature ,
each d.o.f. of each molecule stores an amount of
internal energy equal to
.
• Total internal energy of
molecules:
– Temperature is a measure of internal energy!
– Also expressed as:
.
• Change in temperature implies change in internal
energy:
.
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Isochoric Heat Capacity
• Consider an isochoric process:
– By equip. theorem:
– Work performed:
– By first law:
– By def. of heat capacity:
• Isochoric specific heat capacity is the amount of
heat required to raise the temperature of one mole
(or one kilogram) by one degree at constant volume:
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Isobaric Heat Capacity
• Consider an isobaric process:
– By equip. theorem:
– Work performed:
– By first law:
– By def. of heat capacity:
• Isobaric specific heat capacity is the amount of heat
required to raise the temperature of one mole (or
one kilogram) by one degree at constant pressure:
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Review
• Summary of results from kinetic theory:
– Pressure:
– Temperature:
– Internal energy:
– Molar isochoric specific heat capacity:
– Molar isobaric specific heat capacity:
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Adiabatic Process
• For an adiabatic process:
where
.
• For ideal gas with
degrees of freedom:
– For monatomic ideal gas:
– For cold diatomic ideal gas:
– For hot diatomic ideal gas:
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Practice Problem
Three moles of a cold diatomic ideal gas initially has
volume
and pressure
. The gas is allowed
to adiabatically expand to twice its initial volume.
Calculate the final temperature and pressure of the
gas.
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Confusion with Heat Capacity
• How much heat
is required to
change the
temperature of
something by
?
• Each of the four
quantities above
has an isochoric
and an isobaric
version.
Heat Capacity:
Molar Heat Capacity:
Specific Heat Capacity:
Molar Specific Heat Capacity:
Isochoric
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Isobaric
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