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Lecture 9 – September 23, 2008
Chapter 2 -
Supplemental Handout on Equipartition
Internal Energy
Internal Energy, U,
is the sum of the potential and kinetic energy of the system.
It is a state
function, i.e. it only depends on its current state, not how it got there.
Δ
U = U
f
- U
i
For any gas, the translational kinetic energy can be written as:
2
z
2
y
2
x
K
mv
2
1
mv
2
1
mv
2
1
E
+
+
=
The
equipartition theorem
says that all degrees of freedom for molecular motion have equal
energy.
Quantitatively, it says that each degree of freedom contributes an average energy of 1/2kT to the
total energy where k = Boltzmann's constant = R/N
A
= 1.38x10
-23
J/entity.
For a perfect monatomic gas (He, Ne, Ar, Kr, Xe, .
..) the only degrees of freedom are
translational so:
U = U
t
= U (0) + 3/2 kT
For n moles of perfect monatomic gasses:
U = U
t
= U (0) + 3/2 nN
A
kT = U (0) + 3/2 nRT
and
U =
RT
2
/
3
)
0
(
U
U
t
+
=
Where U(0) is the energy at T = 0 and can be regarded as 0.
For monatomic gases, this is essentially true, and
RT
2
3
U
U
t
=
=
can be regarded as fact.
For Linear and Diatomic molecules (H
2
, N
2
, O
2
, F
2
, Cl
2
, Br
2
, I
2
, .
..), there are two rotational
degrees of freedom, while non-linear molecules have 3 rotational degrees of freedom, hence: