MATLS 2B03 Lecture Notes - Lecture 6: Equipartition Theorem, Vr6 Engine, Frot
29 Aug 2016
School
Department
Course
Professor
Equipartitioning of energy and heat capacities of gases
If there is a molecule made of atoms then 3 motions (movements) are possible. They can be
categorized in
N N
Translations
Rotations
Vibrations
If , then only 3 translations are possible; no rotations and no vibrations.
1N
If , then there are 3 translations. A number of rotations depends on whether a molecule is linear or
not.
1N
If it is non-linear (of course, this is possible if ), then there are 3 rotations and since the total
number of motions is , there are also 3
2N
63N N
vibrations. If the molecule is linear, then only 2
rotations are possible, which means that there are 3N5
vibrations.
22
tr 111
222
2
x
y
U mvmvm
z
v (1)
Apparently, there are three quadratic terms in (1): one quadratic term per each translation.
non-lin 2 2 2
rot 111
222
x
xyyy
UIII
y
lin 2 2
rot
11
22
x
xy
UI I
y
In the case of rotations, there is one quadratic term per each rotation.
The situation is different in the case of vibrations. For each vibration
22
vibr 11
22
Umvkx
which means that there are two quadratic terms per each vibration.
The theorem of equipartitioning of energy states that energy associated with each degree of freedom is
equal to 2kT . A number of degrees of freedom is merely a number of quadratic terms in an expression
for energy. Of course, if there is not a single molecule, but Avogadro’s number of them, then 2
R
T
should be used instead of 2kT . If there are
Z
degrees of freedom, then the molar energy 2
R
T
UZ
and
2
VV
cUTZ R.
Document Summary
Equipartitioning of energy and heat capacities of gases. N atoms then 3 motions (movements) are possible. , then only 3 translations are possible; no rotations and no vibrations. A number of rotations depends on whether a molecule is linear or. If it is non-linear (of course, this is possible if number of motions is rotations are possible, which means that there are 3n 5 vibrations. ), then there are 3 rotations and since the total. If the molecule is linear, then only 2. Apparently, there are three quadratic terms in (1): one quadratic term per each translation. In the case of rotations, there is one quadratic term per each rotation. The situation is different in the case of vibrations. 2 which means that there are two quadratic terms per each vibration. The theorem of equipartitioning of energy states that energy associated with each degree of freedom is equal to kt.
