1. Rotational energy levels for diatomic molecules
l = 0, 1, 2... is angular momentum quantum number
I = moment
of inertia
CO2 I2 HI HCl H2
qR(K)

2. Vibrational energy levels for diatomic molecules
n = 0, 1, 2... (harmonic quantum number) w
w = natural
frequency of
vibration
I2 F2 HCl H2
qV(K)

3. Specific heat at constant pressure for H2
CP = CV + nR
H2 boils w
CP (J.mol-1.K-1)
Translation

4. More on the equipartition theorem
Classical uncertainty:
Where is the particle?
V(x)
V = ∞
V = ∞
V = 0
W = 9 x
x = L

5. More on the equipartition theorem
Classical uncertainty:
Where is the particle?
V(x)
V = ∞
V = ∞
V = 0
W = 18 x
x = L

6. More on the equipartition theorem
Classical uncertainty:
Where is the particle?
V(x)
V = ∞
V = ∞
V = 0
W = 36 x
x = L

7. More on the equipartition theorem
Classical uncertainty:
Where is the particle?
V(x)
V = ∞
V = ∞
V = 0
W = ∞
S = ∞ x
x = L

8. More on the equipartition theorem: phase space
Area h
Cell:
(x,px)
dpx px dx x

9. More on the equipartition theorem: phase space
Area h
Cell:
(x,px)
dpx px dx x

10. More on the equipartition theorem: phase space
Area h
Cell:
(x,px)
dpx px dx x

11. More on the equipartition theorem: phase space
In 3D:
Uncertainty relation:
dxdpx = h
dpx dx

12. Examples of degrees of freedom: