1. Microscopic Model of Gas
Physics I
Microscopic Model of Gas
Prof. WAN, Xin

2. The Naïve Approach, Again
N particles ri(t), vi(t); interaction V(ri-rj)

3. Elementary Probability Theory
Assume the speeds of 10 particles are 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 m/s
When we have many particles, we may denote pa the probability of finding their velocities in the interval [va, va+1].

4. Elementary Probability Theory
Now, the averages become
In the continuous version, we may denote p(v)dv the probability of finding particles’ velocities in the interval [v, v+dv].

5. Assumptions of the Ideal Gas Model
Large number of molecules and large average separation (molecular volume is negligible).
The molecules obey Newton’s laws, but as a whole they move randomly with a time-independent distribution of speeds.
The molecules undergo elastic collisions with each other and with the walls of the container.
The forces between molecules are short-range, hence negligible except during a collision.
That is, all of the gas molecules are identical.

6. The Microscopic Model L x

7. Pressure, the Microscopic View
Pressure that a gas exerts on the walls of its container is a consequence of the collisions of the gas molecules with the walls.
half of molecules moving right
r = N / V

8. Applying the Ideal Gas Law
Boltzmann’s constant

9. Temperature
Temperature is a measure of internal energy (kB is the conversion factor). It measures the average energy per degree of freedom per molecule/atom.
Equipartition theorem: can be generalized to rotational and vibrational degrees of freedom.

10. Heat Capacity at Constant V
We can detect the microscopic degrees of freedom by measuring heat capacity at constant volume.
Internal Energy U = NfkBT/2
Heat capacity
Molar specific heat cV = (f/2)R
degrees of freedom

11. Specific Heat at Constant V
Monoatomic gases has a ratio 3/2. Remember?
Why do diatomic gases have the ratio 5/2?
What about polyatomic gases?

12. Specific Heat at Constant V

13. A Simple Harmonic Oscillator

14. Two Harmonic Oscillators

15. Two Harmonic Oscillators
Assume

16. Two Harmonic Oscillators
Assume

17. Vibrational Mode
Solution 1:
Vibration with the reduced mass.

18. Translational Mode
Solution 1:
Translation!

19. Two Harmonic Oscillators
In mathematics language, we solved an eigenvalue problem.
The two eigenvectors are orthogonal to each other. Independent!

20. A straightforward generalization of the two-atom problem.
Mode Counting – 1D
1D: N-atom linear molecule
Translation: 1
Vibration: N – 1
A straightforward generalization of the two-atom problem.

21. From 1D to 2D: A Trivial Exampley
rotation
translation
vibration

22. Mode Counting – 2D 2D: N-atom (planer, nonlinear) molecule
Translation: 2
Rotation: 1
Vibration: 2N – 3

23. Mode Counting – 3D 3D: N-atom (nonlinear) molecule Translation: 3
Rotation: 3
Vibration: 3N – 6

24. Vibrational Modes of CO2
N = 3, linear
Translation: 3
Rotation: 2
Vibration: 3N – 3 – 2 = 4

25. Vibrational Modes of H2O
N = 3, planer
Translation: 3
Rotation: 3
Vibration: 3N – 3 – 3 = 3

26. Contribution to Specific Heat
Equipartition theorem: The mean value of each independent quadratic term in the energy is equal to kBT/2.

27. Quantum mechanics is needed to explain this.
Specific Heat of H2
Quantum mechanics is needed to explain this.

28. Specific Heat of Solids
DuLong – Petit law
spatial dimension
vibration energy
Molar specific heat
Again, quantum mechanics is needed.

29. Root Mean Square Speed root mean square speed
Estimate the root mean square speed of water molecules at room temperature.

30. Distribution of Speed
slow
fast
oven
rotating drum
to pump

31. Speed Selection
Can you design an equipment to select gas molecules with a chosen speed? ?
to pump

32. Maxwell Distribution

33. number of molecules v  [v1, v2]
Maxwell Distribution
number of molecules v  [v1, v2] dv v N ò 2 1 ) (

34. Total number of molecules
Maxwell Distribution
Total number of molecules

35. Characteristic Speed
Most probable speed

36. Characteristic Speed
Root mean sqaure speed

37. Characteristic Speed
Average speed

38. Varying Temperature T1 T2 T3

39. Boltzmann Distribution
Continuing from fluid statics
The probability of finding the molecules in a particular energy state varies exponentially as the negative of the energy divided by kBT.
potential energy
Boltzmann distribution law

40. How to cool atoms?

41. Laser Cooling
Figure: A CCD image of a cold cloud of rubidium atoms which have been laser cooled by the red laser beams to temperatures of a millionth of a Kelvin. The white fluorescent cloud forms at the intersection of the beams.

42. Bose-Einstein Condensation
Velocity-distribution data for a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate.
Left: just before the appearance of a Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate.

43. For an updated list, check http://ucan.physics.utoronto.ca/
Earlier BEC Research
BEC in ultracold atomic gases was first realized in 1995 with 87Rb, 23Na, and 7Li. This pioneering work was honored with the Nobel prize 2001 in physics, awarded to Eric Cornell, Carl Wieman, and Wolfgang Ketterle.
For an updated list, check

44. BEC of Dysprosium
Strongly dipolar BEC of dysprosium, Mingwu Lu et al., PRL 107, (2011)

45. Brownian Motion

46. Mean Free Path
Average distance between two collisions

47. Mean Free Path
During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.
Volume of the cylinder
Average number of collisions
Mean free path

48. Mean Free Path Relative motion
During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.
Average number of collisions
Relative motion
Mean free path

49. Q&A: Collision Frequency
Consider air at room temperature.
How far does a typical molecule (with a diameter of 2 10-10 m) move before it collides with another molecule?

50. Q&A: Collision Frequency
Consider air at room temperature.
How far does a typical molecule (with a diameter of 2 10-10 m) move before it collides with another molecule?

51. Q&A: Collision Frequency
Consider air at room temperature.
Average molecular separation:

52. Q&A: Collision Frequency
Consider air at room temperature.
On average, how frequently does one molecule collide with another?
Expect ~ 500 m/s
Expect ~ 2109 /s
Try yourself!

53. Transport: Viscous Flow
Fluid flows layer by layer with varying v.
F = h A dv/dy h: coefficient of viscosity A y
F, v A

54. Cylindrical Pipe, Nonviscous
(volumetric flow rate)

55. Cylindrical Pipe, Viscous
V(r) 2R
“current”
“voltage”
(Poiseuille Law)

56. Homework
CHAP. 22 Exercises 7, 8, 10, 21, 24 (P513)