1. Chapter 6 Applications of
Boltzmann Statistics

2. Applications of Boltzmann Statistics
For noninteracting systems, in which the particles are taken to be distinguishable, (localized particles) Boltzmann statistics are very useful. e.g. for atoms who oscillate nearby their positions of equilibrium.
Maxwell-Boltzmann distribution is:

3. 6.1 Thermodynamic quantities in Boltzmann statistics
In Boltzmann statistics, it is convenient for us to define a partition function:

5. The equation above is the statistical formulation of energy
The equation above is the statistical formulation of energy. Additional thermodynamic properties can be also derived according to the partition function.
e.g. the generalized force :Y
The force performed on single particle who is on the energy level is

6. This is the statistical expression for the generalized force, an especially example is for the pressure

7. Y

8. And we have known that 第一项是粒子的分布不变时由于能级的改变而引起的内能的变化,
代表在准静态过程中外界对系统所作的功。
第二项是粒子的能级不变时由于粒子分布发生变化而引起的内能
变化,代表在准静态过程中系统从外界吸收的热量。

9. Boltzmann constant k:

10. Because：

11. （1）
Compare the two equations above, we can see that both and are integrating factor of the nonexact differential So，we can write:

12. (2)
This is the statistical expression of S.

13. At the same time, we can also obtain other statistical expressions of corresponding thermodynamic quantities.
F=U-TS

14. In addition ,because:

16. §6.2 classical approximation
are both called non-degeneracy condition or classical limiting condition.

17. Quantum expression
Classical expression

18. Is a volume cell in the space, so the partition function with the classical approximation can be written as
In classical approximation, the volume cell in phase space is

19. 6.3 The state equation of ideal gas
Example:
6.3 The state equation of ideal gas
For most gases, the condition is always fulfilled. So, we can use Boltzmann statistics to solve the questions of gases.
And especially for the ideal gas, there is no interactions between particles,
So, they can be considered as free particles. And the energy of single particle reads

20. For another reason, in the macroscopic size vessel, the translational energy of an ideal gas particle is quasi-continuum.
So that , the classical approximation can be used in the ideal gas statistics.

21. For monatomic molecule,
Appendix: 3

24. We can see that the expressions of P and U are both in of accord with the formulations we obtained before,

25. classical limiting condition
Addition:
classical limiting condition
For gases:
It should be

26. classical limiting condition
If we assume one gas molecule’s energy is
classical limiting condition

27. Principle of the equipartition of energy:
§6.4 Principle of the equipartition of energy and its applications
Principle of the equipartition of energy:
For a classical system which is in thermodynamic equilibrium and temperature is T, the mean value of each quadratic term in the energy is
e.g.For monatomic ideal gas,

28. Justification:
First ,Kinetic energy can be expressed by

29. = 1 1 Z

30. If the potential energy can be also expressed by quadratic terms, just like

31. (1) For monatomic molecule
According to the principle of the equipartition of energy ,the energy of each molecule is

32. (2) For diatomic molecule
If we neglect the last term

33. (3) For atoms in a solid
(x,y,z)

34. 6.5 energy and heat capacity of the ideal gas
6.5.1 quantum description
For diatomic molecule, the energy includes not only the translational energy but also the vibrational energy and the rotational energy.

36. （2）vibration
For the diatomic molecule’s vibration, it can be approximately held to be same to the linear oscillator’ vibrational action.

37. (1)
(2)
(1)-(2)

38. x

39. Now,we define a vibrational specific temperature:
特征温度实际上是表示能量量子化间距的接近程度

40. （10 3）

42. （3）rotation Analogy,we define a rotational specific temperature:
For heteronuclear diatomic molecule:
Analogy,we define a rotational specific temperature:

45. 6.5.2. classical descriptions
Here, M is the sum of the two atoms’ masses.
And the second term in the equation is the rotational energy which is around the center of mass. The third term is the vibrational energy, in which ,the first part is the vibrational kinetic energy, and is the reduced mass, r is the
distance between the two atoms.

50. 6.6、Entropy of the ideal gas
For monatomic ideal gas:
Boltzmann statistics
But for Fermi statistics and Bose statistics,
(7.1.17’)

51. §6.7 The heat capacity of solid
For atom in the solid who vibrates near its position of equilibrium,

52. Einstein

53. Einstein was the first one who explained the fact that the capacity of solid would reduce with temperature by quantum analyses.
(3 coordinates)
We assume that there are 3N oscillators in the solid, their frequencies are all
And because these oscillators are localized particles, so, they obey Boltzmann statistics, that is:

54. Here, the first term is the zero energy of 3N oscillators, which is independent to the temperature. And the second term is the thermal-excitation energy with temperature T.

55. Now,we define a Einstein specific temperature:

57. Exercise1:
A gas column in the field of gravity, the height is H and the sectional area is S, try to identify that,

58. Exercise2:
For nonrelativistic particles, there is
So, according to
,please prove that

59. Exercise3:
Assume a particle who submits Boltzmann statistics, and we have known the expression of energy is
Here , a and b are constants, now, request the mean energy of the particle.