Chapter 4 Boltzman Distribution in infirm-coupling system

1. Chapter 4Boltzman Distribution in infirm-coupling system Prof. Song http://physics.nankai.edu.cn/grzy/fsong/index.asp Nankai University Song Feng

2. §4.1 The Boltzmann Distribution in Infirm-coupling System. • 1. Infirm-coupling system • number-density of the particles in the system is low enough • mean free paths are longer enough than their interactional distances • Example: Thin gases. Nankai University Song Feng

3. 2. Boltzmann Statistical Distribution • 1877,L.Boltzmann • derived the distribution function when studying the collisions of gas molecules owing to which the distribution set in. • In the thermodynamic system which is made up with discriminable classical particles, if these particles have the same mechanical properties, and they are independent, the system’s most probable distribution is called Boltzmann’s statistical Distribution. Nankai University Song Feng

4. 3. Expression • The expression to Boltzmann statistical distribution in z direction for thermodynamics system: Nankai University Song Feng

5. §4.2 Particles distribution depending on the height in gravity field • not considering the distribution depending on the velocity • Molecule density at Z : Nankai University Song Feng

6. The application of the distribution of ideal gas system in gravity field: ① Isothermal air-pressure formula: (P0 is the air-pressure at z=0) It can be employed to estimate the air-pressure at various height. Nankai University Song Feng

7. Equivalent mass ②Suspended particles’ distribution depending on the height: • In isothermal suspension , Brownian particles’ number-density decreases depend on the height’s increase: • It can be used to compute the constant NA: Nankai University Song Feng

8. §4.3 The Maxwell Velocity Distribution • 1. The Maxwell velocity distribution function: Nankai University Song Feng

9. 2. The Maxwell velocity distribution function in separate direction: Nankai University Song Feng

10. f(v) v vp • 3. The Maxwell Speed (scalar quantity) Distribution Function: • (Vp is the most probable speed) Nankai University Song Feng

11. 4. Some important speeds in the Maxwell speed distribution • The most probable speed vp: • The arithmetic average speed : • The root-mean square speed Nankai University Song Feng

12. 5.Examples for applications of the Maxwell Speed distribution • Doppler Spectra Broadening • Effusion • Pressure difference in thermal molecules • Separation of Isotope (such as U238 and U235) • Molecular ray Nankai University Song Feng

13. 6. Experimental Verification of the Maxwell Distribution Law • The Stern experiment: • 1920, Stern adopt argontum atoms made the experiment to verify the Maxwell Distribution Law • 1930-1933,GeZhengquan and CaiTeman also made the similar experiment. Nankai University Song Feng

14. §4.4 Equipartition of Energy in Classical Statistical Mechanics • Internal Energy : • Kinetic energy+potential energy • Energy of electron • Energy of nuclei • Other energies • For real gas: • Internal energy: Kinetic energy + potential energy • For ideal gas: • Internal energy: Kinetic energy, no potential energy Nankai University Song Feng

15. The internal energy of one molecule is Then, for 1 mol ideal gas: Nankai University Song Feng

16. Heat Capacity of Ideal Gases: • Molar heat capacity • Internal enegry and the heat capacity Nankai University Song Feng

17. Degrees of Freedom of Molecules: • Independent ways in which a molecule can absorb energy • Translational t • Rotational r • Vibrational v Nankai University Song Feng

18. The total degrees of freedom: • a monatomic gas has three degrees of freedom • Such as He, Ar, t=3,r=0,v=0 • A diatomic gas, Such as H2, O2, CO • Rigid body,t=3,r=2,v=0 • Non-body, t=3,r=2, v=1 • A tri-atomic molecules • Linear configuration: CO2, t=3, r=2 • Non-linear configuration: H2O t=3, r=3 • Polyatomic molecules, t=3, r=2 or 3, v=3n-t-r Nankai University Song Feng

19. Energy of the Thermal movement of Molecules Nankai University Song Feng

20. The average value for each term is equal to 1/2kT • Total energy for one molecule is: Nankai University Song Feng

21. Total energy for one mole of molecules: Nankai University Song Feng

22. Heat capacity at constant volume for 1 mole gas Nankai University Song Feng

23. Examples: • For monatomic model: • For diatomic model: • For triatomic model: Nankai University Song Feng

24. Redefinition of the degrees of freedom from the viewpoint of energy • THE NUMBER OF SQUARE TERMS IN ENERGY FOUMULAR Nankai University Song Feng

25. Equipartition of Energy in Classical statisticalMechanics • The principle of equipartitionstates that : the energy of a monatomic gas is distributed equally between the degrees of freedom of translational motion, each contributing an amount of 1/2kT per molecule or 1/2RT per mol. Nankai University Song Feng

26. The Difficulties in Classical Physics from the Direction of Equipartition of Energy Law • ①The Wien and Rayleigh-Jeans Approximation • ② Emission and Absorption of Black-Body Radiation Nankai University Song Feng