Nankai University Song Feng Chapter 4 Boltzman Distribution in infirm-coupling system Prof. Song

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

Nankai University Song Feng §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 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 ② Suspended 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 N A : Equivalent mass

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

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

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

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

Nankai University Song Feng 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 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 ,GeZhengquan and CaiTeman also made the similar experiment.

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

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

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

Nankai University Song Feng 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 H 2, O 2, 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 Energy of the Thermal movement of Molecules

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

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

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

Nankai University Song Feng Examples: For monatomic model: For diatomic model: For triatomic model:

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

Nankai University Song Feng Equipartition of Energy in Classical statistical Mechanics The principle of equipartition states 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 The Difficulties in Classical Physics from the Direction of Equipartition of Energy Law ① The Wien and Rayleigh-Jeans Approximation ② Emission and Absorption of Black-Body Radiation