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Chapter 12 Gas Kinetics Department of Physics Shanghai Normal University.

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Presentation on theme: "Chapter 12 Gas Kinetics Department of Physics Shanghai Normal University."— Presentation transcript:

1 Chapter 12 Gas Kinetics Department of Physics Shanghai Normal University

2 Table of Content F 12-1 The Equilibrium State, the Zero Law of Thermodynamics F 12-2 The Microscopic Model of Matter, the law of Statistics F 12-3 The Pressure Formula of the Ideal Gas F 12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas F 12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas F 12-6 The Law of Maxwell Speed Distribution of Gas Molecules F 12-8 The Average Number of Collisions of Molecules and the Mean Free Path 本章目录

3 Research Object: Thermal Motion : all the small particles (atoms or molecules) are in constant, random motion. Thermal Phenomena: changes of the physical features about the temperature The Features of Research Object: Each molecule : disorder 、 accidental, following Newtonian mechanics Entity (a large number of small particles): obeying Statistical Law. 12-1 The Equilibrium State, the Zero Law of Thermodynamics

4 Macroscopic quantities: the macroscopic state of the entire gas, such as, p , V , T, etc. They can be measured directly Microscopic quantities: describing the individual molecule, such as, its own mass m, velocity,etc. They can not be measured directly Macroscopic quantities Microscopic quantities statistic average 12-1 The Equilibrium State, the Zero Law of Thermodynamics

5 I.The state of gas ( Microscopic quantities) 1. Pressure(p) : the force in per unit area unit: Standard atmospheric pressure: the atmospheric p at 0 ℃ at the sea level of 45°latitude 。 1atm=1.01 ×10 5 Pa 2.Volume(V) : reachable space 12-1 The Equilibrium State, the Zero Law of Thermodynamics unit: 3.Temperature(T) : measure of the coldness or hotness of an object unit:

6 a gas with a certain mass in a container does not transport energy and mass with the environment, after a relatively long time, then the state parameters doo not change with time, such a state is called an ~ 12-1 The Equilibrium State, the Zero Law of Thermodynamics II.Equilibrium State(E-S) Vacuum expansion

7 Characteristic of E-S: (1). oneness: p, T in everywhere are the same; (2). State parameters are stable: independent with time 12-1 The Equilibrium State, the Zero Law of Thermodynamics (3). The final state of a Spontaneous Process (4). thermal equilibrium : different from mechanical equilibrium

8 III.The equation of the state of the Ideal gas The equation of the state : the function connecting the macroscopic quantities of the ideal gas in equilibrium state. Ideal gas: the gas which follows the Boyle’s law, the Gay- Lussac’s law, the charles’ law, and the Aavogadro’s law 12-1 The Equilibrium State, the Zero Law of Thermodynamics

9 Mole gas constant: for the gas with a certain quantity of gas at equilibrium: One equation of the state of the ideal gas: 12-1 The Equilibrium State, the Zero Law of Thermodynamics

10 k is Boltzmann constant n =N/V , the number density of molecules 12-1 The Equilibrium State, the Zero Law of Thermodynamics Another equation of the state of the ideal gas:

11 12-1 The Equilibrium State, the Zero Law of Thermodynamics IV.The Zeroth law of thermodynamics: If A and B are in thermal equilibrium with C, which is in a certain state, respectively, then A and B are in thermal equilibrium each other.

12 I.The scale of molecules and molecular forces: Molecules : including monatomic ~, diatomic ~, polyatomic ~. For example: the oxygen molecules under the standard state Diameter: Distances between gas molecules The diameter of the molecules 12-2 The Microscopic Model of Matter, the law of Statistics Therefore, molecules with different structures have different scales

13 Molecular force 1.when rr 0, the molecular force is mainly attractive; 3. When r  10 -9 m, F  0 12-2 The Microscopic Model of Matter, the law of Statistics II.Molecular force:

14 Thermal motion: large amounts of experimental facts indicate that all molecules move irregularly thermally. for example: oxygen molecules under the normal temperature and normal pressure. 12-2 The Microscopic Model of Matter, the law of Statistics III.The disorder and the statistical regularity of the thermal motion of Molecular

15 ............................................................... 12-2 The Microscopic Model of Matter, the law of Statistics The distribution of the small balls in the Gordon board

16 When the Number of the small balls N  ∞ , the distribution of the small balls shows the statistical regularity........................................... 12-2 The Microscopic Model of Matter, the law of Statistics Statistical regularity

17 12-2 The Microscopic Model of Matter, the law of Statistics Suppose: N i is the number of the small ball in the ith slot, then the total number of the small balls N satisfies: Normalizing condition: Probability : the probability of the small ball which appeared in ith slot

18 1.The size of the molecule itself is negligible compared with the average distance between molecules, and molecules can be viewed as mass points: I.The microscopic model of the ideal gas 12-3 The Pressure Formula of the Ideal Gas 2. Other than the moment of collision, the interaction forces between molecules are negligible. 3. The Collisions between molecules can be viewed as complete elastic collisions. 4.The motion of the molecules follows classical laws.

19 Assume that there is a rectangular container with the side lengths being x, y, and z, in the container there are N gas molecules of the same kind. The mass of each molecule is m. Now we calculate the pressure on wall A 1 perpendicular to the Ox axis 12-3 The Pressure Formula of the Ideal Gas II.The pressure formula of the ideal gas

20 The statistical regularity of the thermodynamic equilibrium: (1 ). the spatial distribution of molecules is uniform: Total effect of all the large quantities of molecules: continuous force. Colliding effect of one molecules: accidental, discrete 12-3 The Pressure Formula of the Ideal Gas

21 The average value of the squares of the velocity components along the Ox aixs: Each molecule moving in any direction is equal: Velocity of the molecule: 12-3 The Pressure Formula of the Ideal Gas (2 ). The probability of each molecule moving in any direction is equal and there is no preferred direction

22 The impulse of the force acted by the molecule on the wall: Momentum increment on the Ox axis: Each molecule follows the mechanical law 12-3 The Pressure Formula of the Ideal Gas

23 Therefore, the total impulse of a molecule acted on the wall in the unit time interval: The time between two consecutive collision: The number of collisions in the unit time interval: 12-3 The Pressure Formula of the Ideal Gas

24 total impulse of N molecules acted on wall in the unit time: total effect of a large quantities of molecules: i.e., the average force on wall A 1 is: 12-3 The Pressure Formula of the Ideal Gas

25 Pressure on wall: Statistical regularity: Molecular average translational kinetic energy: Pressure formula of the ideal gas: 12-3 The Pressure Formula of the Ideal Gas

26 Statistical relationship Physical significance of the pressure Observable macroscopic quantities statistical average value of the microscopic quantity 12-3 The Pressure Formula of the Ideal Gas

27 Observable macroscopic quantities statistical average value of the microscopic quantity Equation of the state of the ideal gas: 12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas Pressure formula of the ideal gas: Molecular average translational kinetic energy:

28 Physical significance of T: (1). Temperature is the measurement of the average translational kinetic energy of large quantities of molecules: 12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas (2). Temperature is the collective behavior of the thermal motion of large numbers of molecules. (3). The average translational kinetic energies in the same temperature are the same.

29 difference between the thermal motion and the macroscopic motion: T is the macroscopic statistical physical quantity expressing the degree of the irregular motion of molecules, and is nothing to do with macroscopic motion of the macroscopic object. Noted: 12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas

30 A.They are in the same temperatures and the same pressures; B.Not only their temperatures but also the pressures are different; C.Temperature is the same, but pressure of He is larger D.Temperature is the same, but pressure of N 2 is larger Solution: Problem 1: two bottles of gas with the same density, one is He, another is N 2, they all in equilibrium state with the same average translational kinetic energy, then ( ) discussion 12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas

31 ( A ) ( B ) ( C ) ( D ) 12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas Problem 2: Ideal gas with state parameters V, p, T, the mass of each molecule is m, k is the Boltzmann constant, R is the mole gas constant, then the total number of the molecules is ( ) Solution:

32 I. Degrees of freedom the average energy of mono-atomic molecules: 12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas

33 rigid diatomic molecule: the average translational kinetic energy: 12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas the average rotational kinetic energy:

34 number of degrees of freedom 12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas Degrees of freedom : the number of independent velocity or coordinate square terms in the energy expression of the molecule as the number of degrees of freedom of the energy of the molecule, or simply degrees o freedom, denoted by symbol i translation rotationvibration

35 Mono-atomic molecule 3 0 3 Diatomic molecule 3 2 5 polyatomic molecule 3 3 6 Degrees of freedom of the energy of the rigid molecules molecule i translation rotation total 12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas

36 the average energy of a molecule can be expressed as: 12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas II. the theorem of equipartition of energy When a gas is at an equilibrium state the average energy of each degree of freedom is equal to the average energy of every other degree of freedom, and it is kT/2, this is the theorem of equipartition of energy per degree of freedom.

37 The internal energy of the ideal gas: the sum of the kinetic energies of the molecules and the atomic potential energies within each molecule The internal energy of one mole of the ideal gas: the change of internal energy of the ideal gas: 12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas II.The internal energy of the ideal gas The internal energy of the ideal gas with the substance quantity ν is :

38 Experimental device Metal vapor Display screen Narrow slit Connect to pump 12-6 The Law of Maxwell Speed Distribution of Gas Molecules I.The experiment of measuring the speed distribution of gas molecules

39 The scenario of molecular speed distribution N : total number of molecules △ N:number of molecules in the speed interval v  v+ △ v 12-6 The Law of Maxwell Speed Distribution of Gas Molecules Ratios of the number of molecules with speeds in between v  v+ △ v to the total number of molecules

40 The distribution function of speed: 12-6 The Law of Maxwell Speed Distribution of Gas Molecules

41 Physical significance of f(v): Under the equilibrium state with temperature T, f(v) represents the ratio of number of molecules in unit speed interval around v to the total number of molecules. Physical significance of f(v)dv: Ratios of the number of molecules with speeds in between v  v+ △ v to the total number of molecules

42 12-6 The Law of Maxwell Speed Distribution of Gas Molecules the number of molecules with speeds in between v  v+ △ v : the number of molecules with speeds in between v 1  v 2 : Ratios of the number of molecules with speeds in between v 1  v 2 to the total number of molecules:

43 The Maxwell speed distribution law The relationship curve between f(v) and v 12-6 The Law of Maxwell Speed Distribution of Gas Molecules II.The Maxwell speed distribution law of gas molecules

44 (1).the most probable speed We get: 12-6 The Law of Maxwell Speed Distribution of Gas Molecules III.The three statistical speeds:

45 Physical significance: 12-6 The Law of Maxwell Speed Distribution of Gas Molecules At a certain temperature, the relative number of molecules distributed in the vicinities of the most probable speed v p is the most.

46 46 12-6 The Law of Maxwell Speed Distribution of Gas Molecules (2). The average speed:

47 12-6 The Law of Maxwell Speed Distribution of Gas Molecules (3). The root mean square speed:

48 Comparison of the three statistical speeds: 12-6 The Law of Maxwell Speed Distribution of Gas Molecules

49 12-6 The Law of Maxwell Speed Distribution of Gas Molecules The speed distributions of N 2 molecules under two different temperature The speed distributions of N 2 and H 2 molecules under the same temperature

50 (1)(1) (2)(2) Solutions: 12-6 The Law of Maxwell Speed Distribution of Gas Molecules Problem 1: one type gas with the total number of molecules N, the mass of each molecule m, and the distribution function f(v), please find out: (1). the number of the molecules in the speed interval (2). the sum of the kinetic energy of all the molecules in the speed interval discussion

51 2 000 12-6 The Law of Maxwell Speed Distribution of Gas Molecules Problem 2: the figure is the Maxwell speed distributions of H 2 and O 2 molecules under the same temperature. Please find out the most probable speed v p for these two gas. 900

52 12-6 The Law of Maxwell Speed Distribution of Gas Molecules Solutions:

53 the free path : the path that a molecule goes through between two consecutive collisions is called ~ 12-8 The Average Number of Collisions of Molecules and the Mean Free Path

54 the mean free path : the average value of the path that the molecule goes through between two consecutive collisions is called ~ the average number of collisions per second(or the average frequency of collisions): the average number of collisions of a molecule with other molecules per unit time is called ~, denoted by 12-8 The Average Number of Collisions of Molecules and the Mean Free Path

55 simplified model 1.the molecules are rigid small balls, all the collisions are completely elastic; 2.the diameter of molecules is d 3.Assume that among all molecules only one molecule moves with the average speed, all others are at rest. 12-8 The Average Number of Collisions of Molecules and the Mean Free Path

56 The average number of collisions per second: 12-8 The Average Number of Collisions of Molecules and the Mean Free Path

57 Taking into consideration of the motion of all other molecules, then we have: The average number of collisions per second: 12-8 The Average Number of Collisions of Molecules and the Mean Free Path

58 the mean free path  when the temperature T of the gas is given, we have: 12-8 The Average Number of Collisions of Molecules and the Mean Free Path  when the pressure p of the gas is given, we have:

59 Solution: 12-8 The Average Number of Collisions of Molecules and the Mean Free Path Problem: estimate the mean free paths of air molecules under the following two circumstances: (1). 273 K and 1.013×10 5 Pa; ( 2 ) 273 K and 1.333×10 -3 Pa. (the diameter of air molecules )

60 12-8 The Average Number of Collisions of Molecules and the Mean Free Path


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