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**Department of Physics Shanghai Normal University**

Chapter 12 Gas Kinetics Department of Physics Shanghai Normal University

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

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

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**Microscopic quantities Macroscopic quantities**

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

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12-1 The Equilibrium State, the Zero Law of Thermodynamics 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 ×105Pa 2.Volume(V) : reachable space unit: 3.Temperature(T) : measure of the coldness or hotness of an object unit:

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**The Equilibrium State, the Zero Law of Thermodynamics**

12-1 The Equilibrium State, the Zero Law of Thermodynamics Equilibrium State(E-S) 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 ~ Vacuum expansion

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**Characteristic of E-S:**

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

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

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**The Equilibrium State, the Zero Law of Thermodynamics**

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

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**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: k is Boltzmann constant n =N/V，the number density of molecules

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**END 12-1 The Equilibrium State, the Zero Law of Thermodynamics**

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. END

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**The diameter of the molecules**

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

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**The Microscopic Model of Matter, the law of Statistics**

12-2 The Microscopic Model of Matter, the law of Statistics Molecular force: 1.when r<r0, the molecular force is mainly repulsive; 2.when r>r0, the molecular force is mainly attractive; 3. When r10-9m, F0 Molecular force

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12-2 The Microscopic Model of Matter, the law of Statistics The disorder and the statistical regularity of the thermal motion of Molecular 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.

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

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**Statistical regularity**

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

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**Normalizing condition:**

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

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**12-3 The Pressure Formula of the Ideal Gas**

The microscopic model of the ideal gas 1.The size of the molecule itself is negligible compared with the average distance between molecules, and molecules can be viewed as mass points: 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.

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**12-3 The Pressure Formula of the Ideal Gas**

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 A1 perpendicular to the Ox axis

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**12-3 The Pressure Formula of the Ideal Gas**

Colliding effect of one molecules: accidental, discrete Total effect of all the large quantities of molecules: continuous force. The statistical regularity of the thermodynamic equilibrium: (1). the spatial distribution of molecules is uniform:

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**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 Velocity of the molecule: Each molecule moving in any direction is equal: The average value of the squares of the velocity components along the Ox aixs:

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**Each molecule follows the mechanical law**

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

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**12-3 The Pressure Formula of the Ideal Gas**

The time between two consecutive collision: The number of collisions in the unit time interval: Therefore, the total impulse of a molecule acted on the wall in the unit time interval:

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**total effect of a large quantities of molecules:**

12-3 The Pressure Formula of the Ideal Gas total effect of a large quantities of molecules: total impulse of N molecules acted on wall in the unit time: i.e., the average force on wall A1 is:

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**12-3 The Pressure Formula of the Ideal Gas**

Pressure on wall: Statistical regularity: Molecular average translational kinetic energy: Pressure formula of the ideal gas:

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**Statistical relationship**

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

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**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: Equation of the state of the ideal gas: Molecular average translational kinetic energy: Observable macroscopic quantities statistical average value of the microscopic quantity

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**Physical significance of T:**

12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas Physical significance of T: (1). Temperature is the measurement of the average translational kinetic energy of large quantities of molecules: (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.

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**difference between the thermal motion and the macroscopic motion:**

12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas Noted: 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.

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**12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas**

discussion Problem 1: two bottles of gas with the same density, one is He, another is N2, they all in equilibrium state with the same average translational kinetic energy, then ( ) They are in the same temperatures and the same pressures; Not only their temperatures but also the pressures are different; Temperature is the same, but pressure of He is larger Temperature is the same, but pressure of N2 is larger Solution:

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**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 ( ) （A） （B） （C） （D） Solution: END

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**12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas**

Degrees of freedom the average energy of mono-atomic molecules:

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**rigid diatomic molecule:**

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

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**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 number of degrees of freedom translation rotation vibration

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**Degrees of freedom of the energy of the rigid molecules**

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

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**the theorem of equipartition of energy**

12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas 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. the average energy of a molecule can be expressed as:

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**12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas**

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 internal energy of the ideal gas with the substance quantity ν is： the change of internal energy of the ideal gas: END

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**The Law of Maxwell Speed Distribution of Gas Molecules**

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

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**The Law of Maxwell Speed Distribution of Gas Molecules**

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

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

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**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

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**The Law of Maxwell Speed Distribution of Gas Molecules**

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 v1 v2: Ratios of the number of molecules with speeds in between v1 v2 to the total number of molecules:

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**The relationship curve between f(v) and v**

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

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**We get: 12-6 The Law of Maxwell Speed Distribution of Gas Molecules**

The three statistical speeds: (1).the most probable speed We get:

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

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**The Law of Maxwell Speed Distribution of Gas Molecules**

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

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**The Law of Maxwell Speed Distribution of Gas Molecules**

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

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

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**The Law of Maxwell Speed Distribution of Gas Molecules**

12-6 The Law of Maxwell Speed Distribution of Gas Molecules The speed distributions of N2 molecules under two different temperature The speed distributions of N2 and H2molecules under the same temperature

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**（1） （2） 12-6 The Law of Maxwell Speed Distribution of Gas Molecules**

discussion 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 Solutions: （1） （2）

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**The Law of Maxwell Speed Distribution of Gas Molecules**

12-6 The Law of Maxwell Speed Distribution of Gas Molecules Problem 2: the figure is the Maxwell speed distributions of H2 and O2 molecules under the same temperature. Please find out the most probable speed vp for these two gas. 2 000 900

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**END 12-6 The Law of Maxwell Speed Distribution of Gas Molecules**

Solutions: END

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

the free path ： the path that a molecule goes through between two consecutive collisions is called ~

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

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

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

simplified model the molecules are rigid small balls, all the collisions are completely elastic; the diameter of molecules is d Assume that among all molecules only one molecule moves with the average speed , all others are at rest.

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

The average number of collisions per second:

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

Taking into consideration of the motion of all other molecules, then we have: The average number of collisions per second:

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

when the temperature T of the gas is given, we have: when the pressure p of the gas is given, we have:

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**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 1.013×105 Pa; （2） 273 K and 1.333×10-3 Pa. (the diameter of air molecules ) Solution:

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

END

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