西南交通大学：《大学物理》课程教学资源（讲稿，双语）CHAPTER 14 Kinetic Theory

UNIVERSITY PHYSICS I CHAPTER 14 Kinetic theor Chapter 14 Kinetic theory The dynamics of article systems is called statistical mechanics The kinetic theory is a special aspect of the statistical mechanics of large number of particles Suitable averages of the physical characteristics and motions of individual particles provide information about the macroscopic behavior of the system as a whole

1 Chapter 14 Kinetic theory The kinetic theory is a special aspect of the statistical mechanics of large number of particles. Suitable averages of the physical characteristics and motions of individual particles provide information about the macroscopic behavior of the system as a whole. The dynamics of many-particle systems is called statistical mechanics

§14.1 The ideal gas 1. Newton's model of gas-static model The corpuscular particles of gas occupy fixed positions and filled the entire space between them Repel force F 1/r2→V个,Pc1/↓ 2. Kinetic model (Daniel Bernoulli, James Clerk Maxwell. Ludwig Boltzmann and others) Ogas is composed of many tiny particles, freely moving at high speed §14.1 The ideal gas @the pressure arises from the innumerable collisions of particles with each other and with the walls of the container. The pressure of a gas in thermal equilibrium inside a container never runs down or decreases with time the collisions must be modeled as completely elastic, the motion Is perpetua B The forces involved in the collisions of th particles of the gas with each other and with walls of the container must conservative briefly acting only during the intervals of the collisions and essentially zero otherwise

2 §14.1 The ideal gas 1. Newton’s model of gas—static model The corpuscular particles of gas occupy fixed positions and filled the entire space between them. Repel force F ∝ 1 r ⇒V ↑, P ∝ 1 V ↓ 2 2. Kinetic model (Daniel Bernoulli, James Clerk Maxwell, Ludwig Boltzmann and others) 1gas is composed of many tiny particles, freely moving at high speed. 2the pressure arises from the innumerable collisions of particles with each other and with the walls of the container. The pressure of a gas in thermal equilibrium inside a container never runs down or decreases with time, the collisions must be modeled as completely elastic, the motion is perpetual. 3 The forces involved in the collisions of the particles of the gas with each other and with walls of the container must conservative, briefly acting only during the intervals of the collisions and essentially zero otherwise. §14.1 The ideal gas

§14.1 The ideal gas o。。。。。 §14.1 The ideal gas @gases can be compressed easily. The space of the particles in gas is the order of 10 times larger than those of liquids and solids 3. Our purpose Find @the connection between temperature Tand pressure P and their microscopic essentials @the average velocity of the particles; specific heat-the microscopic essentials

3 §14.1 The ideal gas 4gases can be compressed easily. The space of the particles in gas is the order of 10 times larger than those of liquids and solids. 3. Our purpose Find: 1the connection between temperature T and pressure P and their microscopic essentials; 2the average velocity of the particles; 3specific heat—the microscopic essentials. §14.1 The ideal gas

§14.1 The ideal gas 4. The properties of ideal gas(model The ideal gas equation of state PV=nRT Othe number of particles N in the gas is very large for instance 10-2mol×6.02×1023/mol=6.02×101l 2NI particle <<y, the volume occupied by the particles themselves is a negligibly small fraction of the volume containing the gas. @all the particles are in random motion and obey Newtons law of motion §14.1 The ideal gas ④ the particles are ally likely to be moving in any direction--symmetry @the gas particles interact with each other and with the walls of the container only via elastic collisions. No force act on a particle except during a collision. all collisions elastic and of negligible duration @the gas is in thermal equilibrium with its surroundings Othe particles of the gas are identical and indistinguishable

4 4. The properties of ideal gas (model) The ideal gas equation of state: PV = nRT 2NVparticle<<V, the volume occupied by the particles themselves is a negligibly small fraction of the volume containing the gas. 1the number of particles N in the gas is very large, for instance 12 23 11 10 mol × 6.02×10 / mol = 6.02×10 − 3all the particles are in random motion and obey Newton’s law of motion. §14.1 The ideal gas 4the particles are equally likely to be moving in any direction--symmetry. 5the gas particles interact with each other and with the walls of the container only via elastic collisions. No force act on a particle except during a collision. All collisions elastic and of negligible duration. 6the gas is in thermal equilibrium with its surroundings. 7the particles of the gas are identical and indistinguishable. §14.1 The ideal gas

814. 2 the pressure and temperature of an ideal gas 1. The pressure of an ideal gas Othe probability that particles in element volume△ will collide Normal to with the wall 1A△L1△L1p△t P 2 AL L ② the change of the momentum of one particle Vibfr =viret mvi j+mvi k -mvi L t mvi)+ m 814.2 the pressure and temperature of an ideal gas Ap: = Piaft -Pi bft =-2mva i @the impulse given to the wall by the particles in the element volume Av F△t=-AD1=2mvai mmy 2 △t 2 Othe total impulse delivered to the wall during the interval At by all the particles that collide with the wall F△t △i=N〈v)A L L LT lotal number of particles

5 §14.2 the pressure and temperature of an ideal gas 1. The pressure of an ideal gas 1the probability that particles in element volume ∆V will collide with the wall L v t L L AL A L p x∆ = ∆ = ∆ = 2 1 2 1 2 1 2the change of the momentum of one particle i ˆ j ˆ m m v mv i mv j mv k v mv i mv j mv k ix iy iz ix iy iz ˆ ˆ ˆ ˆ ˆ ˆ i aft i bfr = − + + = + + r r ∆L A p p p mv i i ix ˆ 2 ∆ = i aft − i bft = − r r r 3the impulse given to the wall by the particles in the element volume ∆V ti L mv mv i L v t F t p mv i ix ix ix i i ix ˆ ˆ 2 2 ˆ 2 2 = ∆ ∆ ∆ = −∆ = r r 4the total impulse delivered to the wall during the interval ∆t by all the particles that collide with the wall N v ti L m v ti L m F t ix x i ˆ ˆ ( ) 2 2 ave∆ = ∑ ∆ = 〈 〉∆ r Total number of particles §14.2 the pressure and temperature of an ideal gas

814. 2 the pressure and temperature of an ideal gas dthe symmetry implies that +1 (2)=(+v2+v2>=3(v2 3 n ave N(v2) L 3 @pressure P exerted by the gas on the wall P= A 3AL N()=mN()=m 814.2 the pressure and temperature of an ideal gas 2. The absolute temperature an ideal gas Othe microscopic interpretation of the absolute temperature NI 2N1 ∴P my 3丿2 ∴P=Mm (v>=NkT T 3 3k KE mv 2 2 T=,KEa0rKEa。=。kT 3k 6

6 5the symmetry implies that = 〈 〉 〈 〉 = 〈 〉 〈 〉 = 〈 + + 〉 = 〈 〉 ⋅ = = + + 〈 〉 = 〈 〉 = 〈 〉 2 ave 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 3 1 3 N v L m F v v v v v v v v v v v v v v v v x x y z x x y z x y z r r 6pressure P exerted by the gas on the wall = = 〈 〉 = 〈 〉 = 〈 〉 ave 2 2 2 A 3 3 3 v V nM N v V m N v AL F m P §14.2 the pressure and temperature of an ideal gas 2. The absolute temperature an ideal gas 1the microscopic interpretation of the absolute temperature = 〈 〉 = 〈 〉 2 2 2 1 3 2 3 m v V N v V Nm Q P v NkT Nm ∴ PV = 〈 〉 = 2 3 = 〈 〉 2 3 v k m T KE KE kT k T KE m v 2 3 or 3 2 2 1 ave ave 2 ave ∴ = = Q = 〈 〉 §14.2 the pressure and temperature of an ideal gas

814. 2 the pressure and temperature of an ideal gas KE kT 2 Conclusions: a. The temperature of all the gas is a manifestation of the average translational kinetic energy of each particle. b. a measurement of temperature is a measurement of the average translational kinetic energy of any particle of the gas. c. Two different gases at same temperature have the same average translational kinetic energy per particle. 814.2 the pressure and temperature of an ideal gas ② the rms speed :T=(v2)∴〈y2) 3kT 3k 3kT 3RT、123RT rms A M The average speed ∑ ∑ 1/2 rms <1

7 KE kT 2 3 ave = Conclusions: a. The temperature of all the gas is a manifestation of the average translational kinetic energy of each particle. b. A measurement of temperature is a measurement of the average translational kinetic energy of any particle of the gas. c. Two different gases at same temperature have the same average translational kinetic energy per particle. §14.2 the pressure and temperature of an ideal gas 2the rms speed m kT v v k m T 3 3 2 2 Q = 〈 〉 ∴ 〈 〉 = 1 2 1 2 1 2 rms ) 3 ) ( 3 ) ( 3 ( M RT mN RT m kT v A = = = The average speed rms 1 2 2 2 rms ( ) v v N v v v N v v i i i i ∴〈 〉 < 〈 〉 = = 〈 〉 = ∑ ∑ Q §14.2 the pressure and temperature of an ideal gas

补充内容:统计方法的一般概念 一、统计规律—大 大量偶然事件整体所遵从的规律 不能预测多次重复 例:伽尔顿板实验 掷骰子 抛硬币 补充内容:统计方法的一般概念 每个小球落入哪个槽是偶然的 伽尔顿板实验了少量小球按狭槽分布有明显偶然性 大量小球按狭槽分布呈现规律性 每掷一次出现点数是偶然的 掷骰子{掷少数次,点数分布有明显偶然性 掷大量次数,每点出现次数约1/6,呈现规律 每抛一次出现正反面是偶然的 抛硬币抛少数次,正反数分布有明显偶然性 抛大量次数,正反数约各1/2,呈现规律性 8

8 补充内容：统计方法的一般概念 一、统计规律 ——大量偶然事件整体所遵从的规律 不能预测 多次重复 掷骰子 抛硬币 例： 伽尔顿板实验 伽尔顿板实验 每个小球落入哪个槽是偶然的 少量小球按狭槽分布有明显偶然性 大量小球按狭槽分布呈现规律性 掷骰子 每掷一次出现点数是偶然的 掷少数次，点数分布有明显偶然性 掷大量次数，每点出现次数约1/6，呈现规律 抛硬币 每抛一次出现正反面是偶然的 抛少数次，正反数分布有明显偶然性 抛大量次数，正反数约各1/2，呈现规律性 补充内容：统计方法的一般概念

补充内容:统计方法的一般概念 共同特点: 群体规律:只能通过大量偶然事件总体显示出 来 2、量变质对数地位 统计规律≠近似规律 统计规律≠个体规律简单叠加 例:理想气体实验定律,传真照片 3、与宏观条件相关 如:伽尔顿板中钉的分布 4、伴有涨落 补充内容:统计方法的一般概念 、统计规律的数学形式—概率理论 定义:总观测次数 出现结果A次数NAN A出现的概率W,=li 2、意义:描述事物出现可能性的大小 两类物理定律<第一类:约束不可能事件 第二类:约束可能性小事件 会沸 例:一壶水在火上腾结冰? 违反能量守恒定律的事件不可能发生 不违反能量守恒定律的事件并不都能发生

9 共同特点： 1、群体规律：只能通过大量偶然事件总体显示出 来， 对少数事件不适用。 统计规律 ≠ 近似规律 统计规律 ≠ 个体规律简单叠加 2、量变—质变：整体特征占主导地位 例： 理想气体实验定律， 传真照片 …... 3、与宏观条件相关 如： 伽尔顿板中钉的分布 4、伴有涨落 补充内容：统计方法的一般概念 二、统计规律的数学形式 —— 概率理论 1、定义： 总观测次数 N 出现结果A次数 NA A出现的概率 N N W A A = lin N → ∞ 2、意义： 描述事物出现可能性的大小 两类物理定律 第一类： 约束不可能事件 第二类： 约束可能性小事件 违反能量守恒定律的事件不可能发生 不违反能量守恒定律的事件并不都能发生 例： 一壶水在火上 会沸 腾？会结冰？ 补充内容：统计方法的一般概念

补充内容:统计方法的一般概念 性质 1)叠加定理 不可能同时出现的事件互斥事件 出现几个互斥事件的总概率等于每个事件单独出 现的概率之和:W=W4+W 出现所有可能的互斥事件的总概率为1 归一化条件: Idw=1 例:掷骰子出现2:W2=1 6 : n 2+3 3 出现16:w=1 补充内容:统计方法的一般概念 2)乘法定理 相容统计独立事件:彼此独立,可以同时发生的事件 同时发生两个相容独立事件的概率是两个事件单 独发生时的概率之积 W A+B =wxW 例:同时掷两枚骰子 其一出现 同时发生 w2 6 1 另一出现 636 W3= 3:

10 3、性质 1)叠加定理 不可能同时出现的事件——互斥事件 出现几个互斥事件的总概率等于每个事件单独出 现的概率之和： WA+B = WA + WB 出现所有可能的互斥事件的总概率为1 归一化条件： = 1 ∫ +∞ −∞ dW 例：掷骰子 出现 6 1 3 6 1 2 3 2 = = W W : : 3 1 W2+3 = 出现1—6： w =1 补充内容：统计方法的一般概念 2) 乘法定理 同时发生两个相容独立事件的概率是两个事件单 独发生时的概率之积 WA+B = WA ×WB 相容统计独立事件： 彼此独立，可以同时发生的事件 例：同时掷两枚骰子 其一出现 2： 6 1 W2 = 另一出现 3： 6 1 W3 = 同时发生 36 1 6 1 6 1 W2+3 = × = 补充内容：统计方法的一般概念