理想气体压强公式oraveragetranslationalkineticenergy平均平动.ppt

1、University Physics,Part Two Thermodynamics 第 二 篇 热学,Chapter 12 The Kinetic Theory of Gases 气体动理论,12-1 Essential Concept of the Kinetic Theory of Gases 气体动理论的基本概念,12-3 Representation of Pressure for Ideal Gas 气体动理论的压强公式,12-4 Average Translational Kinetic Energy and Temperature 理想气体的平均平动动能和温度公式,12-2 S

2、tate Parameters Equilibrium State Ideal Gas Law 状态参量 平衡态 理想气体状态方程,12-6 Maxwell Speed Distribution 麦克斯韦速率分布律,12-7 Mean Free Path & Average Collision Rate分子的平均碰撞次数及平均自由程,12-8 Boltzmann Distribution 玻耳兹曼分布律,12-5 Equipartition Theory of Energy Internal energy 能量均分定理、理想气体的内能,1. 对分子无规则热运动有一个清晰的图景；2. 掌握气体分

3、子运动论的两个基本公式理想气体的压强公式及平均平动动能与温度的关系式，理解压强和温度的微观解释；3.掌握能均分原则和理想气体内能公式；4.理解麦克斯韦速率分布律，明确分布曲线的物理意义；5. 理解分子的平均自由程和平均碰撞次数的规律。,教学基本要求,一、重要人物：,焦耳：J.P.Joule 1818-1889，英国物理学家，职业是酿酒商。是发现能量守恒与转换定律的代表人物，英国皇家学会会员，法国科学院院士。,卡诺：Carnot，17961832，法国物理学家，工程师，热力学奠基人，提出卡诺循环和卡诺定理。,克劳修斯：Clausius,18221888,德国物理学家，提出热力学第二定理（1850

4、），提出熵的概念（1865），给出理想气体压强公式。,麦克斯韦：Maxwell,18311879,英国物理学家，数学家，主要贡献：（1）电磁理论；（2）热学：麦克斯韦速度分布；各态经历假说。,玻耳兹曼：Boltzmann, 18441906,奥地利物理学家，主要贡献：能量分布定律和热力学第二定理的统计解释。,吉布斯：Gibbs,18391903,美国物理学家，化学家，现代化学热力学和统计物理学的奠基人。,热能的应用带来了第一次工业革命！,人类对热的认识：,（1）古代：水、木、金、火、土；,（2）热质说：是没有质量的流质；,（3）热的分子学说：是物质运动的一种表现，即分子运动的表现。,研究方法,

5、The studied object（对象） is :,containing a vast（大量） number of particles:,a system,molecules,The adopted methods are:,(1) Thermodynamics（热力学）: study heat phenomena in the view of energy transformation based on some experimental laws.,(2) Statistical Mechanics or physics（统计力学或统计物理）: based on the mechani

6、cs law and the statistical（统计） theory.,12-1 Essential Concepts of the Kinetic Theory of Gases 气体动理论的基本概念,1. All matters consist of myriad （无数） of tiny微小 molecules which are are separated,A mole of any pure substance contains a definite number of identical molecules , which is called Avogadros consta

7、nt N0 ,denoted by,太多！地球村：60亿！,2.The molecules are moving forever and random(无序）,Famous experimentBrownian Motion.,无规运动,Pollen grain花粉,3. There is interaction between molecules.,The force is attractive when their distance is less than r0; otherwise, it is repulsive(排斥的）,The interaction force between

8、molecules varies with the distance between molecules, as shown in Fig,12-2 State Parameters Equilibrium State Ideal Gas law 状态参量 平衡态 理想气体状态方程,1. State Parameters of Gas 状态参量,For a gas, in order to describe its properties , three parameters（参数） are needed:,(1) volume V: geometric parameter， cubic met

9、ers m3,(2) pressure P: mechanical parameter, Pascal: N/m2,(3) temperature T: thermal parameter, K or C,Note: 1litter （1公升） =,These parameters are macroscopic quantities(宏观量）, which are called as state parameter of the system.,2. Equilibrium state（平衡） and Equilibrium process,The gas, mass of m, is co

10、ntained in a vessel（容器） and has a volume of V. If the gas has a same pressure and temperature everywhere, the gas is said to be in an equilibrium state.,力学平衡：P化学平衡：m热学平衡：T,对气体的平衡态：（P，V，T）,A system(gas) in equilibrium state can be described by its state parameters(P,V,T) among which there is a relati

11、onship(关系). In some cases, it is simple enough that it can be expressed（表示成） in the form of an equation,Hence, there are two variables to be independent.,A equilibrium state can be represented by a dot（点） on the pressurevolume diagram (briefly p-V diagram), shown in the following Figure,Thermodynami

12、c process(热力学过程):the operation(操作） of changing the system from its initial state(a) to its final state(b) is called a thermodynamic process as shown in above figure.,Equilibrium process平衡过程：In an equilibrium process,the system remains approximately in thermodynamic equilibrium at all stages, represe

13、nted by a smooth curve on p-V diagram.,Is a real process considered as Equilibrium process?,3. The ideal gas law 理想气体状态方程,which is the equation of state of ideal gas, and holds for equilibrium state. Here, R is a constant and has the same value for all gases ,called ideal gas constant(理想气体普适常数):,M i

14、s mole mass.,For the ideal gas of mass M, its state parameters have a simple relationship,in SI units.,k 称为玻耳兹曼常量.,n =N/V，为气体分子数密度.,理想气体物态方程二,四 热力学第零定律 如果物体 A 和 B 分别与物体 C 处于热平衡的状态，那么 A 和 B 之间也处于热平衡.,一 分子的线度和分子力,分子有单原子分子、双原子分子、多原子分子和千万个原子构成的高分子.,不同结构的分子其尺度不一样,二分子力,当 时，分子力主要表现为斥力；当 时，分子力主要表现为引力.,分子力为短

15、程力，属于电磁相互作用,利用扫描隧道显微镜技术把一个个原子排列成 IBM 字母的照片.,对于由大量分子组成的热力学系统从微观上加以研究时, 必须用统计的方法.,三分子热运动的无序性及统计规律,热运动：大量实验事实表明分子都在作永不停止的无规运动 .,例 常温和常压下的氧分子,小球在伽尔顿板中的分布规律 .,统计规律 当小球数 N 足够大时小球的分布具有统计规律.,设 为第 格中的粒子数,归一化条件,粒子总数,概率 粒子在第 格中出现的可能性大小,12-3 Pressure Formula for the Ideal Gas 理想气体的压强公式,1. The microscopic model

16、of an ideal gas(理想气体模型）, A gas consists of a very number of rapidly moving molecules which are so small that the volume of the molecules is negligible(忽略的） compared with(与.相比) the volume of gas. That is the size of molecules is more smaller than the distance between molecules so that the volume of t

17、he molecules is negligible.(与分子间的距离相比较，分子的大小可以忽略不计)。, Molecules exert no force to each others except for the instantaneous impulsive force during the collision with the wall of the container and the collision with each others(除碰撞的瞬间外，分子间没有相互作用力，分子自由运动)., Molecules are in constantly random（无规则的） moti

18、on, colliding with each other and with the wall of the container; all collisions are perfect elastic.（分子与分子间，分子与器壁间的碰撞都是完全弹性的）。,This implies that there is not any loss of energy.,. The motion of an individual（单个） molecule obeys （遵守）Newtons law.（单个分子的运动遵从牛顿力学）。,The statistical hypothesis(假设）:,When th

19、e gas is in the equilibrium state, the probability(概率） at which molecule move in all the direction is same.,（1）分子向各方向运动机会相等；,（2）分子数密度分布均匀；,（3）分子速度在各方向分量平均值相等，等于零:,In detail（详细）, it includes,（4）分子速度在各方向分量平方的平均值相等,And it can be proved that,Question: How are the statistical average made?,For a single p

20、article, its motion and state is described by the following: position, displacement, velocity, acceleration, momentum, kinetic energy, et.al，which are called as micro-quantity(微观量）。,For a system of gas, its state is decided by macro-quantity (P,V,T).,Question: what relation is there between macro-qu

21、antities and micro-quantities?,(V,P,T),2. Derivation of pressure equation of ideal 压强公式的推导,推导依据：理气微观模型和统计假设。,立方容器，三边长分别为x,y,z，分子数为 N。,研究对象,个别分子碰撞器壁A是间歇的；大量分子碰撞器壁将产生持续的压力。或说压强是大量分子碰撞器壁，由于每个分子都给予器壁冲量而产生的宏观效果。,统计的观点:,研究方法：,对单个分子用牛顿运动定律求出一次碰撞给面的冲量，再对所有分子给面的冲量求和，用统计的方法求出 P 与 v2 的关系。, The impulse exerted

22、by one molecule during each collision（先考虑一个分子i撞击一次施于器壁A的冲量）:, The times n of collision by a single molecule in one second（平均碰撞一次所用时间）,(单位时间碰撞的次数）, The average force exerted by a single molecule to the wall in one second（单位时间内平均冲力）, The force on the wall by the gas(N molecules):,N 个分子单位时间内对A面的平均冲力:,(

23、5) The pressure on the wall: （统计的方法）,by using statistical hypothesis,理想气体压强公式:,or,average translational kinetic energy平均平动动能,Average translational kinetic energy平均平动动能,The physical meaning of the pressure equation:,单位体积的分子数,分子的平均平动动能,In a word:,The pressure, as a macroscopic parameter of a gas, is a

24、 statistic average quantity because the average translational kinetic energy and n in Eq have definite meaning only for a great number of molecules. Therefore pressure has definite meaning only for a system that consists a vast number of molecules.,12-4 Average Translational Kinetic Energy and Tempe

25、rature平均平动动能和温度的关系,Taking an ideal gas as example, we try to find the relationship with the microscopic quantities of molecules .,What is the temperature of a system associated with?,V the size of container in which the particles move.,P the average translatoinal kinetic energy,On the other hand , t

26、he pressure equation is given by,we can obtain,Thus the average translational kinetic energy per molecule depends only on the temperature, not on the pressure, volume, or molecular species.,(12-5),In other word, that temperature is the measurement of the average translational kinetic energy. This im

27、plies that the higher the temperature of a system, the greater the average translational kinetic energy so that “热运动越剧烈”。,炙热,大小的量度,表征大量气体分子热运动的剧烈程度,是一统计平均量,正如压强一样，温度只对大数分子组成的系统才有确定的值，温度对个别分子无意义。,温度的物理意义,从统计的观点看，温度 T 是分子的平均平动动能,Temperature is a quantity which cannot be defined in terms of mass, lengt

28、h, and time. It is the fourth fundamental quantity.,Remember that T is always absolute temperature in Kelvins in this chapter.,第四个基本量，单位（SI)： 开尔文，简称开，K。,Example : What is the average translational kinetic energy of a molecule of a gas at a temperature of 300K?,Solution:,125 Equipartition Theorem of

29、Energy Internal Energy of Ideal Gas 能量均分定理 理想气体的内能,1. Matter:,In the thermodynamics,The effects of vibration of atoms are ignored in the following.,Lets consider the size of the molecules in the gas to modify the model of ideal gas to approach to the real gas as possible as.,All the molecules are di

30、vided into monatomic（单原子）, diatomic（双原子）, and polyatomic（多原子） molecules.,For simplicity, we assume that the atoms in a molecule are “rigid balls（刚性球）” connected by the rigid rods（刚性棒） at low temperature.,Monatomic molecule: Particle model of Monatomic molecule 单原子理想气体的质点模型 ，只有平动。 Example： He ， Ar,Di

31、atomic molecule（双原子分子） Example: O2 , H2 , N2,Rigid model of diatomic molecule ( Dumbbell model哑铃模型) 双原子理想气体的刚体模型,可以平动和转动.,Polyatomic molecule(多原子分子） Rigid model of polyatomic molecule 多原子理想气体的刚体模型，可以平动和 转动。 Example: H2O , CH4 , NH3,2. The Degrees of Freedom of molecule 分子的自由度,The number of degrees o

32、f freedom is defined as the independent coordinates introduced to determine the position of a moving body in space.(物体的自由度定义为为确定其在空间的位置引入的独立座标数)。,The number of degrees of freedom is labeled as i.,i =3,Only translational motion,(1) Monatomic molecule(单原子分子）,(x,y,z),i = 5 = 3+2,(2) Diatomic molecule（双

33、原子分子）,(3) Polyatomic molecule（多原子分子）,i = 6 = 3+3,From below equations：,We have,It means that on the average(平均来看）, the energy per particle associated with each degree of translational freedom at equilibrium state is,说明：平均而言，每一个平动自由度均具有相等的动能。,The molecule of a gas takes part in rotation and vibration

34、 at the same time with the exception of（除.) the translational motion. How are the contribution（贡献） of these motions to the energy of molecule considered?,转动,The classical statistical mechanics（统计力学） proves that :,This conclusion is called as the principle of the equipartition of energy（能均分定理）.,Hence

35、, the total average kinetic energy of a molecule is ：,自由度: i,单原子分子：He, Ar, .,in which the average rotational energy is,双原子分子：H2,O2,N2,.,in which the average rotational energy is,多原子分子：Co2, NH3, .,3. Internal Energy of Ideal Gas理想气体的内能,For an ideal gas, from the principle of the equipartition of ener

36、gy, the internal energy of one mole ideal gas is,没有势能,所有分子动能之和,Therefore, the internal energy of v mole ideal gas is given by,Note: 内能是状态函数.,A monatomic ideal gas has only three translationl degrees of freedom so that its internal energy is equal to,单原子分子,Note:只有平动!,For the diatomic ideal gas, its i

37、nternal energy without that of vibration is,双原子分子,and the contribution of rotation motion is,The degrees of freedom of polyatomic molecule is six, and,多原子分子,and the contribution of rotation motion is,12-6 Maxwell Speed Distribution,麦克斯韦速率分布律,1. Introduction,(1) Some molecules are moving rapidly, and

38、 some slowly. The speed of the molecule is varying with the time and varies over a wide range of magnitude.,每个分子速度随时间变化,同一时刻不同分子速度不同,(2) The method of photography（摄影）: in a equilibrium state, there is a characteristic（特征） distribution of molecular speeds for a given gas, which depends, as we will se

39、e below, on the temperature.,Some definitions:,N: total number of molecules in a gas.,N: the number of molecules having speeds between v and v+ v.,总分子数N,N：vv+v,the ratio of the number of molecules having speeds between v and v+ v to the total number or probability at which the speeds of molecule lie

40、s between v and v+ v.,总分子数N,N：vv+v,is,where is called as distribution function（分布函数） and represents:,Obviously, we have,N：气体中总分子数；,N：速度位于vv+ v的分子数；,：速率位于vv+ v的分子占总分子的百分比，也可理解为分子速率位于vv+ v的几率；,：速率位于v附近单位间隔的分子占总分子的百分比，也可理解为分子速率位于v附近单位间隔的几率；,非常重要important!,It is called Maxwell distribution function of s

41、peed (麦克斯韦速率分布函数).,2. Maxwell Distribution of Speed 麦克斯韦速率分布律（函数）,Maxwell found the distribution function , that is,Therefore，we have,Some discussions:,(1)The number of molecules between v1 and v2 is,意义：速率位于v1,v2的分子数占总分子数的百分比；或分子速率位于v1,v2的几率。,(2) Normalization（归一化）:,equals to the area below the curv

42、e of .,面积等于1,(3) The characters of the curve:,shape and maximum point,最大点,(4) the relation between the curve and the temperature for a given gas,3. Three speeds:,They are:the most probable speed (最可几速率）the root-mean-square speed （方均根速率） average speed （平均速率）.,is the speed at which the distribution fu

43、nction has maximum value. Using the Maxwell speed distribution, we have:,which relates to the probability.,问题：最可几速率 的意义是什么？,（2）平均速率,Average speed,（3）方均根速率,root-mean-square speed,which relates to the average kinetic energy of molecule,三种速率的比较,12-7 The Mean Free Path,分子的平均碰撞次数及平均自由程,1. The mean free p

44、ath,The distance between the successive(相继） collision is called the free path. The average of free path is called the mean free path, labeled as .,Two factors influencing on the mean free path:,The size of molecule: diameter(直径） d; if the molecules is point, what does it happen?,the density of molec

45、ule of gas.,The average collision rate(平均碰撞次数） is the average number of collision per unit time a molecule suffers as it moves in the gas, labeled as . Obviously,where is the average speed of the molecules.,(1)Assuming the diameter of molecule is d and only a molecule is moving at an average speed ,

46、 others at rest, we have,横截面积,一秒钟走的距离。,(2) Considering all the molecules are moving, it can be proved that the above formula becomes：,分子之间的相对运动！,(3) The mean free path is given by,which shows that depends only on the number of molecule per unit volume.,分子大小,分子密度,Using , the mean free path can also be expressed as,