提升宝宝视觉一空间智力的良方

1、PART ONETHERMODYNAMICSChapter I General Laws of thermodynamics 热力学的基本规律热力学研究的对象是由大量微观粒子组成的宏观系统，它们与外界的相互作用表现为能量的交换和物质（粒子）的交换。由此分为三个系统，如表所示：孤立系孤立系闭闭 系系开开 系系能量交换无有有物质交换无无有QQQQ空间空间http:/www.6665.cchttp:/www.6665.cc1. Equilibrium States and Equilibrium ProcessesWe can use some parameters(参数) to describe

2、the (状态) of the thermodynamic system： a gas or a liquid: P(pressure压强), V(Volume),T(Temperature)a film of a liquid: (surface tension coefficient),(film area), Ta rod(棒): l, (cross-sectional area), f(tensile force), E(Youngs modulus)But, all these parameters describing the systems are based on “Equil

3、ibrium States”What is Equilibrium States?In the figure, a vessel(容器) is divided into two halves by a valve(阀).Left half: containing gas; Right half: evacuated(抽真空的).The question is: when the valve is opened, a stream of gas will flux(流动), what state we call this? What process the system passes? Can

4、we describe this process? If we can, How we describe?The system will finally approach a state in which all the external parameters remain constant. Such state is called equilibrium stateIf the system has gradients(梯度) of macroscopic parameters (P,V,T), such a state is referred to as a non-equilibriu

5、m state The process from a non-equilibrium state to a equilibrium state is called relaxation (弛豫) Relaxation选择物理量“密度密度”。考虑上例中两个部分的密度变化。设初始时刻的密度分别为n = n0和n = 0。平衡后的密度为共同的n。弛豫过程如图：选择物理量“温度温度”。考虑一个长棒，两端初始时刻的温度分别为T1T2。平衡后的温度为共同的T0。如图：当t = 时:)(10nnenn结论结论当时间变化为t = 时，温度的幅度变化为1/e。1）上述的两个例子为“理想的弛豫过程”；2）所选的变

6、量n和T被物理学称之为“物理控制参量”；3）上述过程实际上为“非平衡过程”；4）非平衡过程原则上不能用物理量描述；5）热力学的主要目的是研究“热力学过程”；6）实际的热力学过程会是非平衡的，怎么办？If we slow down the rate of relaxation process, and assume that there will be enough time for the parameters to equalize over the entire system. Therefore: 非平衡过程是由连续不断的无穷小间隔的平衡态构成。这种用平衡态描述非平衡过程的方法被称之为：

7、 准 静 态 quasi-state其目的是使我们可以用热力学参量描述热力学过程，因为在非平衡状态，系统的不均匀性使我们不能使用热力学参量描述热力学过程。今后在描述热力学过程的时候，均默认了其过程为准静态过程。 系统特征：由于分子处于不停的热运动之中，所谓的平衡只是一种热动平衡。宏观系统的物理量是在平衡值附近涨落的统计结果。By virtue of symmetry(由于对称性)，the process may operate in a forward and reverse direction. For reverse process an equilibrium process can b

8、e reversed in time. During the reverse process the system will pass through the same succession of states as during the forward process, but in a reversed sequence. This equilibrium process is referred to as “reversible process”.在时间上相反的过程为“逆过程”，一个过程可正可反，则为“可逆过程”。在线性代数中，一个3*3的方阵，如果其秩为2，其结论？1.行列式的值为0；

9、2.两个独立变量；3.第三个变量可以用以上两个变量线性表示。一个热力学系统可用三个变量P、V、T描述，两个独立.任意选取两个独立变量，这两个变量为“状态参量”。三个变量之间的关系称之为状态方程：f (P,V,T ) = 0 .对于均匀的(homogeneous)系统，状态参量选 P、VThe state are depicted(描绘) by points and processes by lines on a PV plane. (系统的状态用PV平面的点及过程用线描述)。Thermodynamic Units 热力学单位牛(N) =kg m s-2压强：帕斯卡Pa (N m-2) 1大气压

10、强 (pn)= 101325 Pa能量： 焦耳(J) 1J = 1N m(见中文教材P6)2. Temperature. The temperature principle 热平衡定律和温度Temperature is a rather subjective (主观的) term(术语)-degree of hotness of body. But, the term can be given a more objective meaning through a easily measured physical parameters depending on the degree of hot

11、ness.Examples: (测量温度的物质)height of the column of liquid mercury in a glass tube (汞柱); the pressure of a gas in a constant volume vessel; the resistance of a conductor; the emissive power of a red-hot body, etc.The limit of applicability(应用的限制):Mercury solidification point(凝固点);gas condensation point(

12、凝聚点)； fusion point of metal(金属熔点)。The fields of application of these thermometers overlap(重叠)。Thermostat, 恒温热源The concept of the thermostat: a thermostat has heat capacity(热容) greater than that of any reference bodies. When contact, the temperature of thermostat does not change and , on the other ha

13、nd, any reference body acquires its temperature after a short relaxation time.(恒温热源的温度不变，与其接触的物体瞬时达到恒温热源的温度)Adiabatic wall(绝热壁)：热量不能透过的壁。将物体A和B同时与热源C接触，且A、B之间用绝热壁隔开，A、B与C达到平衡后，换为透热壁，此时A与B的状态不会发生变化平衡状态下的热力学系统当A、B、C三个系统处于热平衡时，用参量p、V可以表达系统的平衡状态。当A、C和B、C两个系统分别处于热平衡时，整体可以分别描述为：0),;,(CCAAACVpVpf分别取三个变量，得到

14、的方程是：0),;,(CCBBBCVpVpf);,(CAAACCVVpFp);,(CBBBCCVVpFp上面两个方程的右边均含有Vc项。由此得到0),;,(BBAAABVpVpf归纳上式得到：),(),(BBBAAAVpgVpg结论：如果两个物体各自与第三个物体达到热平衡，它们彼此也必处在热平衡。处在热平衡状态下的热力学系统存在一个状态函数，表示它们具有相同的冷热程度 -温度 （热平衡定律，热力学第零定律）。取一个物体作为比较的标准，该物体就是温度计。与恒温热源(thermostat)接触的系统的P-V图Placing a vessel containing a gas into a ther

15、mostat, and changing the volume of the vessel with the help of a piston (活塞), we measure simultaneously the pressure(压强). Then, plotting the results on a P-V plane, a isotherm curve is obtained.The form of isotherm curve depends on the conditional temperature, of the thermostat. Let the isotherm cor

16、responding to the conditional temperature be described by the equation ( | P,V) = 0. Solving this equation with respect to , we obtain = (P,V) . 恒 温 热 源gas in vessel piston123PV与恒温热源接触的系统有什么性质呢？Changing the in a infinitely small and repeating the procedure(流程) above, different isotherm curves are ob

17、tained as the Fig.121020321PVIt has been shown experimentally that the isotherms of the same gas never intersect(相交) and, consequently, the conditional temperature is a single-valued function of state specified by parameters P and V.Experiments also shown that at levels far from the gas liquefaction

18、 points the isotherms can be approximately represented by equiangular hyperbolas PV = const (2.1)Consequently, the equation = (P,V) will be referred to as the equation of state (状态方程). If a gas satisfies Eq.(2.1) and equation of state, this gas is called thermally ideal gas or ideal gas. The real ga

19、ses close to it, when the temperature is high and density is low.Result:A postulate(基本原理) is introduced - temperature principle: There exists (more then one) function of states of a system that remains constant for any process operating in a thermostat, referred to as the conditional temperature.(在恒

20、温热源所进行的过程，保持不变的状态函数被称之为：条件温度)。3. Entropy. The Entropy PrincipleA vessel with ideally heat-insulated walls is called: adiabatic vessel (绝热容器)An equilibrium process in conditions of heat insulation is called adiabatic process. 绝 热 过 程 pistonHeat-insulated walls 绝 热 壁Varying the volume of the gas by me

21、ans of the piston and measuring the pressure, we can plot a curve on P-V plane, which is a adiabat curve.If we take the vessel out temporarily and contact with a body at higher or lower temperature, then restore the vessel and repeat the procedure described above, we can plot another adiabat curve o

22、n the P-V plane, then the other. piston T = T0T T0T 1Each adiabat intersects each isotherm at one and only one point. At that point, PV = a, and PV = b. Solution:01202040intersecting pointPV1111,baPabVResult:A postulate(基本原理)- the entropy principle: There exists a single-valued function of state, re

23、maining constant for any processes operating in a thermally-insulated vessel, called the conditional entropy. It depends only on the product PV . “绝热过程中存在着一个单值的、不变的状态函数 - 条件熵,它仅仅依赖于乘积项PV ”4. Absolute Temperature and Absolute Entropydefine accurately the transformations = f1( ) and = f2(), which is m

24、onotonically increasing function of their arguments.Preceding 前面的， monotonically 单调地， argument 自变量According to the one-to-one correspondence of the pairs of variables P-V and - on the P-V and - planes. The difference of the Jacobian transformation is applicable),(),(VPDThe geometrical meaning of the

25、 Jacobian lies in that its module gives the variation factor from P-V plane to - plane. There is a complete equivalence for the two planes, and the | D | = 1. This normalization (归一化) leads to a defined selection of the temperature and entropy scale.PV均匀有确定值，根据| D | = 1，可以推断，- 也应该有确定值，即绝对值:即PVD的确定导致

26、了、的确定值为ST。 absolute temperature and absolute entropy.ddDdPdV|由 (4.1)可以确定T、S：在等温方程和绝热方程已知的条件下， 由P和V可以确定T和S。以理想气体作为工作介质，确定温度和熵： 理想气体的温度依赖于PV，设 PV = x理想气体的熵依赖于PV，设 PV = y则有：T = T(x), S = S (y) x和y被看成是相互无关的独立变量。由此导出：1),(),(1),(),(VPSTSTVP)2 . 4() 1(1, 1) 1(),(),(ySTySTVPST(4.2)式的意义：T 和S 表示它们对自变量的偏导数: T

27、= T / x, S = S / y.(4.2)式的右边方程中，the right-hand side depends only on y and left-hand side only on x. Each side is equal to 1/R)3 . 4(,) 1()(,1)(yRySRxT(4.3)式的求解：)4.4()ln(1,CPVRSRPVT如果我们知道了任意一点的状态S0、P0、V0、T0，就可以求出常数C，导出S的表达式；由热容量的测量结果，清楚地知道 = 0。最后得到结论：由此确定了理想气体的绝对温度和绝对熵Now, look at on P-V and T-S plan

28、es the isochore, isobar, isotherm, isoentropic curves (adiabat) for a perfect gas.(V、P、T、S = const)6.4(ln1,10010PTPTRSSRTPVS = constT = constP = constV = constPVS = constT = constP = constV = constTS On the T-S plane, P and V are expressed by the equation:Eliminating V or P from above equations, the

29、 isochores and isobars on the T-S plane:RSeconstPVRTPV/) 1(,RSRSePTTconstPeVTTconstV/) 1(2/) 1(1)(,)(,Which is the final expressionHomework : question one and two. 5. Work 功In mechanics(力学)， the work is determined by formula:dxfA In thermodynamics, a perfect gas undergoes expansion or compression(压缩

30、), the work is determined by the quantities P and V, formula:21)2.5(,VVPdVAPdVAV1V2PV此表达式为系统对外作功。不同的路径，系统对外作功是否相同？The pressure of a gas is not determined only by V, but by the state of the gas(V and T) .So, the pressure is concerned with the process, consequently, the work involved in the process is

31、 not determined by the initial and final states of the gas but depends on the entire course of the process. A denoting by the infinitely small amount of work.Results:T,S,P,V are functions of the state of the gas.Work is a function of the process, expressing the compression or expansion of the gas .几

32、个过程所作的功1)the isothermal process: 4)the isochoric process: 3)the isobaric process: 2)the adiabatic process: VVPPeiPVVPVP112211.,)(11112111121VVRTdVVVPAVVSVVPPeiPVVPVP112211.,1211ln12121VVRTdVVVPPdVAVVVVTVVV2121)(12VVPVVPdVPAPPP210VA作业：英Page 25 Answer the Problem 中P66：1.6，1.7， 1.96. Adiabatic and Isot

33、hermal PotentialsIn one dimension case of the mechanics, the force f is a function of coordinate x, the work , A = f d x, can be represented as the decrease of a function U(x), which is referred to the potential of force f or potential energy.Mechanics:力学 , coordinate: 坐标, analogy:类比 A = f d x= - d

34、U(x), f = -U/ xU(x) is an opposite sign of work; 因为是对外作功In thermodynamics, analogy, fP and xV,there is a thermodynamics potential: P = -U/ V 结论：系统对外作功会改变系统的势能! 几种绝热作功的过程焦耳从1840年开始，在20多年的时间内，反复进行了大量的(多达400多次)热与功的实验。下面看两个实验：实验1. 在绝热的水箱中，重物下降带动叶片转动，使其在水中搅动使水温升高。保持过程为绝热。实验2. 在绝热的水箱中，电流通过电阻器使水温升高。保持过程为绝热

35、。结果发现：使物体升高一定的温度，所需的功在实验误差范围内相等。由此定义一个态函数U在终态B和初态A之差：WUUAB此态函数被称之为“内能”Adiabat process 绝热过程Equate the work done in an adiabatic process to the decrease in the adiabatic potential U, denoted by绝热过程做的功绝热势的减少In an arbitrary non-adiabatic process, the general equation is:SSSdUPdVA)2 . 6(dSPdVdUConsiderin

36、g a total differential expression(全微分表达式)VSSPVTranslated to Jacobian form, and use its property, we find1),(),(,),(),(),(),(VPSSVVPSVS(在非绝热过程中，应该考虑熵S的变化量。)In ( / T)S, S is fixed. So, (S) is an arbitrary function of entropy, and (S) dS is an arbitrary function too.)(.,1STeiTS( (S) 是任意的， (S) dS 也是任意的)

37、For normalizing U, assuming (S) = 0, = T, then dU = T dS P dV (6.3)Adiabat potential,U, has its own variables S and V, the derivatives with respect to two variables areT = ( U/ S)V , P = ( U/ V)S (6.4)Isothermal processEquating the work done in an isothermal process to the decrease in the isothermal

38、 potential F, denoted by等温过程做的功 等温势的减少TTTdFPdVAIn an arbitrary non-isothermal process, the general equation is:PdVdTdFConsidering a total differential expression(全微分表达式)VTTPVTranslated to Jacobian form, and use its property, we find1),(),(,),(),(),(),(VPTTVVPTVT)(.,1TSeiSTIn ( / S)T, T is fixed. So,

39、 there is an arbitrary function of (T), and then isothermal potentials F.( (T) 是任意的， 是任意的， F 也是任意的)For normalizing F, assuming (T) = 0, = - S. thendF = -S dT P dV (6.5)Isothermal potential,F, has its own variables T and V, the derivatives with respect to two variables areS = - ( F/ T)V , P = ( F/ V)

40、TFrom (6.3) and (6.5), we findd(U-F) = T dS +S dT = d(TS) F = U - TS7.The Energy Principle. Supply and Removal of HeatIntroducing a concept of the “internal energy”(内能), defining it as the total energy of the thermodynamic system.改变 内能的两种方法：(1)performing work A on a gas by applied forces;(2)contacti

41、ng with a body heater or cooler, the system will obtain or loss the amount of energy, the quantity of heat, denoted(定义为) by the symbol Q.热力学第一定律This assertion, being in essence the law of conservation of energy, will be referred to as the energy principle or the first law of thermodynamics.这个基于能量守恒原

42、理的论断被称之为“能量原理”或“热力学第一定律”。值得注意的是，在绝热过程中，系统对外作功消耗了内能。因系统对外作的功等于绝热势能的减少。故绝热势能与内能是一致的。The adiabatic potential U is identical to internal energy.) 1 . 7(PdVQAQdU热量与微“熵”的关系是：)2 . 7(TdSQ 因此有：) 3 . 7(PdVTdSdU热力学第一定律的意义A special form of the law of conservation.It is valid for any processes: equilibrium and n

43、on-equilibrium方程(7.3)所涉及的内容比方程(7.1)更加广泛。因为熵具有更深刻的含意。(7.3) is often referred to as the basic thermodynamic identity for equilibrium processes.(7.3)式被称之为热力学恒等式。(T,S)/(P,V)=1 is a corollary(推论) of Eq.(7.3)。比较两个公式：TdSQ PdVA Gibbs(吉布斯) suggested that V is a mechanical coordinate(力学坐标) in work, S and T is

44、 a thermodynamic coordinate and a thermodynamic force, respectively.为系统加热相当于为系统增加热力学坐标-熵。平衡过程的热力学相当于力学的扩展-被认为是引入了“附加的非力学的自由度”。理想气体的内能焦耳的实验：1840年焦耳做了自由膨胀的实验：两个容器均浸没在水中。实验的目的是要检测气体自由膨胀导致的水温变化。其结论是：水 温始终保持不变。分析：打开活门，气体扩散。在扩散过程中，不受任何阻力，即不与外界做功W = 0。温度没有变化，说明不存在热交换Q = 0。由热力学第一定律得到内能U = 0。在理想气体的自由扩散过程中，内能

45、与体积无关，U(T，V) = U(T)。焦耳定律1VUTUTTVVU焦耳系数：0TVU导致0UVT自由能FThe isothermal potential F is also called the free energy(自由能), the difference U F = TS,the bound energy(束缚能).F 被看成是在等温过程中内能花在产生功上的一部分。在等温过程中，理想气体在容器中被活塞压缩，由于温度保持不变，气体的内能U不变。因此，仅用内能不能表示该过程的做功与放热。外界对气体作的功PdV转变成热量TdS释放给周围的恒温热源。自由能F仅仅描述内能中作功的一部分。 恒 温

46、 热 源gas in vessel piston熵S如何变化？系统释放了热量, 为TdS,T不变，相当于系统释放了熵外界做功补偿了TS的减少。Since temperature is a function not only of entropy, but also of the state of a gas (for instance, S, V). The area (quantity of heat) depends not only on the initial and final states, but also on the entire course of the process.

47、S1S2TS温度T不是S的单值函数，而是S和V的函数，因此，在TS平面T相对于S的变化不是单值的，而是与过程相关的。中间曲线I的吸热不同于其它两条曲线，即吸热与积分路径相关。同功一样，被积函数 (integrand)Q不是气体的态函数，而是过程量，为无穷小量。即“the quantity of heat is a function of a process”PdV and TdS are not total differentials(全微分), only t he difference(差) TdS-PdV is a total differential of the internal en

48、ergy, that a function of the state of a gas.S1S2TSV1V2PV设两个图的初、终两点表示相同的两个状态，虽然三条曲线的过程均不相同，但曲线下的面积之差必须相同。即过程不同，初、终两点相同。热力学第一定律的物理含义等温势和绝热势的讨论P-V图上的曲线为过程量，因为隐含的S、T不确定。一旦参量确定，则过程确定。但描述过程的是用热或功，则不合适，因为它们是过程量。在等温过程中，作功的过程是确定的，可以用状态量描述，该参量为“等温势F”。在绝热过程中，作功的过程也是确定的，可以用状态量描述，该参量为“绝热势U”。其中，dF=-SdT-PdV, dT=0时

49、,作功为dF=-PdV。dU=TdS-PdV,dS=0时，作功为dU=-PdV.试想，若试想，若dP=0,或或dV=0,类似的讨论应该如何？类似的讨论应该如何？不同热力学过程的计算Isothermal process T= constantIsobaric process P = constantIsochoric processAdiabatic process)ln(1),ln(1)(, )(1200221112021VVRTRTQVPPVRSSlawBoyleVPVPSSTTdSQTSST)(1,112TTRQdVRdSp1)(,)1(12TTRQPRdPdSV0SQ作业：证明上述结论理

50、想气体的三大定律1. 玻意耳定律(1662年)：等温条件下，PV为常数。2. 阿伏加德罗定律(1811年)：相同的T、P条件下，相等体积所含的摩尔数相同。3. 焦耳定律(1852年)：内能仅仅是温度的函数。8. Heat Capacity of Gases(热容量)Define: the quantity of heat supplying to the gas to raise its temperature by one degree.TQCFunction: State or Process ?Process Function !There are different heat capa

51、cities depending on different process.Such as CP, CV, CS, CT 。Infinite set .But CS = 0 and CV = 1/0Is negative heat capacity possible ?Yes, it is possible for two having opposite signs.For example, heat a gas, and do the work, but WQ, here Q 0, and T Q2热力学温标开尔文发现热机的效率与吸热放热比有关，且与工作的物质无关。将热量比对应于某种参量比：

52、1212QQ他将此参量称之为热力学温标：该温标与理想气体的绝对温度是一致的。1212TTQQ由此得到结论：低温热源的温度愈低，传给它的热量就愈少；在极限时，传给它的热量为零：“绝对零度”Nernsts law(the third law): at absolute zero the entropy of any equilibrium system is constant, independent of any varying parameters. (绝对零度不可能达到，只能无限接近。)克劳修斯等式和不等式由卡诺定理：任何热机的效率均低于可逆热机的效率：121211TTQQ则有02211TQ

53、TQ02211TQTQ取吸热为正：此式为“克劳修斯等式和不等式克劳修斯等式和不等式”对于一个普遍的热循环，将会有： 0iiTQ其积分表达式为：0TdQ克劳修斯对熵的原始定义两个可逆过程：0TdQ定义态函数：BArABTdQSS对于任意可逆过程，态函数相同。若有一个过程是不可逆的（设为若有一个过程是不可逆的（设为R）：先经过不）：先经过不可逆的过程到可逆的过程到B，再从，再从B经过可逆过程到经过可逆过程到A：0TdQBAABBArTdQSSTdQTdQdS 在绝热条件下:0dS熵增加原理熵增加原理0ABRBARTdQTdQ证明麦克斯韦关系说明：每个偏微分代表一个“物理效应物理效应”vsSpVT)

54、()(psSVpT)()(vTTpVS)()(pTTVpS)()(作业：中文作业：中文P68：1.19, 1.21, 1.22, 1.23, 1.25, 1.27The van der Waals GasFor the equation: PV=RT if T 0K, then P or V 0. Problem? the molecule of perfect gas has not volume.The Clausius equation: P (V-b) = RTVan der Waals: there are attration and repulsion forces between

55、 molecules, at low temperatures the repulsion force is large, which will increase pressure and is proportional to NA/V.So, van der Waals equation is RTbVVaP)(2作业：作业：2.10实际气体物态方程的计算结果根据物态方程：RTbVVaP)(2RbVTPV)()(dVTPdTTCdVVSdTTSdSVVTV)()()(dVbVRdTTCdSV微商：内能：dVVadTCPdVdVbVRdTTCTPdVTdSdUVV2)(自由能，利用：PdVdT

56、bVbVRTTCSPdVSdTdFV)ln()ln(000 xxeydeydxxyy) 1)(ln() 1(lnVadbVRTdPdVdTbVR1)ln()ln(算出S得：Onnes(昂尼斯) equation有许多实际气体的近似表达式，其中之一为.)()(1 (2VTCVTBRTPV此方程将影响内能的温度关系、熵、热容量等。Homework, 英P60: 1右图为B(T)T的关系图。其特点是：1）低温时为负，高温为正。2）B(T)的一阶导数为正。3）高温时B(T)趋向恒定值。低温下，分子的动能小，分子间的吸引力(a/V2)使压强降低，B(T)0。几个与物态方程有关的参量几个与物态方程有关的参

57、量体胀系数是：压强系数是：等温压缩系数T是：利用关系式：得到三者的关系：pTVV)(1VTpp)(1TTpVV)(11)()()(pVTVTTppVpT内容见中文书内容见中文书 P11,作业：作业：1.1,1.2,1.3A reversible adiabatic expansion of a gas can be used to cool it.Gay-Lussac first carried out this experiment.Thermal equilibrium sets in in the vessel, and the change in temperature ,T2-T1,

58、can be measured. The experiment show that T2=T1, i.e. U is independent of volume.It should be noted that this process is an irreversible adiabatic process, and we cannot claim that the entropy remains constant. Later, we will know that the entropy increases in irreversible adiabatic process.The mean

59、ing of the derivative(导数) (T/V)U。TPVSPdVTdSdUVSVUVSSVSUUSVUSTTPVTUSVUSTSUVTUVSVSVUTUVUTVTUUVSVSVSVVSU/)(,0)(),(),(),(),()()()()()()()()()(),(),(),(),(),(),(For the real gas, and van der Waals equationVVVUCVaTPTPCTVT2)()(The gas filling a heat-insulated tube is forced through a porous(多孔的) plug with

60、the aid of two pistons.Constant pressures P1 P2 are maintained by two pistons.A steady-state(稳态的) gas flow through the porous partition。Now, analyze the process: abrupt volume change from V1 to V2. The process is not a quasi-static state process.The Joule-Thomson process is adiabatic.PVUHVdPTdSdH,内能