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Hong Kong Polytechnic University Heat & Light (AP306) Part I: Thermal Physics Dr.Haitao Huang (黄海涛) Department of Applied Physics, Hong Kong PolyU Tel: 27665694; Office: CD613 Lecture Notes can be downloaded from http://ap.polyu.edu.hk/apahthua Textbook: Thermal Physics, 2nd Edition, C.B.P.Finn Assessment Weighting: Continuous Assessment: Quiz (and midterm exam): 35% Final Exam: 65% (Minimum requirement: D) Minimum requirement for overall grade: D Heat & Light----by Dr.H.Huang, Department of Applied Physics 1 Hong Kong Polytechnic University Temperature Some Concepts: Wall System; Surroundings; Piston Gas system Boundary / Wall Surroundings An equilibrium state is one in which all the bulk physical properties of the system are uniform throughout the system and do not change with time. State Variables / Thermodynamic Variables / Thermodynamic Coordinates Stat Functions / State Properties Two systems in thermal contact can finally reach thermal equilibrium. Zeroth Law of Thermodynamics: If each of two systems is in thermal equilibrium with a third, they are in thermal equilibrium with one another. A B C A B A C Heat & Light----by Dr.H.Huang, Department of Applied Physics 2 Hong Kong Polytechnic University Temperature Temperature: A property of a system that determines whether or not that system is in thermal equilibrium with other systems. If two systems have the same temperature so that they are in thermal equilibrium, this does not necessarily mean that they are in complete or thermodynamic equilibrium. For this condition to hold, in addition to being in thermal equilibrium, they would also have to be in mechanical equilibrium and chemical equilibrium. Indicator Diagram: P T1 T2 T3 V Equation of State: All the bulk properties can be fixed by specifying two independent state variables. f P, V , T 0 PV nRT Heat & Light----by Dr.H.Huang, Department of Applied Physics 3 Hong Kong Polytechnic University Temperature Temperature Scale: It is customary to choose a fixed point at which ice, water and water vapor coexist in equilibrium; this is known as the triple point of water. The temperature at this point is defined as 273.16K. The gas scale temperature is determined from, P Tgas 273 .16 PTP where Tgas is the temperature of an ideal gas with constant volume at pressure P. PTP is the pressure of the constant volume gas at the triple point of water. The unit of temperature on the ideal gas scale is K (Kelvin). The Celsius scale t is measured in C and is related to the ideal gas scale T by, t C T K 273 .15 Homework: exercises 1.1-1.3 Heat & Light----by Dr.H.Huang, Department of Applied Physics 4 Hong Kong Polytechnic University Reversible Process and Work Process / End Points / Reversible Process: P P2, V2 Reversible processes are quasistatic processes where no dissipative forces such as friction are present. P1, V1 How to recognize reversible process? V 1 V Volume thermal expansivity V T P Linear expansion coefficient 1 L 3 L T F Bulk modulus P 1 and compressibility K V V T L F Young’s modulus Y A L T Heat & Light----by Dr.H.Huang, Department of Applied Physics 5 Hong Kong Polytechnic University Reversible Process and Work Example: Calculate the increase in tension F of a wire clamped between two rigid supports, a distance L apart, when it is cooled from T1 to T2. Known parameters: cross-section area, Young’s modulus and linear expansion coefficient L F F F F Y F F L, T dF dT dL dT A L T T L L T T L 1 L F T2 F T L L T F F F2 F1 dT 1 T1 T L T L L F F T F F L AY T L L T T F T2 F dT AY dT AY T2 T1 T2 F T1 T L T1 Heat & Light----by Dr.H.Huang, Department of Applied Physics 6 Hong Kong Polytechnic University Reversible Process and Work Example: Work done by an ideal gas in a cylinder with a P 1 frictionless piston. (Work is path dependent.) Suppose that the pressure is P and the balancing force is F at the equilibrium state, 2 F PA 3 V Infinitesimal small work done by the gas dW Fdx PAdx PdV V2 V2 Total work: W dW PdV V1 V1 W PdV nRT dV V nRT ln V2 V1 V2 V2 Along 12: V1 V1 Along 13 2: W=P2(V2-V1) Heat & Light----by Dr.H.Huang, Department of Applied Physics 7 Hong Kong Polytechnic University Reversible Process and Work Sign convention for work: Movable bar When the surroundings do work on the system, that l work is positive; conversely, when the system does work on the surroundings, that work is negative. dx Wire frame More examples on infinitesimal small work: The work done to stretch a wire at a tension F though a infinitesimal distance dx is, dW Fdx The work done to stretch a surface film (increase the dW ldx dA surface by dA) is, In a reversible electrolytic cell, the work performed by dW dZ the external charging circuit is, For a uniformly magnetized material, the external work required to increase the magnetic moment by dM in the dW B0 dM applied induction field B0 is, The work required to increase the overall dipole moment dW EdP of a uniformly polarized dielectric material by dP in an electric field E is, Heat & Light----by Dr.H.Huang, Department of Applied Physics 8 Hong Kong Polytechnic University Reversible Process and Work Example: Calculate the work done in changing the state of a compressible fluid from (P1, T1) to (P2, T2) in a reversible process. V V ( P, T ) P K V V T V V V dV dP dT dP VdT P T T P K 1 V V T P PV dW PdV dP PVdT K 2 P2 PV T2 W dW dP PVdT 1 P1 K T1 In the case of an isothermal change of liquid: W P1 P2 PV K dP V 2K P22 P 2 1 Homework: exercises: 2.1-2.9 Heat & Light----by Dr.H.Huang, Department of Applied Physics 9 Hong Kong Polytechnic University First Law of Thermodynamics Joule’s Experiment Determined the mechanical equivalent of heat, that is, it required 4.2kJ of work to raise the temperature of 1kg of water through one degree Kelvin. First Law: If a thermally isolated system is brought from one equilibrium state to another, the work necessary to achieve this change is independent of the process used. There must exist a state function whose difference between the two end points (1 and 2) is equal to the adiabatic work. Such a function is called the internal energy U Wadi U 2 U 1 Heat & Light----by Dr.H.Huang, Department of Applied Physics 10 Hong Kong Polytechnic University First Law of Thermodynamics If a system is not thermally isolated, the work done in taking the system between a pair of equilibrium states depends on the path. The change of the internal energy can be expressed as, U U 2 U1 W Q where Q is heat. This is the mathematical statement of the first law. This means that the internal energy can be increased either by doing work on or by supplying heat to the system. It is true for all processes whether reversible or irreversible. In closed systems, heat is the non-mechanical exchange of energy between the system and the surroundings because of their temperature difference. The sign convention for Q is that it is positive when heat enters the system. For an infinitesimal process, the first law takes the form, dU=đW+đQ In general, W and Q are path dependent. We call they are inexact differentials. For a compressible fluid or gas where đW=-PdV for an infinitesimal reversible process, the first law becomes, đQ=dU+PdV In differentiating between heat and work, it is very important to be clear as to what constitutes the system. Heat & Light----by Dr.H.Huang, Department of Applied Physics 11 Hong Kong Polytechnic University First Law of Thermodynamics Microscopic View of First Law: Energy can be added to the system in the form of heat by increasing the randomness of the motion of the constituent molecules. We can also increase the energy of the system by performing work. In this way we displace the molecules in an ordered way. As a result of quantum mechanics, each of the N particles in the system can exist in a series of discrete energy levels. If there is a population of ni particles in the ith energy level i, the total internal energy of the system will be N U ni i i 1 Now U can be changed in two ways: either the energies i can be changed, with the populations remaining the same—this is work; or the populations ni can be changed with the energies remaining the same—this is heat. Heat & Light----by Dr.H.Huang, Department of Applied Physics 12 Hong Kong Polytechnic University First Law of Thermodynamics Heat Capacity C the limiting ratio of the heat introduced reversibly into the system divided by the temperature rise, Q đQ C lim T 0 T dT C 1 đQ Specific Heat: c m m dT Since there are a large number of possible reversible paths between the end points with a temperature difference T, it follows that there are many possible heat capacities. đQ U Heat Capacity at constant volume CV V dT T V dQP dU PdV H Heat Capacity at constant pressure C P dT dT T P Enthalpy: H=U+PV For ideal gas: U=U(T) Heat & Light----by Dr.H.Huang, Department of Applied Physics 13 Hong Kong Polytechnic University First Law of Thermodynamics Free Expansion: partition break partition gas For an ideal gas, there will be no temperature change. Real gases decrease their temperatures slightly when they undergo a free expansion. T Joule Coefficient: J V U Heat & Light----by Dr.H.Huang, Department of Applied Physics 14 Hong Kong Polytechnic University First Law of Thermodynamics Adiabatic Process: P Adiabatic U dU CV dU CV dT T V dT The first law becomes: đQ CV dT PdV Isothermal V C P đ P CV P Q V dT T P For ideal gas: C P CV nR dT dV For a adiabatic process: 0 CV nR T V After integration: TV 1 const PV const n2a van der Waals gas equation of state: P 2 V nb nRT V Heat & Light----by Dr.H.Huang, Department of Applied Physics 15 Hong Kong Polytechnic University First Law of Thermodynamics Throttling Process: We do work on the left-hand side: 0 W Pi dV PiVi Vi The gas does work on the right-hand side: Vf W Pf dV Pf V f 0 Consider the gas on both sides as a whole system U f U i 0 PiVi Pf V f U i PiVi U f Pf V f Hi H f This is a isenthalpic process. T Joule-Kelvin coefficient JK P H Heat & Light----by Dr.H.Huang, Department of Applied Physics 16 Hong Kong Polytechnic University First Law of Thermodynamics Steady Flow Process: We treat this unit mass as the system. Suppose the shaft work done by the turbine is w, we can summarize the following energy changes: The internal energy changes by u2-u1. The kinetic energy changes by ½(v22-v12). The potential energy changes by g(z2-z1). The net work done on the fluid is P1v1-P2v2-w. The heat flow into the system is q. KE PE bulk U W Q 1 2 v 2 v1 g z 2 z1 u 2 u1 P v1 P2 v2 w q 2 2 1 w h1 h2 1 2 v1 v 2 g z1 z 2 q 2 2 This is the general equation for steady flow. Heat & Light----by Dr.H.Huang, Department of Applied Physics 17 Hong Kong Polytechnic University First Law of Thermodynamics Steady Flow through a Nozzle: for fast flow q=0 w h1 h2 1 2 v1 v 2 2 2 since w=0 v1 v 2 2h2 h1 2 2 1 T1 P Assuming an ideal gas and an adiabatic process 1 T P 2 2 Heat & Light----by Dr.H.Huang, Department of Applied Physics 18 Hong Kong Polytechnic University Second Law of Thermodynamics Carnot Cycle: hot body Q1 E W Q2 cold body Efficiency: W Q1 U Q1 Q2 W 0 W Q1 Q2 Q2 1 Q1 Heat & Light----by Dr.H.Huang, Department of Applied Physics 19 Hong Kong Polytechnic University Second Law of Thermodynamics The Kelvin-Planck statement: It is impossible to construct a device that, operating in a cycle, will produce no effect other than the extraction of heat from a single body at a uniform temperature and the performance of an equivalent amount of work. In a concise form: A process whose only effect is the complete conversion of heat into work is impossible. Key points: There is no net change of the working system (substance). The hot body is only a source of heat. It will not do work on the substance. There is only a single hot body (or heat source). The Clausius statement: It is impossible to construct a device that, operating in a cycle, produces no effect other than the transfer of heat from a colder to hotter body. Q reservo ir at T Heat & Light----by Dr.H.Huang, Department of Applied Physics 20 Hong Kong Polytechnic University Second Law of Thermodynamics Schematic representation of the Kelvin-Planck (left) and Clausius (right) statements. hot body hot body Q Q W=Q R Q cold body cold body The statements are equivalent. hot body hot body Q1 Q1+Q2 Q2 Q1 W=Q1 W=Q1-Q2 E R R E Q2 Q2 Q2 cold body cold body Heat & Light----by Dr.H.Huang, Department of Applied Physics 21 Hong Kong Polytechnic University Second Law of Thermodynamics Carnot’s Theorem: No engine operating between two reservoirs can be more efficient than a Carnot engine operating between those same two reservoirs. hot body hot body Q1 Q1 Q1 Q1 W W E W E R C Q2 Q2=Q1-W Q2=Q1-W Q2=Q1-W cold body cold body C All Carnot engine operating between the same two reservoirs have the same efficiency. Heat & Light----by Dr.H.Huang, Department of Applied Physics 22 Hong Kong Polytechnic University Second Law of Thermodynamics Temperature Scale: The efficiency of a Carnot engine operating between the two reservoirs is dependent only on the temperatures of the reservoirs. This gives us a meaning of defining a temperature scale, which is independent of any particular material. The thermodynamic temperature T can be so defined that T1 and T2 for the two reservoirs in a Carnot engine are related as, T1 T2 T C 1 2 1 Q2 T1 T1 Q1 T1 Q1 T2 Q2 Heat & Light----by Dr.H.Huang, Department of Applied Physics 23 Hong Kong Polytechnic University Second Law of Thermodynamics Temperature Scale: T1 Q1 For Carnot engine C12, T1 T2 Q1 Q2 C12 W12 For Carnot engine C23, T2 T3 Q2 Q3 Q2 Composite T2 engine C13 T1 T3 Q1 Q3 Q2 C23 W23 We use the symbol Tg for the gas scale temperature Q3 and T for the thermodynamic temperature as just T3 defined. We will show that these two temperatures are identical. Heat & Light----by Dr.H.Huang, Department of Applied Physics 24 Hong Kong Polytechnic University Second Law of Thermodynamics For isothermal bc: PV nRTg1 since đ Q dU PdV PdV we have nRTg1 ln Vc Vb Vc Vc dV Q1 PdV nRTg1 Vb Vb V Similarly, Q2 nRTg 2 ln Vd Va Q1 T1 Tg1 ln Vc Vb Q2 T2 Tg 2 ln Vd Va For ab and cd adiabatics, Tg1Vc 1 Tg 2Vd 1 Tg1Vb 1 Tg 2Va 1 Vc Vb Vd Va T1 Tg1 Tg const T Tg T T2 Tg 2 Heat & Light----by Dr.H.Huang, Department of Applied Physics 25 Hong Kong Polytechnic University Second Law of Thermodynamics Efficiency: Q2 T C 1 1 2 Q1 T1 It is interesting to note that the efficiency would be 100% were we able to obtain a lower temperature reservoir at absolute zero. This is forbidden by the third law. Imagine now that the Carnot engine is run backwards to act as a refrigerator. The coefficient of performance is defined as the heat extracted divided by the work expended, T1 Q Q2 T2 C 2 R Q1 W Q1 Q2 T1 T2 C W=Q1-Q2 One interesting application of the Carnot refrigerator is the so-called heat pump. The efficiency of a Carnot heat pump Q2 is, Q Q T 1 T2 C HP 1 1 1 W Q1 Q2 T1 T2 1 T2 T1 Since the efficiency of a Carnot heat pump rises as the temperature difference between the reservoirs decreases, heat pumps are best used in providing background heating. Heat & Light----by Dr.H.Huang, Department of Applied Physics 26 Hong Kong Polytechnic University Second Law of Thermodynamics Otto-cycle: P c ab: TaV1 1 TbV2 1 Adiabatic bc: Q1 CV Tc Tb b Q1 cd: Td V1 1 TcV2 1 d Q2 a da: Q2 CV Td Ta V2 V1 V Q2 T Ta 1 1 d Q1 Tc Tb Td Ta V1 1 Tc Tb V2 1 1 V2 1 1 V 1 1 rc 1 where rc is the compression ratio (=V1/V2). It is important to have as high a compression ratio as possible in order to get a high efficiency. Heat & Light----by Dr.H.Huang, Department of Applied Physics 27 Hong Kong Polytechnic University Entropy Clausius inequality : đQ 0 T P f R2 For a reversible cycle, the equality sign holds. R1 đ QR f đ QR i đQR 0 i R T R1 i T R2 f T V f đ QR i đ QR R1 i T R2 f T for reversible process R2: f đ QR i đQR R2 i T R2 f T f đQ R f đQR đ QR f R1 i T R2 i T S S f S i R i T This state function is called entropy. For an infinitesimal reversible process, đQR TdS Heat & Light----by Dr.H.Huang, Department of Applied Physics 28 Hong Kong Polytechnic University Entropy Example: Determine the entropy change of a beaker of water when it is heated at atmospheric pressure between 20C and 100C. Example: Calculate the entropy change of an ideal gas undergoing a free expansion doubling its volume. Example: A beaker of water is heated at atmospheric pressure from 20C and 100C by a reservoir at 100C. Determine the total entropy change of water and reservoir. Heat & Light----by Dr.H.Huang, Department of Applied Physics 29 Hong Kong Polytechnic University Entropy Suppose now the path R1 is irreversible, according to the Clausius inequality, đQ f đQ i đQ T i T R f T R 0 P f R2 f đ Q i đQR f đQR S S i T R f T T R i f i R1 i For an infinitesimal part of the process, đQ dS T V The equality sign holds if the process is reversible. For an isolated system, dS 0 or for a finite process, S f S i S 0 Principle of increasing entropy: The entropy of a thermally isolated system increases in any irreversible process and is unaltered in a reversible process. For a system thermally isolated from the surroundings: S approaches to a maximum; For a system totally isolated from the surroundings: S approaches to a maximum with U remains constant. Heat & Light----by Dr.H.Huang, Department of Applied Physics 30 Hong Kong Polytechnic University Entropy The thermodynamic state of a system can be specified by any pair of independent state functions. In particular, a state is equally well specified by the pair S and T as by the pair P and V. T A Carnot cycle shown in a T-S diagram. a b T1 For any reversible process Q1 adiabatic Q R TdS adiabatic T2 d c the net heat absorbed in a Carnot cycle is Q2 given by the area shaded. The differential form of the first law is, S dU đ Q đ W For an infinitesimal reversible process, đ W PdV đQ TdS dU TdS PdV TdS dU PdV This is the central equation of thermodynamics, or thermodynamic identity. Example: Calculate the entropy of an ideal gas. Heat & Light----by Dr.H.Huang, Department of Applied Physics 31 Hong Kong Polytechnic University Thermodynamics Revision Ideal Gas: Equation of state: pV nRT Molar specific heat: c p cV R Heat capacity: Q C T f Ti cm T f Ti Internal energy change: U CV T Thermodynamics: Work by the gas : đW=pdV U S H S Entropy: dS đQ CV T CP T T T V T V T P T P The first law: dU=đQ-đW dU TdS PdV The second law: S 0 Principle of increasing entropy W Q T2 Efficiency of a thermal engine: 1 2 Carnot cycle: 1 Q1 Q1 T1 Heat & Light----by Dr.H.Huang, Department of Applied Physics 32 Hong Kong Polytechnic University Thermodynamics Revision Special processes: free expansion U=0 throttling process H=0 Basic concepts: state function; reversible process; irreversible process; Thermal dynamic potentials (Legendre transformations): H=U+PV; F=U-TS; G=H-TS Maxwell Relations: T P T V P S V S V S S V P S S P T V V T T P P T dU TdS PdV dH TdS VdP dF PdV SdT dG VdP SdT Cyclic Relation Heat & Light----by Dr.H.Huang, Department of Applied Physics 33 Hong Kong Polytechnic University Heat & Light----by Dr.H.Huang, Department of Applied Physics 34 Hong Kong Polytechnic University Heat & Light----by Dr.H.Huang, Department of Applied Physics 35

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