TEMPERATURE I am teaching Engineering Thermodynamics to

TEMPERATURE • I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. • These slides follow closely my written notes (http: //imechanica. org/node/288). • I went through these slides in four 90 -minute lectures. Zhigang Suo, Harvard University

The play of thermodynamics ENTROPY energy temperature space pressure heat capacity compressibility Helmholtz function enthalpy matter chemical potential charge electrical potential capacitance Gibbs free energy thermal expansion Joule-Thomson coefficient 2

The basic algorithm of thermodynamics • (entropy) = log (number of sample points). • Entropy is additive. • When a constraint internal to an isolated system fixes an internal variable at a value x, the isolated system flips in a subset of quantum states. • The number of quantum states in the subset is W(x). • Call S(x) = log W(x) the entropy of the configuration of the isolated system when the internal variable is fixed at x. 1. Construct an isolated system with an internal variable, x. 2. When the internal variable is constrained at x, the isolated system has entropy S(x). 3. After the constraint is lifted, x changes to maximize S(x). 3

Classify systems according to how they interact with the rest of the world Exchange matter Exchange energy by work Exchange energy by heat Open system yes yes Isolated system no no no Closed system no yes Thermal system no no yes Adiabatic system no yes no 4

An open system modeled as a family of isolated systems • The fire, the weights and the valve make the water an open system. • Insulate the wall, jam the piston, and shut the valve. Make the water an isolated system. • A system isolated for a long time flips to every quantum state with equal probability. • Entropy S = log (number of quantum states). • Isolating water at various (U, V, N), we obtain a family of isolated systems of three independent variables • For the family of isolated systems, the entropy is a function, S(U, V, N). weights O vapor 2 open system vapor valve liquid fire a family of isolated systems S(U, V, N) liquid 5

Basic problem of thermodynamics adiabatic, stationary, impermeable wall U’, V’, N’ S’(U’, V’, N’) open system A’ diathermal, moving, leaky wall U’’, V’’, N’’ S’’(U’’, V’’, N’’) isolated system open system A’’ Isolated system conserves energy, space, and matter U’ + U’’ = constant. U’ is an internal variable V’ + V’’ = constant. V’ is an internal variable N’ + N’’ = constant. N’ is an internal variable How does the system isolated for a long time choose the values of the three internal variables? System isolated for a long time maximizes entropy Entropy is additive, but not constant. Choose U’, V’, N’ that maximize S’(U’, V’, N’) + S’’(U’’, V’’, N’’) 6

Calculus 7

Conditions of equilibrium adiabatic, stationary, sealing wall diathermal, moving, leaky wall U’, V’, N’ S’(U’, V’, N’) open system A’ U’’, V’’, N’’ S’’(U’’, V’’, N’’) open system A’’ isolated system 8

The goal: understand the relation • We understand everything in this equation, except for temperature. • Temperature is a child of entropy and energy. 9

Count the number of quantum states by experimental measurement • For a closed system, entropy is a property, • According to calculus, • In later lectures we will show that • Measure entropy incrementally. weights No quantum mechanics. No theory of probability. vapor liquid fire 10

Circular statements What is temperature? Answers from teachers in kindergartens: • Temperature is the quantity measured by a thermometer. • Thermometer is an instrument that measures temperature. Answers from textbooks of thermodynamics: • Temperature is a property shared by two bodies in thermal contact, when they stop exchanging energy by heat. • Heat is the transfer of energy caused by difference in temperature. 11

Heat and temperature and are distinct quantities, and can be measured by separate experiments. • Calorimetry. The art of of measuring heat. • Thermometry. The art of measuring temperature. 12

What can we do for temperature? • Temperature as an abstraction from everyday experience of thermal contact. • Temperature as a consequence of the two great principles of nature: an isolated system conserves energy and maximizes entropy. And so, my fellow enthusiasts of thermodynamics: ask not what temperature can do for you—ask what you can do for temperature 13

Plan • Calorimetry. Find a method to measure heat without the concept of temperature. • Empirical observations of thermal contact • Theory of thermal contact • Refinements and applications 14

Thermodynamic states of equilibrium • • A closed system changes under the fire and the weights. The system isolated for a long time reaches a state of thermodynamic equilibrium. A fixed amount of matter can be in many thermodynamic states of equilibrium. For a fixed amount of a pure substance, specify all thermodynamic states of equilibrium using two thermodynamic properties, P and V. weights O closed system vapor 2 vapor isolated system P state liquid V fire 15

Experimental determination of internal energy Internal energy is a thermodynamic property, U(P, V). • • • Seal and insulate a system, making it an adiabatic system. Do work Wadiabatic to the system. The system changes from state (PA, VA) to state (PB, VB). The internal energy changes by U(PB, VB) - U(PA, VA) = Wadiabatic Reach many states to determine the function U(P, V). P state B state A V force x displacement torque x angle voltage x change How do we know that we have sealed and insulated the system well enough? 16

Calorimetry: the art of measuring heat • • • We have measured the function U(P, V). The fire and the weights change the closed system from state (PA, VA) to state (PB, VB). We know that the internal energy changes by U(PB, VB) - U(PA, VA). We also know that the weights do work W = force x displacement. We determine the heat Q from U(PB, VB) - U(PA, VA) = W + Q. weights O vapor 2 state B vapor O vapor liquid P state A fire V Isolated system A U(PA, VA) closed system W, Q isolated system B U(P B, VB) 17

Plan • Calorimetry • Empirical observations of thermal contact. Name all places of hotness by a single, positive, continuous variable. • Theory of thermal contact • Refinements and applications 18

Everyday experience of hotness • (temperature) = (hotness) • (a value of temperature) = (a place of hotness) • Hot, warm, cool, cold. • Words are not enough to name many places of hotness. • Name all places of hotness by a single, positive, continuous variable. • Why is hotness so different from happiness? 19

Ranking Universities Two-dimensional ranking average salary of graduates One-dimensional ranking • Harvard • Princeton • Yale citations to papers Why stop at two dimensions? 20

Ranking places of hotness Imagine you were born 500 years ago. Galilei (1564 -1642) Middleton, A History of the Thermometer and Its Use in Meteorology 21

Air thermometer • • Problem: Visualize hotness. Invention: Air thermometer (Galileo and others, 1612) Science: Air expands when heated. Engineering: Map hotness to height. 22

Liquid-in-glass thermometer • • Problem: Gas thermometer is bulky and sensitive to pressure. Invention: liquid-in-glass thermometer (Ferdinando 1654) Science: Liquid expands when heated. Engineering: Volume of liquid is insensitive to pressure. Glass is transparent. Mark the glass. Middleton, A History of the Thermometer and Its Use in Meteorology Wikipedia page on thermometer 23

Thermal contact stationary, impermeable, but diathermal wall heat • • isolated system The two systems together form an isolated system. The two systems do not exchange matter (impermeable wall) The two systems do not exchange energy by work (stationary wall). The two systems exchange energy by heat (diathermal wall). 24

Observation 1 Two systems in thermal contact for a long time will stop transferring energy. thermal contact isolated system The two systems are said to have reached thermal equilibrium. 25

Observation 2 (zeroth law, Fowler 1931) If two systems are separately in thermal equilibrium with a third system, the two systems are in thermal equilibrium with each other. Use thermal contact to discover places of hotness. 26

Observation 3 For a fixed amount of a pure substance, once pressure and volume are fixed, the hotness is fixed. A weights P B fire C V 27

Observation 4 For a pure substance in a state of coexistent solid and liquid, the hotness remains fixed as the proportion of liquid and gas changes. This place of hotness is specific to substance, but is insensitive to pressure. liquid For water, this place of hotness has many names • Melting point • Freezing point • 0 Celsius • 32 Fahrenheit • 273. 15 Kelvin 28

Name places of hotness The same way as we name streets • Harvard, Cambridge, Oxford… • Washington, Lincoln, … • 5 th Avenue, 6 th Avenue, … After physical events. • Streets are physical objects. • WATER at the melting point • Names of the streets are arbitrary. • LEAD at the melting point • ALUMINUM at the melting point • GOLD at the melting point • Steam at pressure 0. 1 MPa and specific volume 2000 m 3/kg Relative terms • Cold • Cool • Warm • Hot 29

Thermometry: the art of measuring hotness Match system X in thermal equilibrium with a system in the library. X An isolated system at a unknown place of hotness A library of isolated systems preserved at previously named places of hotness 30

Observation 5 (Fermi’s improved version of the Clausius statement of the second law of thermodynamics) When a system of hotness A and a system of hotness B are brought into thermal contact, if energy goes from B to A, energy will not go in the opposite direction. Places of hotness are ordered. • When two systems are in thermal contact, a difference in hotness gives heat a direction. • By convention, the system losing energy is said to be hotter than the system gaining energy. Fermi, Thermodynamics 31

Hotness “WATER” is lower than hotness “LEAD” heat liquid solid Water at melting point Lead at melting point Calorimetry determines the direction of heat and the quantity of heat. Thermometry uses only the direction of heat, not the quantity of heat. 32

In thermodynamics, the word “hot” is used strictly within the context of thermal contact. It makes no thermodynamic sense to say that one movie is hotter than the other, because the two movies cannot exchange energy. 33

Observation 6 (A generalization of the zeroth law) If hotness A is lower than hotness B, and hotness B is lower than hotness C, then hotness A is lower than hotness C. heat liquid solid WATER LEAD ALUMINUM GOLD hotness Order all places of hotness in one dimension. 34

Scale of hotness An ordered array of places of hotness Scales of other things • Scale of earthquake • Scale of hurricane • Scale of happiness • Scale of terrorism threat 35

Observation 7 Between any two places of hotness there exists another place of hotness. heat liquid solid WATER X LEAD hotness Name all places of hotness by a continuous variable. 36

Numerical scale of hotness • Problem: Every thermometer is unique. Newton’s thermometer disagreed with Galileo’s thermometer. • Invention (Fahrenheit 1720): Name two places of hotness after physical events. Name other places by thermal expansion of mercury • Science: Melting point. Boiling point. Name all places of hotness by a single, continuous variable. • Engineering: Why mercury? hotness 32 Freezing point of water 212 boiling point of water 37

Map one numerical scale of hotness to another Any increasing (linear or nonlinear) function will do. C = (5/9)(F – 32) 38

Long march toward naming places of hotness using a single continuous variable • • No useful way to name all streets by an ordered array. Cannot name streets with a continuous variable. We don’t know how to name all places of happiness. We laugh at rankings of universities. A library of isolated systems of previously named places of hotness 39

Non-numerical vs. numerical scales of hotness • A non-numerical scale of hotness perfectly captures all we care about hotness. • Naming places of hotness by using numbers makes it easier to memorize that hotness 80 is hotter than hotness 60. • Our preference to a numerical scale reveals more the nature of our brains than the nature of hotness. 40

Numerical values of hotness do not obey arithmetic rules • Adding two places of hotness has no empirical significance. It is as meaningless as adding the addresses of two houses. House number 2 and house number 7 do not add up to become house number 9. • Raising the temperature from 0 C to 50 C is a different process from raising temperature from 50 C to 100 C. 41

Observation 8 All places of hotness are hotter than a certain place of hotness. • • There exists a coldest place of hotness, but not a hottest place of hotness. Name the coldest place of hotness zero. Name all other places of hotness by a single, positive, continuous variable. Such a scale of hotness is called an absolute scale. Under rare conditions, however, negative absolute temperature has been attained. We will not consider these conditions in this course. 42

Observation 9 Thin gases obey the law of ideal gases. thermal contact Gas A’’ P’, V’, N’ P’’, V’’, N’’ Experimental discovery: The two gasses reach thermal equilibrium when P’V’/N’ = P”V”/N” Ideal-gas scale of hotness: t = PV/N. This scale of hotness has the same unit as energy, J 43

Observation 10 For a pure substance, its solid phase, liquid phase and gaseous phase coexist at a specific hotness and a specific pressure. 44

Kelvin scale of hotness 1. 2. 3. The Kelvin scale of hotness, T, is proportional to the ideal-gas scale of temperature. Write k. T = PV/N. The unit of the Kelvin scale, K, is defined such that the triple point of pure water is T = 273. 16 K exactly. Experimental value: k = 1. 38 x 10 -23 J/K. k is the conversion factor between the two scales of hotness, and is known as Boltzmann’s constant. • Experimental value: melting point of water at 0. 1 MPa: 273. 15 K. • Modern definition of the Celsius scale: C = T - 273. 15 45

Today’s temperature is… • • • 20 degree Celsius 68 Fahrenheit 293 Kelvin 404. 34 x 10 -23 J 0. 0253 e. V C = (5/9)(F – 32) K = C + 273. 15 1 K = 1. 38 x 10 -23 J 1 e. V = 1. 6 x 10 -19 J 46

Thermometry is a growing art Temperature affects everything. Everything is a thermometer. Today’s opportunity: The Internet of things. Air thermometer resistance thermometer liquid-in-glass thermometer pyrometer bimetallic thermometer thermocouple 47

Plan • Calorimetry • Empirical observations of thermal contact • Theory of thermal contact. Identify temperature as a child of entropy and energy. • Refinements and applications 48

The play of thermodynamics ENTROPY energy temperature space pressure heat capacity compressibility Helmholtz function enthalpy matter chemical potential charge electrical potential capacitance Gibbs free energy thermal expansion Joule-Thomson coefficient 49

The basic algorithm of thermodynamics • (entropy) = log (number of sample points). • Entropy is additive. • When a constraint internal to an isolated system fixes an internal variable at a value x, the isolated system flips in a subset of quantum states. • Denote the number of quantum states in the subset by W(x). • Call S(x) = log W(x) the entropy of the configuration of the isolated system when the internal variable is fixed at x. 1. Construct an isolated system with an internal variable, x. 2. When the internal variable is constrained at a value x, the isolated system has entropy S(x). 3. After the constraint is lifted, x changes to maximize S(x). 50

A thermal system modeled as a family of isolated systems • • The fire heats up a thermal system. Insulate the wall and an isolated system. A system isolated for a long time reaches a state of thermodynamic equilibrium. Isolating at various values of U, we obtain a family of isolated systems of one independent variable. • Use U as a coordinate. • Each state of thermodynamic equilibrium corresponds to a point on the coordinate. state U fire thermal system Fire changes U Fix V and N a family of isolated systems of a single independent variable, U 51

Entropy is a thermodynamic property • Entropy S = log (number of quantum states). • Isolating at various values of U, we obtain a family of isolated systems of one independent variable. • For this family of isolated systems, entropy is a function of a single variable, S(U). S S(U) fire thermal system U a family of isolated systems 52

A bit of high-school mathematics three ways to represent a function of a single variable, f(x) • Table • A curve in a plane • An equation 53

Construct an isolated system with an internal variable Diathermal wall. Thermal contact U’ S’(U’) thermal system A’ U’’ S’’(U’’) isolated system themal system A’’ Isolated system conserves energy U’ + U’’ = constant. U’ is an internal variable How does the isolated system partition energy between the two subsystems? Isolated system maximizes entropy Entropy is additive, but not constant. Change U’ to maximize S’(U’) + S’’(U’’) 54

Calculus of changes U’ S’(U’) heat d. U’ thermal system A’ U’’ S’’(U’’) themal system A’’ Isolated system conserves energy. Isolated system changes entropy • Entropies of the two subsystems: • Entropy of the isolated system: 55

Thermodynamic scale of temperature The internal variable U’ changes to maximize the entropy of the isolated system: U’ S’(U’) heat d. U’ thermal system A’ U’’ S’’(U’’) thermal system A’’ Thermal equilibrium • If , the entropy of the isolated system does not change when energy flows either way. Irreversibility and direction of heat: • If , the energy of the isolated system increases when energy flows in the direction d. U’ < 0. • If , the energy of the isolated system increases when energy flows in the direction d. U’ > 0. Thermodynamic scale of temperature defined (Clausius-Gibbs equation): 56

Theory explains empirical observations Thermal contact U’, S’(U’) U’’, S’’(U’’) 1/T’ = d. S’(U’)/d. U’ 1/T’’ = d. S’’(U’’)/d. U’’ thermal system A’ • • • Observation 1. Observation 2. Observation 5. Observation 6. Observation 7. Observation 8. isolated system themal system A’’ Thermal contact leads to thermal equilibrium Zeroth law Arrow of heat (Second law) Temperature is ordered Temperature is continuous Temperature is positive 57

An open system modeled as a family of isolated systems weights open system O vapor 2 vapor valve liquid A family of isolated systems S(U, V, N) fire Clausius-Gibbs equation 58

Thermodynamic scale of temperature coincides with ideal-gas scale of temperature Thermodynamic scale Ideal-gas scale • Historically the law of ideal gases was discovered empirically. • Later we will derive the law of ideal gases theoretically. 59

Unit of entropy Temperature in the unit of energy • Definition: Entropy = log (number of sample points), s = log W. • Entropy is dimensionless. • Define temperature (Clausius-Gibbs equation): 1/t = ds/d. U. • Temperature has the same unit as energy. Temperature in the unit of Kelvin • The committee adopts Kelvin as the unit for temperature T. t = k. T. • To preserve the Clausius-Gibbs equation 1/T = d. S/d. U, we write S = k log. W. • The unit for entropy is the same as that of k: J/K. 60

Experimental determination of thermodynamic scale of temperature Division of labor 1. For a simple system (e. g. , a thin gas), create a theory to relates thermodynamic scale of temperature to measurable quantities (theoretician). 2. Use the simple system to calibrate a thermometer by thermal contact (manufacturer). 3. Use thermometer to measure the temperature of any other system by thermal contact (patient). Theoretician manufacturer patient 61

Experimental determination of Energy-temperature curve T U fire thermal system • • a family of isolated systems Properties: T(U), S(U) Calorimetry measures internal energy U. Thermometry measures temperature T. Both U and T are thermodynamic properties. A thermal system is a system of a single independent variable. T(U). 62

Practical calorimetry Increase internal energy by a known amount without changing volume. liquid solid fire Thermal system How much heat? Thermal system Ice calorimetry Adiabatic system Frictional heating Adiabatic system Joule heating 63

Count the number of quantum states • U, T, S are thermodynamic properties. • A thermal system is a system of a single independent variable. T(U), S(U). T(U) Clausius-Gibbs equation T U S S(U) 1 T U 64

Heat capacity T T(U) 1 fire thermal system C U A family of isolated systems of a single independent variable, U 65

Plan • • Calorimetry Empirical observations of thermal contact Theory of thermal contact Refinements and applications. Ask what temperature can do for you. 66

Thermal system state U fire thermal system A family of isolated systems • Many states of thermodynamic equilibrium • One independent variable: U • Four thermodynamic properties: USTC • Three equations of state: S(U), T(U), C(U) T T(U) U Define temperature (Clausius-Gibbs equation): Define heat capacity: Given a thermal system, we obtain the equations of state as follows. • Perform calorimetry and thermometry to measure T(U). • Integrate the Clausius-Gibbs equation to obtain S(U). • Differentiate T(U) to obtain C(U). 67

4 properties: USTC 6 curves: US, UT, UC, ST, SC, TC Measure one curve. Calculate the other five. T S T(U) C S(U) U C(U) U intensive-extensive-extensive • • • U extensive-extensive Cross-plot T(U) and S(U) to obtain T(S). Cross-plot S(U) and C(U) to obtain C(S). Cross-plot T(U) and C(U) to obtain C(T). 68

Entropy as independent variable O vapor 2 liquid U T liquid 1 fire S thermal system isolated system T(S), U(S) T U S 69

Temperature as independent variable • Perform thermometry and calorimetry to measure U(T). • Differentiate U(T) to obtain C(T) = d. U(T)/d. T. • Integrate the Clausius-Gibbs equation to obtain S(T) T U(T) S T S(U) 1 C S(T) 1 T U U S 70

Experimental control of temperature water ice Ice-water mixture thermal bath (heat reservoir) thermostat 71

Thermodynamic model of reservoir of energy • • • A reservoir of energy is a thermal system, and has a single independent variable, UR. Entropy of the reservoir of energy is a thermodynamic property, SR(UR). The reservoir of energy has a fixed temperature, TR. Clausius-Gibbs equation: Integration: Reservoir of water Reservoir of energy Potential energy PE Internal energy UR Height h Temperature TR Weight w Entropy SR DPE = h. Dw DUR = TRDSR 72

A small system in equilibrium with a reservoir of energy • • (isolated system) = (small system) + (reservoir) Internal variable: the internal energy of the small system U • The isolated system conserves energy: • The isolated system changes entropy: • In equilibrium, the isolated system maximizes entropy: reservoir SR(UR) = SR(Ucomposite) – U/TR small system S(U) heat U 73

A small system in thermal contact, but of equilibrium, with a reservoir of energy • • (isolated system) = (small system) + (reservoir) Internal variables: the internal energy of the small system U, and something else Y • Entropy is additive: • Internal variables change to increase entropy (Clausius inequality): • Irreversible change in U because • Irreversible change in Y. S(U, Y) vapor reservoir SR(UR) = SR(Ucomposite) – U/TR heat U = Q liquid 74

Free energy • • (isolated system) = (small system) + (reservoir) Internal variables: the internal energy of the small system U, and something else Y • Entropy is additive: • Thermal equilibrium: TR = T. U(T, Y). • At a fixed T, the internal variable Y changes to maximize S(U, Y) – U/T. Function (T, Y) • Define the Helmholtz free energy (or Helmholtz function): F = U-TS. • Helmholtz function is an extensive property. • At a fixed T, the internal variable Y changes to minimizes F(T, Y). S(U, Y) vapor reservoir SR(UR) = SR(Ucomposite) – U/TR heat U liquid 75

Free energy, entropy, and temperature A thermal system modeled by a function S(U) Define temperature by the Clausius-Gibbs equation: d. U = Td. S Define the Helmholtz free energy: F = U - TS Calculus: d. F = d. U - Td. S - Sd. T d. F = -Sd. T U F(T) T 76

Graphic derivation of the condition of equilibrium Entropy is additive: In equilibrium, the isolated system maximizes entropy: entropy 1 energy U reservoir SR(UR) = SR(Ucomposite) – U/TR small system S(U) heat U 77

Convex function Define temperature (Clausius-Gibbs equation): S S(U) 1 Define heat capacity: T U T Equivalent statements • S(U) is convex • T(U) is an increasing function • C(U) > 0 U(T) 1 C U 78

Non-convex S(U) causes phase transition (i. e. , discontinuous change of state) S U T U 79

Co-existent phases of a pure substance water T ice mixture of ice and water ice Tm latent heat U fire thermal system isolated system T(U), S(U) 80

Thermodynamic theory of co-existent phases water ice s'(u’) s’’(u’’) water ice Each phase has its own s(u) function. 81

Rule of mixture corresponds to a line in the s-u plane water ice isolated system 82

Isolated system conserves energy and maximizes entropy water T mixture of ice and water Tm ice latent heat u Coexistent phases correspond to a common tangent. 83

Free energy and co-existent phases • Each phase has its own free-energy function. f’(T), f’’(T). • Mixture obeys the rule of mixture, f = (1 -x)f’(T) + xf’’(T) • x changes to minimize the free energy. f’ = f’’ u’ - Tms’ = u’’ –Tms’’ f f '(T) f ’’(T) Tm T 84

summary • Heat and temperature are distinct concepts, and are measured by separate experiments: calorimetry and thermometry. • Define temperature by the Clausius-Gibbs equation: • Define heat capacitance: • Define free energy: F = U – ST. • Thermal system: many states of thermodynamic equilibrium, one independent variation, five properties (USTCF) • Thermodynamics provides a theory of co-existent phases. 85