TEMPERATURE

1. 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

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

3. The basic algorithm of thermodynamics • Construct an isolated system with an internal variable, x. • When the internal variable is constrained at x, the isolated system has entropy S(x). • After the constraint is lifted, x changes to maximize S(x). • (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.

4. Classify systems according to how they interact with the rest of the world

5. An open system modeled asa 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 2O vapor a family of isolated systems S(U,V,N) vapor open system valve liquid liquid fire

6. Basic problem of thermodynamics adiabatic, stationary, impermeable wall diathermal, moving, leaky wall 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’’) U’, V’, N’ S’(U’, V’, N’) U’’, V’’, N’’ S’’(U’’, V’’, N’’) isolated system open system A’ open system A’’

7. Calculus

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

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

10. 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

11. 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.

12. 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.

13. 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

14. 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

15. 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 twothermodynamic properties, P and V. weights 2O P state vapor vapor isolated system closed system liquid liquid V fire

16. 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?

17. 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 2O state B vapor P O vapor vapor liquid liquid state A liquid V Isolated system A closed system isolated system B U(PA,VA) W, Q U(PB,VB) fire

18. 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

19. 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?

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

21. 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

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

23. 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

24. Thermal contact stationary, impermeable, but diathermal wall • 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). heat isolated system

25. Observation 1Two 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.

26. 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.

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

28. Observation 4For 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. • For water, this place of hotness has many names • Melting point • Freezing point • 0 Celsius • 32 Fahrenheit • 273.15 Kelvin liquid

29. Name places of hotness The same way as we name streets • Harvard, Cambridge, Oxford… • Washington, Lincoln,… • 5th Avenue, 6th Avenue,… After physical events. • WATER at the melting point • 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 m3/kg Relative terms • Cold • Cool • Warm • Hot • Streets are physical objects. • Names of the streets are arbitrary.

30. 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

31. Observation 5 (Fermi’s improved version of the Clausius statement of thesecond 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

32. Hotness “WATER”is lower than hotness “LEAD” heat liquid liquid solid 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.

33. 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.

34. Observation 6 (A generalization of thezeroth 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 heat heat liquid liquid liquid liquid solid solid solid solid WATER LEAD ALUMINUM GOLD hotness Order all places of hotness in one dimension.

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

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

37. 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 32 212 Freezing point of water boiling point of water

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

39. 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

40. Non-numerical vs. numerical scales of hotness • Anon-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.

41. 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 0C to 50C is a different process from raising temperature from 50C to 100C.

42. Observation 8All 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.

43. Observation 9Thin gases obey the law of ideal gases. thermal contact Gas A’ P’,V’,N’ Gas A’’ 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

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

45. Kelvin scale of hotness • The Kelvin scale of hotness, T, is proportional to the ideal-gas scale of temperature. Write kT = 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 = 1.38x10-23 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: 273.15K. • Modern definition of the Celsius scale: C = T - 273.15

46. Today’s temperature is… • 20 degree Celsius • 68 Fahrenheit • 293 Kelvin • 404.34x10-23 J • 0.0253 eV C = (5/9)(F – 32) K = C + 273.15 1 K = 1.38x10-23 J 1 eV = 1.6x10-19J

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

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

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

50. The basic algorithm of thermodynamics • Construct an isolated system with an internal variable, x. • When the internal variable is constrained at a value x, the isolated system has entropy S(x). • After the constraint is lifted, x changes to maximize S(x). • (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.