Energy: Mysterious and Amazing, Conserved and Conserving

1. Energy: Mysterious and Amazing, Conserved and Conserving R. Stephen Berry The University of Chicago Nerenberg Lecture The University of Western Ontario 21 March 2006

2. An Outline • The mystery and history of energy • Thermodynamics: Not quite what we were taught it is, in unusual regimes • Going beyond, to more efficient ways to use energy

3. Easy Question: What is Energy?

4. Easy Question? Oh, is it? What is Energy?

5. Can you say, tersely, what energy is? • Energy is one of the most incredible concepts to emerge from the human mind • Is it a discovery or an invention?

6. Energy is an abstract concept that ties together a remarkable range of dissimilar human experiences • And does it in a way with astounding quantitative predictability!

7. It seems an obvious concept, even to science students today • But it wasn’t obvious at all for a long, long time • Bacon, Galileo: heat is motion • Rumford: mechanical work converts into heat • But is heat a fluid, “caloric,” or is it matter in motion?

8. Commonality of heat and light • Scheele (1777) identified “radiant heat” to establish an equivalence between heat and light • But he recognized two kinds of transfer, essentially radiation and convection • Lavoisier & Laplace (1783): whether caloric or motion, there is a “conservation of heat”

9. An indication of the problems: A controversy • What is the ‘measure of motion’? • Is it mass x velocity, or mass x (velocity)2 ? • This was the conflict between the Leibnitzians and Cartesians • At that time, it was inconceivable that both could be valid!

10. How to account for heat that doesn’t change temperature • Recognize latent heats of phase changes, and role of heat in changing densities • Rumford: heat has no weight • Young: heat and light are related • Leslie (1804): distinguishes conduction, convection and radiation and uses the term “energy” without defining it

11. Fourier: Quantifies Heat • Heat capacity • Internal conductivity • External conductivity (radiation, convection) • Quantification of heat flow and transfer, with differential eqns.

12. The Steam Engine: Watt • The external condenser • The direct measure of pressure as a function of volume, to determine efficiency (the Indicator Diagram, p vs. V) • The use of high pressures and therefore of high temperatures

13. Carnot: The Breakthrough, stimulated by applications • Heat is ‘motive power’ that has changed its form • “The quantity of motive power in nature is invariable” • In effect, Energy is conserved!

14. More from Carnot • The invention of the reversible engine and the demonstration that it is the most efficient engine possible • The determination of that maximum efficiency, and that no engine can do better

15. Aha! Conservation of Energy! • J. R. Mayer (1842-48) stated the principle explicitly, and included energy from gravitational acceleration • Quantified the mechanical equivalent of heat • Included living organisms

16. Joule, of course! (1840’s,’50’s) • Brought electromagnetic energy into the picture • Measured mechanical equivalent of heat • Showed that expansion of a gas into a vacuum does no work

17. Creation of Thermodynamics • Motivation: How little fuel must I burn, in order to pump the water out of my tin mine? • Carnot confronted and solved this problem, but the great generalization came later

18. The First Law • Two kinds of variables: State variables, e.g. pressure p, volume V, temperature T • Process variables, energy transferred either as heat Q, or as work W. • The Law: the change of energy, E = Q – W, whatever the path

19. This law states conservation of energy • Whatever the path, only the end points determine the energy change • If the final and initial states are the same, the energy of the system is unchanged

20. The Second Law • The randomness--or entropy--or the number of microstates the system can explore--never decreases spontaneously • Decreasing entropy requires input of work • Corollary: Max efficiency is (Thigh–Tlow)/Thigh

21. The Third Law • There is an absolute zero of temperature, 0o K or –273o C • You can never get there; it is as unreachable as infinitely high temperature • But we can now get pretty cold, as low as 10–8 o K

22. Einstein: Thermodynamics is, among all sciences, the one most likely to be valid • Hence we can think of thermodynamics as the epitome of general scientific law • But we sometimes lose sight of what is truly general and what is applicable for only certain kinds of systems or conditions

23. A common, elegant presentation • Thermodynamics has two kinds of state variables: • Intensive, independent of amount, e.g. Temperature, pressure • Extensive, directly proportional to amount, e.g. mass, volume

24. Also two kinds of relations • General laws, the Laws of Thermodynamics • Relations for specific systems, e.g. equations of state, such as the ideal gas law, pV = nRT, giving a third quantity if two are known (Remember that one?)

25. Degrees of freedom • How many variables can we control? For a pure substance, we can change three, e.g. pressure, temperature and amount of stuff • Fix the amount and we can vary only two • The equation of state tells us everything else

26. But Equations of State are usually not simple • The equation of state for steam, used daily by engineers concerned with real machines, requires several pages to write in the form they use it! • Not at all like pV=nRT!

27. Generalize to find optimal performances • Thermodynamic Potentials are the quantities that tell us the most efficient possible energy use for specific kinds of processes, different potential for different processes • All use the infinitely slow limit, as Carnot did, to do best

28. Some jargon • Names for some thermodynamic potentials are “free energy,” “availability,” “enthalpy,” “exergy,” and energy itself; • The change in the appropriate potential is the minimum work we must do, or the maximum we can extract, for that process

29. The subtle profundity of thermodynamics • The Gibbs phase rule: relates the number of degrees of freedom, f, to the number of components c (kinds of stuff) and the number of phases present in equilibrium, p: • f = c – p + 2, the simplest equation in thermodynamics, perhaps in all science

30. A simple relation • The amount of each component can be varied at will • Each phase, e.g. liquid water, ice or water vapor, has its own equation of state, implying a constraint for each phase • One substance, one phase, yields two degrees of freedom, as we saw

31. Water vapor: any T or p is okay

32. But look now, if there is liquid also:

33. What’s profound about the Gibbs phase rule? • The f comes by definition • The c is obviously our choice • The p is the number of constraints • Hence all these are easy and obvious • It’s the 2 that is profound! Only experience with nature tells us what that number is!

34. The real generality of thermodynamics • Very big systems--galaxy clusters--and very small systems--atomic clusters--should all be describable by thermodynamics • What’s the predominant energy of a galaxy cluster? Gravitation, of course

35. What’s the gravitational energy of two objects? • Inversely proportional to distance of the objects, • Directly proportional to the product of their masses, m1 x m2 ! • This is notlinear in the mass! • Astronomers created nonextensive thermodynamics to deal with this.

36. Another case where thermodynamics holds, but not as it’s usually taught • Very small systems, e.g. nanoscale materials, composed of thousands or even just hundreds of atoms • The distinction between component and phase can be lost, so the Gibbs phase rule loses meaning

37. With very small systems, • Two phases may coexist over a bandof pressures and temperatures, not just along a single coexistence curve • More than two phases can exist in equilibrium over a band of conditions • Phase changes are gradual, not sharp

38. Can we do thermodynamics away from equilibrium? • Close to equilibrium, Lars Onsager showed a fine way to do it, back in the 1930’s • Further away from equilibrium, one needs more variables to describe the system • Can we guess what variables to use? Sometimes, not always

39. Create a thermodynamics for processes that must operate in finite time • We can, for many kinds of finite-time processes, define quantities like traditional thermodynamic potentials, whose changes give the most efficient or effective possible use of the energy for those processes

40. Finite-time potentials • It is possible to define and evaluate these, for specific processes, to learn how well a process can possibly perform • It is then possible to identify how, in practice, we can design processes to approach the limits that are those ‘best performances’

41. Example: the automobile engine • The gas-air mix burns, the heat expands the gas, driving the piston down, so the pistons go up and down • The connecting rod links piston with driveshaft, changing up-down motion into rotation • Does the piston, in an ordinary engine, follow the best path to maximize work or power? NO!

42. So how can we do better? • Change the time path to make the piston move fastest when the gas is at its highest temperature!

43. Changing the mechanical link would improve performance about 15% • Red: conventional time path of piston; black: ideal, given a maximum piston speed

44. One other example • Distillation, a very energy-wasteful process • But make the temperature profile along the column a control variable and the energy waste goes way down • One such column is going up now,in Mexico

45. So what have we seen? • Energy is an amazing concept, subtle, powerful, elegant, general, • Isn’t it incredible that we found it! • Its quantitative, predictive power is perhaps the epitome of what science is about! • It is important for all its aspects, from the most basic to the most practical and applied

46. Thank you!