Energy: Mysterious and Amazing, Conserved and Conserving

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Energy Mysterious and Amazing, Conserved and
Conserving

• R. Stephen Berry
• The University of Chicago
• Nerenberg Lecture
• The University of Western Ontario
• 21 March 2006

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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

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

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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!

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It seems an obvious concept, even to science
students today

• But it wasnt 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?

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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

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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!

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How to account for heat that doesnt 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

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Fourier Quantifies Heat

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

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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

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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!

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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

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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

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Joule, of course! (1840s,50s)

• Brought electromagnetic energy into the picture
• Measured mechanical equivalent of heat
• Showed that expansion of a gas into a vacuum does
  no work

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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

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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

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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

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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
  (ThighTlow)/Thigh

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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 108 o K

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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

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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

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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?)

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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

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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 pVnRT!

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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

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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

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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

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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

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Water vapor any T or p is okay
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But look now, if there is liquid also
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Whats 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
• Its the 2 that is profound! Only experience with
  nature tells us what that number is!

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The real generality of thermodynamics

• Very big systems--galaxy clusters--and very small
  systems--atomic clusters--should all be
  describable by thermodynamics
• Whats the predominant energy of a galaxy
  cluster? Gravitation, of course

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Whats 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 not linear in the mass!
• Astronomers created nonextensive thermodynamics
  to deal with this.

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Another case where thermodynamics holds, but not
as its 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

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With very small systems,

• Two phases may coexist over a band of 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

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Can we do thermodynamics away from equilibrium?

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

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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

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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

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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!

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So how can we do better?

• Change the time path to make the piston move
  fastest when the gas is at its highest
  temperature!

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Changing the mechanical link would improve
performance about 15

• Red conventional time path of piston black
  ideal, given a maximum piston speed

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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

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So what have we seen?

• Energy is an amazing concept, subtle, powerful,
  elegant, general,
• Isnt 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

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Thank you!