Energy: Mysterious and Amazing, Conserved and

ConservingR. Stephen Berry

The University of Chicago

Nerenberg LectureThe University of Western Ontario21 March 2006

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

Easy Question: What is Energy?

Easy Question? Oh, is it?

What is Energy?

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?

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!

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?

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”

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!

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

Fourier: Quantifies Heat

• Heat capacity

• Internal conductivity

• External conductivity (radiation, convection)

• Quantification of heat flow and transfer, with differential eqns.

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

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!

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

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

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

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

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

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

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

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

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

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

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?)

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

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!

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

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

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

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

Water vapor: any T or p is okay

But look now, if there is liquid also:

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!

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

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 not linear in the mass!• Astronomers created

nonextensive thermodynamics to deal with this.

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

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

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

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

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’

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!

So how can we do better?• Change the time path to make the

piston move fastest when the gas is at its highest temperature!

Changing the mechanical link would improve performance

about 15% • Red: conventional time path of piston;

black: ideal, given a maximum piston speed

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

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

Thank you!