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H A&S 220c Energy and Environment: Life Under the Pale Sun
Fall 2004 3.01
14x04
Note on Essay 1. The essays you produced have described scenes, often from your
youth, often in woods or forest or on a beach. The ‘functional’ part of these landscapes, part 2 of
the essay, described natural cycles of many kinds: animals, plants, insects sharing a common turf,
sometimes supporting, sometimes feeding one another. I was interested to see in many of these
portraits that the author was present, a part of the function. That is an important lesson about
landscapes, that the observer affects them, either (i) because the observer’s reaction to Nature
colors the description, or (ii) because the observer by being there materially alters that landscape,
now or in past or future. The latter idea is important to environmental studies, because it is the
beginning of a realization that human alteration of Nature can be subtle, hard to see, yet is very
pervasive. Observing and acknowledging this alteration is important to building a sustainable
future. In Farley Mowatt’s Never Cry Wolf (book and film) the central character, an idealistic
biologist studying wolves in Alaska, turns away from his beloved animals as he realizes that the
most kindly, benign visitor to the wilderness brings in his/her wake a parade of human
interventions. Even at the level of simple tourism, as some critics of the Sierra Club put it, we can
‘love the wilderness to death’ by making it a poster child, inviting hordes of visitors to famous
beauty spots.
In the extreme, the functions found in the natural world involve energy cycles we have
been studying, beginning with the Big Bang, or whatever cosmic event marked the beginning of
time knowable to us. No one went quite so far as this in the essays, but it comes to mind when
reading a quotation that you have seen; I repeat it here because it fits so well with the science of
energy transformation, carbon cycle and photosynthesis:
"The world looks so different after learning science. For example, trees are made of air, primarily. When
they are burned, they go back to air, and in the flaming heat is released the flaming heat of the sun which was
bound in to convert the air into tree. And in the ash is the small remnant of the part which did not come
from air, that came from the solid Earth, instead.
These are beautiful things, and the content of science is wonderfully full of them. They are inspiring and
they can be used to inspire others." -- Richard Feynman, physicist, California Inst. of Technology.
Trees are made of air, plus some nutrients, built into a massive store of chemical
energy...which of course came from the energy of the sun.
II. SCIENCE CORE: PHYSICS OF ENERGY, continued
We are beginning to fill in the list of ‘forms of energy’ and to give examples of the
magnitudes...the amounts...of energy present in familiar objects. The kinetic energy in a 1 kg.
rock moving at 1 meter/sec is ½ Joule. As shown earlier, Ch. 2 sec. 2.11, a flowing river can
contain a large kinetic energy by virtue of its size...its mass (although there it makes sense to
consider the power flowing past an observation point on the river, as well as the kinetic energy
per kg. of water). We will see again that thermal energy is ‘rich’, or concentrated and chemical
energy is even more so.
Thermal energy and the heat engine. In sections 3, 10 and 11 of Ch. 2 we introduced
the 1st law of thermodyamics, and mentioned the idea of an engine that produces mechanical
energy by converting some thermal or chemical energy. If you squeeze a gas it becomes
warmer…instantly. This happens because ‘squeezing’ means exerting a force and
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changing the volume of gas. Recall that exerting a force on a moving object requires
work in the amount:
work = force x distance traveled
This is equal to the change in energy of the moving object. In our case the object is the
gas, and compressing it requires work, and the work goes into internal heat energy of the
gas: it warms up. Thinking of a gas like air as billiard-ball like molecules flying around,
bouncing off each other and bouncing off the walls of their container, you can imagine
that squeezing the gas could make the molecules move faster. We illustrated this by
squeezing the air in a spherical glass vessel, measuring a 20% increase in pressure, and
seeing the temperature rise (by the change in color of a liquid crystal thermometer).
When you hit a tennis ball with a racquet, the spped of the ball is greatly increased. Now think of a billiard ball
bouncing off a wall. If no energy is lost (the collision is called ‘elastic’), then the speed of the ball perpendicular to the
wall after rebounding, U2, is the same as before rebounding, U1, yet in the opposite direction: U2=U1. If, instead, the
wall is moving steadily toward the ball before the collision, then the ball’s rebound speed will be greater, U2 = U1 + 2
Uwall . We can see why by imagining that we are riding with the moving wall. In this frame of reference, the ball
bounces simply, conserving its velocity which is U1+Uwall both before and after the collision. An observer not riding
on the moving wall thus sees speeds U1 and U1 + 2Uwall before and after the collision.
In the 1st Law of Thermodynamics, that is the thermal energy equation,
∆Eint = ∆ ' Q + ∆ 'W
we just described an event where the internal energy change (left-hand side) was
produced by work done by squeezing, ∆’W, which is equal to P∆v, or the product of
pressure and the change in volume, v. The middle term, ∆’Q is the heating, for example
by a candle, which was not active. In the simple case above, P (force per unit area)
multiplied by ∆’v (change in volume of gas) is the same as force multiplied by distance
traveled. Check the units: force m-2 x m3 = force x m.
The idea of temperature. Temperature arose as an intuitive feeling, and it was
then noticed that some clear physical events correspond well to this feeling. The height
of a column of mercury or alcohol, for example, provides a useful scale that seems about
right. In this way temperature could be measured. The expansion of a material as it is
warmed can be analyzed in the simplest case: an ideal gas. This perhaps should be called
‘idealized gas’. It is a model of a real gas based on having molecules far apart (a ‘dilute
gas’) and simplifying their collisions as being those of billiard balls; no energy loss
(known as ‘elastic collisions’). The kinetic molecular theory (KMT) for this gas provides
a theoretical model of the distribution of velocity among the molecules. In doing so it
relates their kinetic energy to the pressure that is felt on the walls containing the gas.
Working models illustrating the KMT model of a gas can be found at
http://comp.uark.edu/~jgeabana/mol_dyn/KinThI.html
http://www.chm.davidson.edu/Chemistry/Applets/KineticMolecularTheory/BasicConcepts.html
Of great importance, the Kinetic Molecular Theory tells us that temperature is
proportional to the kinetic energy of molecules making up an ideal gas. And, what we call
thermal energy is the kinetic energy of molecules of an ideal gas.
Equation of state for an ideal gas. The results of the kinetic-molecular theory of
gases (based on a billiard-ball model of molecules and their collisions) include an
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equation which connects the pressure, P, temperature, T, and volume, v, of a gas. [Don’t
confuse the volume v (m3) with the speed V (m sec-1). It grew out of experimental
discoveries, particularly those of Boyle and Charles. Boyles’ experiments showed that if
the temperature of a gas is held constant, its volume v varies inversely with its pressure,
P, or
v ∝ 1/P (for constant temperature, fixed mass of gas).
Charles’ experiments found that, for a fixed amount of gas at constant pressure, the
volume varies in proportion to temperature,
v ∝ T (at constant pressure, with fixed mass of gas)
If n is the amount of the substance (measured in moles, a chemical measure that specifies
the number of molecules), then clearly
V ∝ n (at constant temperature and pressure)
All three relationships are combined to give the equation of state,
Pv = nR* T
where R* is a constant, equal to 8.3145 Joule mole-1 K-1. The units here are:
pressure P (=force/area) in kg m-2 sec-2 ≡ Newton m-2, also as a ‘Pascal’.
temperature T in K (degrees Kelvin or just ‘Kelvins’); the equation of state doesn’t work
if degrees Celcius or Fahrenheit are used.
volume in m3. Instead of the volume, v, we can introduce the mass density, ρ, which is
the mass divided by the volume. Then the equation of state becomes
P = ρRT
where R = nR*/M = R /molecular weight of the gas.
For dry air, R = 287.04 Joules kg-1 K-1 .
The complete derivation can be found in most chemistry texts.
Given this relationship between P, v and T (or P, ρ and T) is seems natural to plot
the pressure and volume of a gas on a graph of P against v.
Specific heat capacity. The result of this model of a gas is that the thermal
energy, Eint, which is the kinetic energy of the molecules of gas, is a constant multiplied
by the temperature. We write this, for a gas made up of single atoms, as
Eint = 3/2 RT
where R is the gas constant given above, and T the absolute (Kelvin) temperature (that is,
the temperature in degrees Celcius + 273.15; a balmy Seattle day is about 300K). The
constant 3/2 R is also known as Cv, the ‘specific heat capacity’ of the gas, relating
temperature change to energy changes when the volume of gas is held constant.
It is useful also to consider energy changes due to heating at constant pressure;
there we use the symbol Cp and find that
Cp – Cv = R
so C p = 5/2 R for this one-atom gas. These numbers change for a gas like air which is
99% made of molecules with two atoms (a molecule of oxygen has two oxygen atoms
(21% of air) and a molecule of nitrogen also has two atoms of nitrogen (78.1% of air).
They are then
Cv = 5/2 R, Cp = 7/2R {specific heat capacities for 2-atom molecules}
For dry air, Cp = 1004.6 J kg-1 K-1 and Cv = 717.6 J kg-1 K-1 {here J stands for Joules}.
For fresh water, Cp = 4184 J kg-1 K-1 at temperature of 20C.
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These relationships between temperature and energy are in a sense the ‘bottom
line’ of our discussion of thermal energy. Consider how remarkable this is: just as
Benjamin Franklin, waving his hollowed out cane over the water at Clapham Common in
London, came close to measuring the size of a molecule of oil, (Spherical Cow, p.5) we
here can know the average speed of molecules simply by observing their energy…taking
their temperature or ‘heat storage’. The calculation is this:
kinetic energy of molecules, KE = ½ MV2 (based on the average of V2);
1 kg of dry air at 200 C (that is 293.15K) has internal thermal energy
equal to CvT, or 717.6 x 293.15 = 2.1 x 105 J kg-1. Setting this equal to the KE, we find
that
V = √(2 KE/M)
= √(2 x 2.1 x 105/1)
= 648 m sec-1.
This is a high speed, but not beyond human experience. For example it is not very much
greater than the speed of sound, which is equal to √ (RT Cp/Cv), or 343 m/sec. Of course
the molecules have a range of speeds; this example gives just their average. The
simulations of a box of molecules on the Web illustrate this.
The essence of the ‘richness’ of thermal energy is seen in these numbers: the
molecules of air are moving faster than the typical speeds of the winds (perhaps 10 m
sec-1), hence the microscopic kinetic energy is greater typically than the kinetic energy
we see.
In doing calculations of any kind it is good to check for mistakes by comparing
the units on either side of an equation. Here V is in meters per second, m sec-1. The
right-hand side is in uints of √(kg m2 sec-2/kg) which indeed is just m sec-1. So there is a
better chance that we are correct.
Solids and liquids do not have a simple Kinetic Molecular Theory as do gases. In
solids, atoms are bound together rather rigidly, for instance in a geometrically regular
crystal. In liquids, molecules are loosely bound together. In either case, vibration of
atoms or molecules is responsible for what we sense as temperature. The specific heat
capacity of common minerals is about 1300 J kg-1, and other solids range widely:
Cp = 910 J kg-1 K-1 (aluminum)
390
(copper)
Metals conduct heat very well but as you see they don’t ‘store’ it well. The amazing
substance water has high heat capacity:
Cp = 4184 J kg-1 K-1 (water)
and we saw that air (with no moisture in it) has
Cp = 1005 J kg-1 K-1 (dry air)
While physicists have not found an accurate theory to give these numbers there is
a tendency for lighter molecules or atoms to imply higher heat capacity: this is seen in the
formulas above for the case of an ideal gas, in which Cp and Cv vary inversely with
molecular weight. If a gas is made up of a number of different molecules, each with a
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different molecular weight, there is a strong tendency for the mean speeds to vary in just
such a way as to make the kinetic energies, KE, the same for each kind of molecule.
Thus hydrogen molecules fly around much faster than oxygen molecules, in air.
Heat engine. If the gas is allowed to expand it could lift a weight at the same time.
Lifting the weight means giving it some extra potential energy. Or, if the gas expands it
could do some useful mechanical work as in turning a crank or a propellor.
A heat engine takes these two ideas and combines them:
•1 heat an enclosed volume of gas at constant pressure (with a weight
sitting on top of it…see sketch): the gas expands, lifting the
weight
•2 stop heating and remove the weight, which is higher than when it
started; the gas expands some more, with no energy input
(‘adiabatically’).
•3 now cool the gas down, removing just the same amount of heat added
in step 1. It will shrink in volume.
•4 place another weight on top, which will compress the gas further.
We are back where we started, after a 4-stage cycle.
The figure below is a wonderful way to represent this heat engine cycle. Plot the
pressure of the gas, P, on the vertical axis and the volume of the gas, V on the horizontal
axis. This so-called P-V diagram will show you a closed curve as the engine moves
cyclically. The heat-engine curve can differ in detail, but it always has one key feature:
gas expands when pressure is relatively high and contracts when the pressure is relatively
low. When the weight is sitting on the gas, pressure is higher, when the weight is
removed, pressure is lower.
How much mechanical energy is this little engine producing? That is how much
work can it do? As we described earlier, lifting a weight to a height h requires an amount
of work;
work is quantitatively given by force times the distance traveled by the forced object.
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This work creates potential energy which can then be utilized (by dropping the weight for
example) to make kinetic energy.
In the gas experiment, the force per unit area is the pressure P, and in the twodimensional sketch above, the distance is proportional to volume V. Since force times
distance is proportional to P times V, during each piece of the cycle the work done is
proportional to the area beneath the PV curve. When it travels left the sign is negative,
subtracting off from the work done on the rightward part of the cycle. Thus
the work done by the gas during a complete cycle is equal to the area inside the closed
curve on the P-V diagram.
A surprising feature of this heat engine is that it doesn’t really ‘use up’ any heat
energy. That energy goes from a warm place (like the candle) to a cold place (whatever
is cooling the gas down, say an ice cube). It goes to a lower temperature. We sense that it
is the contrast in temperature, cold next to hot, that makes engines run. Having a lot of
hot things around, all at the same temperature, would not allow us to do much of
anything. Plenty of energy but since heat flows from a hotter body a colder body and it is
heat flow that makes things go, it is also temperature differences that also make things
go. Each time we run our heat engine we bring the Universe close to a uniform
temperature. When and if we get there, there will be no more heat engines.
The statement above in italics is the simplest form I know of the ‘2d Law of
Thermodynamics’ that talks about a quantity called entropy. It has deep philosophical
connections as you might guess from the paragraph above. It also has a connection with
patterns, information, chaos…lots of trendy ideas about ‘order’ and ‘disorder’. You may
encounter these ideas far away from science, and find them interesting. The Universe,
we imagine is heading from order to disorder, from hot and cold bodies to a single, cool,
uniform temperature. We as human beings represent very unlikely combinations of
atoms; we are examples of organization, islands of ‘low entropy’ trying to fight off the
forces that would disorder us, turning us into seas of randomly distributed atoms. It is
enough to build a religion on.
The Stirling Cycle engine in the lab takes this kind of heat-work-cycle and puts it
all together. It is complicated to figure out all its parts, but worth the effort. Notice that
‘mechancal energy produced’ or ‘work done’ by this engine is pretty hard to see: it is in
the rolling marbles in the glass tube back and forth. We could instead drive a propellor
which can be observed and measured…yet each cycle the mechanical energy is lost as the
marbles hit the end of the tube and stop (ironically losing their kinetic energy to a slight
amount of heat…pity). That’s a confusing ‘error’ in this demonstration. But the rest is
convincing. It happens that this PV cycle can be quite efficient in a practical application.
Let’s define efficiency as the mechanical work done, divided by the heat energy input to
the engine. For the record, the heat engine has a maximum possible efficiency (the
‘Carnot’ efficiency) of
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efficiency =
T1 − T2
T1
where T1 is the absolute (degrees Kelvin) temperature of the heat put into the engine (the
‘intake’ temperature) and T2 is the temperature at which it is taken out (the ‘exhaust’
temperature). What this represents is the mechanical power (energy per unit time)
produced by the engine, divided by the heat energy flowing into it. On the PV diagram,
if the gas is an ‘ideal’ gas like air, without moisture in it, then P and V are related by the
equation of state,
PV = nRT
where T is temperature and n = mass of gas divided by molecular weight and R is a
constant, the ‘universal gas constant’. Thus curves PV = constant on the diagram
correspond to constant temperature curves. By making intake and exhaust temperatures
very different, we make the heat engine’s cyclic curve larger (reaching larger range of
temperature) and thus it will enclose more area, and correspond to more efficiency
This efficiency is greatest if the difference between intake and exhaust
temperatures is large. It shows why we want to concentrate sunlight before using it to
drive an engine: we will get more mechanical energy out. Your car is more powerful on
a cold winter’s day than a hot summer’s day (other things being equal).
The heat engine takes heat from hot regions and moves it to colder regions, taking
out some mechanical energy on the way. Like many of our labs, the heat engine is
‘reversible’, meaning that it can run backward. We can put in mechanical work and
make heat flow away from a cold region to a hot region. This increases the temperature
difference rather than reducing it. It is a simple refrigerator! Lord Kelvin (otherwise
known as William Thompson, from northern Ireland) discovered many things in physics
and fluid dynamics; the absolute temperature scale commemorates him. But to the
general public he was made famous by the Kelvinator, a popular brand of refrigerator.
Kelvin promoted the use of electricity in England, donating the money to have his college
at Cambridge University ‘electrified’. He also was an early advocate of chilling foods to
preserve them, hence his memorial in the form of a refrigerator.
Summary . Thermal energy is a ‘hidden’, microscopic form of mechanical
energy. Kinetic Molecular Theory (KMT) is a ‘model’ of a real gas which quite
accurately gives us relationships between molecular motion and the
temperature, pressure and volume we experience at the human scale. It is a
mathematical ‘microscope’ allowing us to ‘see’ the speed of molecules that make
up a gas like air. In this theory, molecules act like billiard balls, occupying a
small part of the volume of the gas, flying around without loss of energy,
colliding with one another after moving 10 molecular diameters or so. In the
simplest ideal gas the internal energy is given by
Eint = CvT
where Cv is a constant known as the ‘specific heat capacity’ (its units are
Joules/(kg 0K)). The temperature, T, is simply proportional to the kinetic
energy, KE, of the gas molecules. They bounce around and exert a force on the
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walls of their container. The pressure is the force, per unit area, and from
Newton’s 2d Law, P is found to be simply proportional to the molecular KE.
Given that both T and P are proportional to KE, they are proportional to
one another. This gives us the equation of state of an ideal gas, Pv = nR*T or in
terms of fluid density, ρ,
P = ρ RT
where R is a constant for a given gas, R = R*/molecular weight of gas. While it
is derived purely with theory, this equation agreed well with prior experiments
relating P and v, and P and T.