THE UNIVERSE is a violent place, and its violence is irreversible. Stars
flare into being and burn out, bombs explode and cause devastation, we are
born, grow old and die, time moves on and all that is left to mark the
maelstrom of what passed before is an invisible glow in the night sky, the
embers of actions, cool dark matter and heat radiation spreading into the
void. Cars, power stations, people and plants feed on concentrated sources of
highly organised energy and transfer it to heat 鈥� always heat at ever lower
temperatures.
Long ago it was believed that expended effort had been 鈥渦sed up鈥� and that
was an end to it, but in the 1840s the British physicist James Joule
demonstrated that mechanical work can be converted to heat and that the heat
generated is equal to the work done. His ideas paved the way for the First Law
of Thermodynamics: 鈥淗eat is a form of energy and energy is conserved鈥�.
Advertisement
We are surrounded by processes whose net effect is the transfer of energy
to ever colder and more diffuse heat. But what about the reverse process,
converting heat to work? This is not ruled out by the First Law and would be
of great benefit to society. Imagine a generator producing electricity
directly from the thermal energy of the atmosphere or oceans. Nature and the
history of technology suggest there are problems in doing this.
Impossible processes
Time鈥檚 arrow
THE SECOND Law began life as a statement of what cannot happen. The British
physicist William Thomson claimed, in 1851, that: 鈥淎ny process whose sole
effect is the conversion of heat to work is impossible鈥�. Cold water poured
into a warm drink cannot spontaneously freeze and cause the drink to boil; we
cannot warm ourselves with snow; a stone block basking in warm sunshine cannot
suddenly cool and use the energy released to hurtle off into space. This is a
pity because the thermal energy of an object at room temperature is enough to
propel it at several hundred metres per second 鈥� and it can easily he warmed
up again.
On the other hand we can convert work entirely into heat (think about
rubbing your hands together), so the Second Law defined an asymmetry in
nature, a direction of change which distinguishes the past (lots of
concentrated energy sources capable of delivering energy at high temperature)
from the future (low temperature thermal equilibrium) 鈥� energy flows in nature
reveal a thermodynamic 鈥渁rrow of time鈥�.
Time鈥檚 arrow is obvious in the irreversible processes that surround us.
Glasses shatter when dropped, but we never see the shattered fragments coming
together to form a glass which leaps back onto the table; burnt fuel is lost
forever; mixed paint stays mixed however much we stir it; buildings crumble
and the Universe expands and cools.
Given a shuffled sequence of time-lapse photographs of an ink drop mixing
in water we could easily arrange them in their correct temporal order. We know
what happens when things mix 鈥� there is a spreading, an evening out of
concentrations, a gradual evolution from order to disorder. Applied to the
Universe as a whole the Second Law implies a gradual running down as energy
from chemical and nuclear reactions is transferred to heat and becomes
unavailable for conversion to useful work. The German physicist Hermann
Helmholtz called this 鈥渢he heat death of the Universe鈥�.
The early descriptive versions of the Second Law needed to be tightened up.
How much work can be extracted from heat? How can we optimise the performance
of heat-powered engines? The incentive to answer these questions came from the
development of the steam engine but the purpose of any heat engine is the
same, to harness some of the heat that flows from a hot source (for example,
the heat supplied to steam from burning fuel) to a cold sink (for example, the
condenser) and to use it to do work (for example, pump water out of mines, or
generate electricity).
Of heat and work
Steam engines
IN THE early 19th century the French physicist Sadi Carnot analysed the
general case of an ideal reversible heat engine operating between two fixed
temperatures (a source at T1 and sink at T2), and proved
that no heat engine could better its efficiency. All real heat engines are
irreversible because factors like friction in the pistons and unwanted heat
losses cannot be eliminated, so Carnot鈥檚 value of: efficiency n = 1
鈭� T2/T1 (the temperatures here are measured in
kelvin) represents a theoretical upper limit for any heat engine. Power
stations and internal combustion engines are heat engines. In both cases heat
is supplied at a high temperature (by burning fuel) and ejected to a low
temperature (the atmosphere). The source temperature is limited by the
materials used to contain the hot, high-pressure gases and the sink
temperature by ambient atmospheric temperature. Typical values might be 900 K
and 300 K respectively, giving an efficiency of 1 鈥� 300/900 = 2/3 or 67
per cent. The fact that practical systems are not reversible and do not
achieve these efficiencies does not lessen the importance of Carnot鈥檚 result.
Increasing T1 and/or decreasing T2 would increase
efficiency.
This is clearly linked to the Second Law. A fraction (at least
T2/T1) of the heat supplied must be delivered to the
sink. So any attempt to extract work from heat must produce more heat at the
lower temperature. Also, if T1 = T2, n = 0
and no energy is available for work, it is not possible to extract work from
heat with no other processes occurring. And if we want heat to flow from cold
to hot (that is, T2 鈫� T1) n is negative and we must
pump it (that is, put work in). This is the idea behind the refrigerator and
the reason why, if you leave the door open, it acts as a room heater.
Carnot proved that, to extract work from heat efficiently, we need sources
at high temperature and sinks at low temperature. The world energy problem is
not caused by a lack of total energy but a lack of available energy sources
which can provide heat at high temperatures. So far such sources have been
mainly fossil and nuclear fuels. We are now running out of these energy dense
supplies and the energy that surrounds us is mainly low-grade heat. To make
our atmospheric or oceanic heat engine generate worthwhile quantities of
electricity we would need a large local and very low temperature sink.
A heat engine extracts more heat from its source than it delivers to its
sink (since some is converted to work). In the Carnot engine, the ratios of
heat flow to absolute temperature at the source and sink are equal. The German
physicist Rudolf Clausius suggested that this ratio (heat over temperature)
represents a new type of quantity, entropy: 鈥淓ntropy gain/loss is equal to
heat gain/loss divided by absolute temperature鈥�.
A reversible heat engine transfers as much entropy to the sink as it
extracts from the source. However, real processes involve irreversible steps,
delivering more entropy to the sink than is extracted from the source so
reversible processes conserve entropy whereas all irreversible processes
increase it. No process is possible whose net effect is to decrease the
entropy of the Universe. This is another statement of the Second Law, but in
terms of a property which can be calculated, entropy.
Entropy is a property of a system just as its energy or mass, except it is
not conserved, it tends to a maximum value. But what is entropy? To answer
this question we need to look at the microscopic structure of matter.
Large scale to small
Molecular chaos
EVERYTHING we have said so far is a macroscopic (large-scale) description.
Presumably this macroscopic picture could be explained in terms of atomic
interactions on a microscopic scale. We could then relate entropy to particles
and energy. The first person to do this was the Austrian physicist Ludwig
Boltzmann. In a hot body the individual particles possess energy and can
exchange it with one another. Boltzmann鈥檚 approach was to assume molecular
chaos and let the laws of chance dictate how the system evolves.
Consider a pair of similar bodies, one hot and the other cold, placed in
thermal contact. If energy is exchanged freely and randomly between particles
then there will be a greater chance of energy flow from hot to cold simply
because there are initially more quanta of energy in the hot body. This bias
remains until the amount of energy in each body is about equal, they then have
the same temperature, and the probability (and hence quantity) of energy
passing in either direction is also the same. This is thermal equilibrium.
(Fluctuations from equilibrium will certainly occur but everyday objects
contain such incredibly large numbers of particles 鈥� 1025 molecules
in a glass of water 鈥� that fluctuations are normally tiny and
insignificant.)
If chance and the Second Law both predict that heat flows only from hot to
cold then surely chance explains the law and accounts for the arrow of time?
Boltzmann thought so but how? To understand this we need to examine the
meanings of 鈥渙rder鈥� and 鈥渄isorder鈥�.
Order and disorder
External influences
LET US re-examine our hypothetical block which cools and flies off into
space. The warm block contains particles moving randomly, in a state of
maximum attainable disorder, whereas the moving block has less thermal energy,
but a great deal of ordered motion (particles moving with equal speeds in the
same direction). Although total energy is conserved it is daunting to
contemplate the sophistication and complexity of the mechanism required to
impose this degree of order onto the disordered thermal motions of particles
in the warm block. The reverse process, however, in which a rapidly moving
block undergoes an inelastic collision and converts its ordered energy of
motion into disordered heat, is easy to contemplate and often occurs. A stone
dropped into sand, for example.
Boltzmann realised that the microscopic distinction between order and
disorder is simply the number of ways in which a particular macroscopic state
(for example, a warm stationary block, or a cold moving block) can be
obtained.
When the cold moving block collides with something, a proportion of its
kinetic energy is 鈥渞andomly reshuffled鈥� among its particles. This inevitably
results in the collection of uncoordinated motions we interpret as 鈥渉eat鈥�. The
reverse process does not occur because the probability that this reshuffle
produces a large majority of parallel motions is, to all intents and purposes,
zero. States in which energy is randomly distributed among the available
particles (disorder) are far more numerous than those in which the energy is
distributed in a very special way in order to create some distinct large-scale
effect (order) such as translational motion.
External influences (for example, a Bunsen burner that makes one end of a
body hot, or a catapult that projects a body at high speed) may cause a system
to start off in a relatively ordered state, but molecular chaos and collisions
produce random energy exchanges between particles and they all end up as
disordered heat. For macroscopic bodies, the number of ways of spreading the
particles and energy in a highly disordered manner so far exceeds the number
of ordered configurations that all closed systems inevitably evolve toward
thermal equilibrium (the macrostate corresponding to the maximum number of
microstates).
Molecular chaos makes the evolution inevitable and accounts for the
irreversibility of the Second Law. Furthermore, it links entropy to disorder.
Microscopically, entropy is a measure of the number of ways in which the
energy of a system can be spread among the available particles. Processes
which increase the 鈥渘umber of ways鈥� (W)are more likely to occur (because there
are more of them and we are selecting at random) and so entropy tends to
increase. The equation linking entropy with number of ways is carved on
Boltzmann鈥檚 grave in Vienna:
S = k loge W
Here S is entropy, k is Boltzmann鈥檚 constant and W is the number of
microstates corresponding to a particular macrostate.
The idea that statistical mechanics explains the arrow of time, and thus
the asymmetry of nature, was soon challenged. How could a theory based on
time-symmetric laws of Newtonian mechanics result in the time asymmetric
Second Law of Thermodynamics? Some extra ingredient must have been sneaked
in.
That extra ingredient is the initial condition of the system. If we start
with a highly ordered state it is bound to evolve toward equilibrium, whereas
a system started in a state of near equilibrium fluctuates a little but
basically stays put. Statistical mechanics is reversible 鈥� chance is blind and
does not force entropy to increase but the initial conditions do. It is like
placing a marble in a dish. If placed centrally the symmetry is preserved and
the marble stays there. If it is placed off centre the symmetry is broken and
it rolls toward the centre. The laws governing the marble鈥檚 motion are
symmetric, but its starting position need not be.
Birth of the Universe
Background radiation
IF WE live in a universe of increasing entropy it must have started off in
a highly ordered state. We need only look at the night sky to realise that the
Universe as a whole is far from equilibrium. Large quantities of high
temperature energy reside in stars which radiate into cold dark space. This is
a kind of heat engine. Hot thermal radiation from stars and galaxies is
absorbed and re-radiated at a much lower temperature (the surface temperature
of the Sun is about 6000 K, whereas the Earth鈥檚 surface re-radiates at about
300 K) until eventually it contributes to the 2.7 K background radiation that
bathes the Universe. The energy that was contained in a few high-energy
photons is spread over an enormous number of low-energy photons resulting in a
massive increase in the way the energy can be distributed, and hence in
entropy.
The background radiation is a remnant of the big bang (see Inside Science
No. 69 鈥淏irth of the Universe鈥�) and gives a good idea of the entropy of the
early universe: an estimated 1088 units. Radiation from stars and
galaxies has done little to change this value so, until recently, cosmologists
believed it also represented the total entropy now. However, in the 1970s two
physicists, Jacob Beckenstein in the United States and Stephen Hawking in
Britain, realised that the collapse of matter into black holes is a powerful
entropy generating process. According to the Beckenstein-Hawking formula the
entropy of a black hole is proportional to the square of its mass. As the
number and mass of black holes increases they make an increasingly dominant
contribution to the total entropy of the Universe.
It is not known what the ultimate fate of our Universe will be, but it is
fairly certain that gravitation will continue to cause matter to collapse. The
British mathematician Roger Penrose considers a model of the Universe in which
all matter collapses into black holes and everything ends in a big crunch. The
B-H formula predicts a final entropy of around 10123 units in such
a universe. It dwarfs the original 1088 units and implies that a
state like the big crunch could be realised in about 10 to the
10123 times more configurations than the big bang. Its like saying
that the chance of selecting an initial state like the big bang from all the
possible states of the constituents of our Universe is only 1 in 10 to the
power 10123. Penrose comments that the Creator, if there was one,
must have had a pretty good idea of how he wanted the Universe to begin.
At first glance, living things seem to violate the Second Law. If I offered
a shuffled series of snapshots of a growing flower, or of a developing fetus,
I am sure that once again you would have no problem arranging them correctly.
This time, however, the sequence is one of increasing order, not increasing
disorder. Does life somehow cheat thermodynamics?
A legacy of disorder
With some order 鈥�
THE SECOND Law tells us that entropy must increase for all irreversible
processes. Life is certainly irreversible, but living things are not closed
systems, they are open, exchanging energy and matter with their surroundings.
Eating consumes energy dense, low entropy matter which is used in many ways in
a body, and most of the energy is output to the environment as low-
temperature, high-entropy heat. During development, entropy generated by this
discarded heat more than pays for the lowering of entropy in the living system
itself.
Energy-dense-low-entropy food exists because the Universe began in a very
special way and because gravitational forces have since provided a mechanism
for intergalactic gases to collapse into stars and galaxies and radiate. Life
owes its existence and survival to this cosmic heat engine and the great
entropy that it generates, diverting a little of the heat flow to do work and
to create and maintain an ephemeral local state of low entropy.
Evolution produces temporary structures of subtle and beautifully ordered
complexity, but the Second Law ensures that their net contribution will be a
permanent increase in the entropy and disorder of the Universe (see also
Inside Science No. 20, 鈥淟ife and the Universe鈥�).
1: Why heat flows from hot to cold
TWO identical bodies, one at 500 K and the other at 400 K, are placed in
thermal contact. Assuming that both bodies have a large energy content, the
removal or addition of 20 joules of energy makes no significant difference to
either temperature. Now consider the entropy change when the 20 joules of heat
flows between the bodies:
Entropy change = heat flow/absolute temperature
In the first case, HOT to COLD, the hot body loses 20/500 = 0.04
units of entropy while the cold one gains 20/400 = 0.05 units. Thus
there is an overall increase of 0.01 units of entropy.
In the second case, COLD to HOT, the only difference is in the sign of the
entropy changes. The result is an overall decrease of 0.01 units of entropy.
The Second Law forbids processes which cause total entropy to decrease. This
prevents isolated heat flow from cold to hot.
The British physicist James Clerk Maxwell suggested in 1867 an ingenious
thought experiment to 鈥渃heat鈥� the Second Law and convert heat directly into
work. He imagined a tiny demon capable of opening and closing a valve
connecting two chambers, each filled initially with a similar gas at the same
temperature and pressure. Maxwell knew that in any gas the molecules have a
range of velocities and hence energies (described by the Maxwell-Boltmann
distribution). The demon鈥檚 job is to open the valve when a fast molecule
approaches from the left or a slow molecule approaches from the right.
The rest of the time it is kept closed. By doing this repeatedly the
temperature of the gas on one side should fall while that on the other side
should rise. This produces a temperature difference which could be used to
drive a heat engine. The net effect has been to convert some of the heat
energy of the gas into work.
The 鈥減aradox鈥� was resolved by the Hungarian physicist Leo Szilard in 1929,
when he analysed the problem for a gas consisting of just one molecule. The
loophole in this is that the demon has to interact with the gas in order to
鈥渒now鈥� when to open and close the valve. This interaction involves an exchange
of energy and information which, when considered in detail, is in complete
agreement with the restrictions of the Second Law (the demon may heat up to
such an extent that, when he is at equilibrium with the gas, he is hopping
about so rapidly that he can no longer control the valve).
One interesting consequence of Szilard鈥檚 approach was to develop the first
strong link between entropy and information theory 鈥� a field which continues
to grow and to attract controversy.
2: Order, disorder and chance
A SIMPLE experiment with coins clearly illustrates the distinction between
order and disorder. If 50 coins were tossed into the air together and all
landed 鈥渉eads鈥� we would either accuse the coin tosser of cheating or feel we
have witnessed an event bordering on the miraculous. On the other hand, any
near-even split of heads and tails would be unsurprising. And yet, a little
reflection tells us that any outcome of heads and tails among 50 coins has
exactly the same probability of occurring. This probability is
1/2050, about 1 in a million billion (1015). If we
tossed the coins once every 10 seconds a particular outcome would occur on
average only once every hundred million years. So there is a sense in which we
should be utterly startled whatever happens. The difference between our actual
response to the two results is due to the different number of ways that can be
achieved. There is only one way of getting all the coins to land heads whereas
an even 25:25 split can be achieved about 1014 different ways,
about 1 in 10 experiments.
Outcomes achievable in many ways are called disordered states whereas those
achievable in few ways are ordered. Thermal equilibirum is disordered because
it is represented by an enormous number of microscopic distributions of energy
among particles. The evolution from order to disorder occurs because states
produced in many distinct microscopic ways are more likely to occur than those
produced in just a few.
If we had the time and patience to continue tossing coins eternally we
would notice that all configurations of heads and tails occur over and over
again. Although chance makes it extremely likely that order dissolves into
disorder, it doesn鈥檛 forbid the reverse process and must sometimes allow a
spontaneous transition from disorder to order.
Boltzmann, like most 19th-century physicists, assumed that the Universe as
a whole was in a state of static equilibrium but thought local fluctuations
from equilibrium would occur. He suggested that our Universe occupies a region
in which, in the past, a large local fluctuation had occurred to produce the
degree of order in which we evolved and which we now observe to be moving
inexorably toward disorder and equilibrium. He speculated that our perception
of the arrow of time is determined by this process and that other distant
regions of the Universe which are fluctuating toward order may contain beings
for whom our past is their future and their arrow is reversed. This view is
not generally accepted nowadays, mainly because astronomical observations
reveal that the distant universe is pretty much like the local Universe as far
as thermodynamics is concerned.
FURTHER READING
The Enigma of Time, edited by P. T. Landsberg (Adam Hilger, 1982); The
Arrow of Time, by P. Coveney and R. Highfield (W. H. Allen, 1990); Order out
of Chaos, by I. Prigogine and I. Stengers (Heinemann 1988); The Emperor鈥檚 New
Mind, by R. Penrose (Oxford University Press, 1989). The Second Law, by P.
Atkins (W. H. Freeman, 1984).