v3.1.6 / chapter 9 of 14 / 01 jan 08 / greg goebel / public domain
* One of the simplest, and most difficult to grasp, concepts in elementary physics is that of "heat", and that is the starting point for the study of "thermal physics" or "thermophysics". This chapter provides an introduction to thermal physics.
* "Heat" and "temperature" may seem to be intuitive concepts, and certainly we do understand them intuitively from daily experience. We can go outside and notice if the temperature is high on a hot day, or low on a cold one. We put food in the microwave or in the oven to heat it up, and we put icecubes in a drink to cool it down. However, properly defining these terms is tricky. Formally speaking, heat is not a property of an object, but is instead a transfer of energy between objects. It follows a few simple rules:
The great French chemist Antoine Lavoisier (1743:1794) postulated that a hot object contained a concentrated amount of a hypothetical "fluid" that he called "caloric" that was the agent of heat transfer. It could be generated by friction, for example rubbing sticks together to start a fire, and could be converted back into mechanical work, for example using a steam engine.
The idea seems a bit silly in hindsight, but the notion of "imponderable fluids" was popular at the time, and considering electricity as a fluid, as was the case in that era, was not so ridiculous, since an electric current can be thought of as the flow of a "fluid" of electrons. However, the problem with caloric was that its only identifiable characteristic was that it transferred heat. It was not visible in any way and had no other recognizable properties. What transferred heat? Caloric. What was caloric? It was what transferred heat. Physicists felt they weren't on very solid ground.
It wasn't long after Lavoisier introduced the idea of caloric that scientists began to understand that heat was simply the consequence of the motion of the molecules in an object. Benjamin Thompson (1753:1814) was a Loyalist who backed the wrong side in the American Revolution and had to flee to Europe, where he acquired the title of "Count Rumford". Thompson was working in the arsenal at Munich, and noticed that boring out a cannon generated a good deal of heat. That wasn't any new observation, but he observed that the rate of heating didn't depend at all on the rate at which the cannon bore was ground out, only on how much work was put into the grinding process. A dull tool that ground the bore slowly generated about as much heat at any one time as a sharp tool that ground quickly, if both tools were rotated as fast. From the point of view of caloric theory, the sharp tool that ground quickly should extract more caloric and should release more heat. That wasn't the case, and Thompson realized that he could make the tool duller and duller until he could get heat out of the process indefinitely. That would imply that the cannon blank stored a infinite supply of caloric.
Thompson deduced that the work of the grinding process was being converted into heat, and that caloric was just a fantasy. It would take time for his ideas to be widely accepted, since his writings on the subject were general in nature and not backed up by detailed studies. Others would take up his ideas and put them on a stronger foundation.
James Prescott Joule conducted a series of elegant experiments to demonstrate the link between work and heat, culminating in an 1845 paper in which he described an apparatus in which weights were connected to a cord and a pulley arrangement, to spin a set of paddles in a closed vessel full of water. Using a sensitive thermometer to measure the heat produced by the paddles showed it was predictably proportional to the energy released in the fall of the weight.
It is now known that force generated by friction over time exerts work that produces heat energy, in a cannon or anything else that is affected by friction forces. In more modern terms, adding energy to an object -- by friction or another heating process -- increases the velocity and the kinetic energy of the molecules, increasing the "internal energy" or, more informally, the "thermal energy" of the object. The temperature of the object was then a measure of the thermal energy, essentially an average of the kinetic energy of all the molecules of that object.
Heat in turn became a transfer of this thermal energy. If a hot object was brought in contact with a cooler object, collisions between the energetic molecules of the hot object increased the velocity of the slower molecules of the cooler object. One of the interesting implications of the view of temperature as a measure of molecular motion was that if all molecular motion in an object ceased, then there would be no way for the object to get any colder. This implied the existence of an "absolute zero" temperature.
* There are several different temperature scales in use today. The "Celsius" scale, once known as the "Centigrade" scale, is in use over most of the world, while the "Fahrenheit" scale is used in the United States. The "Kelvin" scale, which is closely related to the Celsius scale, is in common use for scientific purposes.
In the Celsius scale, the freezing point of water is specified as 0 degrees
Celsius, while the boiling point is specified as 100 degrees Celsius. In the
Fahrenheit scale, the freezing point of water is 32 degrees Fahrenheit and
the boiling point is 212 degrees Fahrenheit. Conversions between the two can
be performed as follows:
degrees_Fahrenheit = ( 9/5 ) * degrees_Celsius + 32
degrees_Celsius = ( 5/9 ) * ( degrees_Fahrenheit - 32 )
The Kelvin scale is the same as the Celsius scale, except that the 0 point is
at absolute zero, which is equivalent to -273.15 degrees Celsius. This makes
conversion between the Kelvin and Celsius scales simple:
degrees_Kelvin = degrees_Celsius + 273.15
degrees_Celsius = degrees_Kelvin - 273.15
The Celsius and Fahrenheit scales are useful for describing temperatures in
our daily environment, since they give small values for most Earthly
temperature ranges. The Kelvin scale is much more convenient for scientific
work, since it eliminates negative temperature values that are clumsy in
calculations. Incidentally, there is also a "Rankine" scale, now almost
never seen, that is the same as the Fahrenheit scale, except that the 0 point
is at absolute zero, equivalent to -459.67 degrees Fahrenheit.
* In metric units, heat and thermal energy are given in terms of the "calorie", which is the amount of heat required to raise a gram of water from 15 degrees Celsius to 16 degrees Celsius at a pressure of one atmosphere. Since heat is equivalent to energy, one calorie is also the same as 4.186 joules. Since the calorie is a somewhat small unit for most practical purposes, it is often expressed in terms of "kilocalories", and just to confuse matters, the "Calorie" used to rate food energy actually means a kilocalorie.
In English units, heat is given by the "British thermal unit (BTU)", which is the amount of heat required to raise a pound of water one degree Fahrenheit. The BTU has been generally obsoleted by the calorie. One BTU is equivalent to 252 calories or 1,055 joules.
* Heat of course has a number of effects on matter, most visibly on gases.
An "ideal gas", meaning one where the molecules that are visualized as
perfectly elastic particles that have no tendency to attract each other, has
a neat relationships between temperature, volume, and pressure. This can be
summarized as the "ideal gas law":
_________________________________________________________________________
pressure1 * volume1 pressure2 * volume2
--------------------- = ---------------------
temperature1 temperature2
_________________________________________________________________________
For example, suppose Dexter stores a gas in a rigid airtight container with
solid walls -- meaning its volume is constant -- and that the temperature of
this gas is the same as the ambient temperature of the outside world. If
Dexter then heats up the gas inside the container to twice its original
temperature (using the absolute Kelvin scale), them the final pressure is
twice the original pressure. Similarly, if Dexter cools the container to
half its original temperature, the pressure falls to half of its original
pressure. This is a "constant volume" process:
pressure1 / temperature1 = pressure2 / temperature2
One interesting example of this process at work is to pipe hot steam into a
can and then seal it. If the can's walls are thin, once the steam cools off,
the pressure in the can will drop, and the external atmospheric pressure will
crush the walls of the can.
Now suppose the top of Dexter's rigid container is really an airtight piston
that can slide up or down in the container. Dexter again heats up the gas
inside the container to twice its original temperature, but this time he lets
the piston out so the pressure remains constant as the heat increases.
Assuming that he does this quickly so that none of the heat leaks out, the
end volume is twice that of the original volume. Similarly, if Dexter cools
the gas and pushes in the piston to maintain a constant pressure, assuming no
heat leaks in, the end volume is half the original volume. This is a
"constant pressure" process:
volume1 / temperature1 = volume2 / temperature2
Now suppose Dexter pulls out the piston to expand the container to twice
volume and then lets the gas inside warm up to the outside ambient
temperature. The end result will be that the pressure falls by half.
Similarly, if he pushes in the piston to compress the container to half
volume and lets the gas cool off to the outside ambient temperature, the
pressure doubles. This is a "constant temperature" process:
pressure1 * volume1 = pressure2 * volume2
The relationship between pressure and volume, temperature being constant, was
observed by the English chemist Robert Boyle (1627:1691), who suggested it in
a book published in 1661. The volume and temperature relationship, pressure
being constant, was suggested by the French chemist Jacques Charles
(1746:1843) in 1787, and the pressure and temperature relationship, volume
being constant, was proposed by another French chemist, Joseph Gay-Lussac
(1778:1850), who published it early in the 19th century and came up with the
combined gas law to cover all three laws.
The relationships in the combined gas law become more complicated if temperature, volume, and pressure are varied all at the same time -- this is one of the reasons why calculating the effects of compressible flows is so difficult -- but the general effects of changes in pressure, temperature, and volume remain apparent. It should be remembered that gases at typical Earthly conditions are a good approximation of ideal gases, but at extremes of pressure or temperature gases may depart from this nice neat behavior.
* The ideal gas law was derived empirically, determined from experiments not too much different conceptually from the experiments Dexter performed above. The idea that heat was due to molecular motion did obviously imply a link between the ideal gas law and classical Newtonian mechanics. Newton's PRINCIPIA MATHEMATICA pointed out that the pressure-volume relationship discovered by his contemporary Robert Boyle could be explained if a gas was considered as a collection of tiny elastic particles, flying around in free space. However, making a rigorous connection between the two was tricky.
It was possible to model the behavior of a single molecule using Newtonian mechanical principles, envisioning it as a particle with a given velocity that undergoes collisions. The problem was that a gas is made up of a vast number of particles that all have different speeds. Trying to measure or model the individual behavior of each molecule in a gas was obviously impossible.
The solution to this problem was discovered by three 19th-century physicists: James Clerk Maxwell (1831:1879), a Scotsman ("Clerk" is pronounced "Clark" in this case, incidentally); Ludwig Boltzman (1844:1906), an Austrian; and Josiah Willard Gibbs (1839;1903), one of the first major American theoretical physicists. Their approach was to perform a statistical analysis on the large number of particles, using the basic laws of Newtonian mechanics to obtain a "probability distribution" that gave the relative proportions of particles in a given range of velocities for a given temperature. Although tracking all the particles was impossible, it was possible to statistically determine their average behavior.
The analysis, now known as "Maxwell-Boltzman statistics", allowed the ideal gas law to be derived from Newtonian mechanics, linking the "microscopic" world of particle interactions with the "macroscopic" world of the ideal gas law. The details of Maxwell-Boltzman statistics are not very relevant to an introductory physics text, since the ideal gas law is all a casual student of physics really needs to know. The idea is relevant in the broader sense that it introduced the concepts of statistical analysis to fundamental physics: the notion that, although there might be no way to determine the action of the specific particles of a system, the "law of large numbers" would still allow the behavior of that system to be predicted. This powerful technique would be exploited in new directions by 20th-century physics.
* Liquids and solids are generally not very compressible, and so do not follow the same laws as gases. Heat still has an effect on them. Most solids and liquids increase in volume when heated, and decrease in volume when cooled. This is why bridges have "expansion joints" at intervals, to compensate for the changing length of the span due to temperature changes. The change in the size of an object made of a particular material is called its "coefficient of thermal expansion".
One simple practical application of this property is the "bimetallic coil" used in some thermometers. This is a coiled strip of metal with brass on one side and iron on the other. Since the two metals have different coefficients of thermal expansion, as the strip is heated or cooled it will bend one way or another in a predictable fashion, and the raveling or unraveling of the coil could be used to turn a thermometer needle.
The same principle is used in traditional thermostats, though they use an uncoiled bimetallic strip. The thermostat can be set so that if the strip bends to a particular position, it will complete an electric connection and turn on the heat or air conditioning as need be.
* Heat of course can change a substance from solid to liquid and from liquid to gas. For example, water is solid ice at low temperatures, but changes to liquid at higher temperatures, and steam at even higher temperatures. Different materials go through such "changes of state" or "phase changes" at specific temperatures. Some materials don't go through a liquid phase, converting directly from solid to vapor, at least under typical Earthly atmospheric pressures, a process that is known as "sublimation". Frozen carbon dioxide sublimates, which is why it is used to cold-pack parcels that have to be shipped, since such "dry ice" doesn't create puddles. Incidentally, water ice sublimates at a low rate: leave an ice-cube tray in a freezer for many months, the ice cubes will gradually fade away, or at least they will if the freezer is opened often and the climate is dry.
The amount of heat that must be added to a substance produce a phase change is called the "latent heat". Different phase changes have different latent heats: a particular material will have different latent heats of melting, vaporization, or when it applies, sublimation.
At standard atmospheric pressure, water cannot be heated above 100 degrees Celsius, because any additional energy simply vaporizes the water. Boiling water is actually, in a sense, a "cooling" process since it prevents the accumulation of energy. A pressure cooker is used to obtain higher water temperatures. When water condenses again, the latent heat that vaporized the water is released.
The proportion of water vapor in the air is referred to as "humidity". There is a maximum level of humidity that air can support, which increases with temperature since water condenses back to liquid more easily at low temperatures than high. When the air can accommodate no more water vapor, it is said to be "saturated". Humidity is usually given in terms of "relative humidity", or the ratio of actual water vapor to the saturation level. At 50% relative humidity, the air contains half the amount of water vapor that it is capable of supporting, and at 100% relative humidity the air is saturated.
The evaporation of water is used for cooling in the "evaporative cooler" or "swamp cooler", a simple household cooling device used in dry, warm climates. It consists of little more than a fan pulling air through a filter through which water is pumped. The air causes the water to evaporate, moistening and cooling the air. It is substantially cheaper to buy and operate than a conventional air conditioner, whose operating principles will be discussed in the next chapter. An evaporative cooler is ineffective in hot humid climates since the rate of evaporation slows, and in fact it may simply make a living space more humid and sticky.
Incidentally, water is an unusual substance in that it actually expands when frozen, due to the way its molecules rearrange themselves. This is a fortunate circumstance for life on Earth, since if it were not so, all ice forming on the ocean would sink to the bottom and gradually build up a reservoir of ice that would make the planet a permanent "icebox".
* Gases, liquids, and solids that are being heated, but not changing state, have a "specific heat" that defines how much energy must be added per unit mass to raise the temperature of unit mass of the substance one degree. The specific heat of water, for example, is by definition one calorie per gram per degree Celsius. Incidentally, the specific heat of water is unusually high, and so it takes more energy to heat water than most other substances. In the case of materials that are compressible, specific heat has to be rated in terms of whether the materials are held at constant pressure or at constant volume.
The concept of specific heat leads to the notions of "thermal mass" and "heat reservoir". For example, a lake has a thermal mass in that it will require energy and involve a certain delay to heat it up, and once heated up will act a reservoir of heat, slowly releasing it back to the environment.
* Heat is transferred by three processes:
Solid and opaque objects support heat transfer primarily through conduction. Different materials have specific "thermal conductivities", or rates of heat flow through them. For example, metals generally have high thermal conductivities: if Dexter wears metal-rimmed sunglasses on a cold day, they will painfully drain the heat right out of his nose. Plastics generally have lower thermal conductivities, and it is much more comfortable to wear plastic-rimmed sunglasses on a cold day.
Materials with low thermal conductivities are, naturally, used as thermal insulators for homes and buildings. Gases generally have low thermal conductivities, and most forms of home insulation are basically designed as "gas traps". The most vivid example of this principle is sheet plastic foam insulation, which is filled with bubbles of gas. Incidentally, the insulating capability of commercial insulation is specified by an "R-value", which is inversely related to the thermal conductivity. The higher the R-value, the better the insulator. For example, thick fiberglass insulation has an R-value of almost 20, while double-paned glass has an R-value of a little over 1.
For a high degree of thermal insulation, a "dewar flask" is used. The common consumer "Thermos bottle" is a cheap type of dewar flask. A lab-quality dewar flask consists of a double-walled vessel made of Pyrex glass with silvered surfaces and the space between the two walls evacuated. A vacuum has no thermal conductivity; of course it can't support convection; and the silvered surfaces reflect radiation and so limit loss by that route. The neck and cap of the flask end up providing most of the thermal conduction, and so are generally made as small as possible.
Extreme cooling requires a "double dewar" scheme, with one Dewar flask contained in a second, and the space between the two filled with a cryogenic fluid such as liquid nitrogen (with a boiling point of 77 degrees Kelvin) or, for really cold applications, liquid helium (with a boiling point of 4 degrees Kelvin).
Infrared telescopes launched into orbit around the Earth require extreme sensitivity to observe distant and cool cosmic objects. One of the problems with infrared telescopes and other infrared imagers is that they can't pick up a target that's cooler than they are, since their own thermal emission will drown out the infrared emitted by the target. As a result, space infrared telescopes are often built as (very large) double-dewar flasks known as "cryostats", with an infrared telescope built inside the inner dewar flask. They may also have a reflector shield made of multiple layers of plastic, separated by spaces exposed to the vacuum, to reduce solar heating and conserve their cooling helium. As mentioned in an earlier chapter, they are often put into halo orbits at the Earth-Sun L2 point so they can always keep the shield pointed at the Sun.
* The physical configuration of an object affects its ability to retain heat. The simple general rule is that an object retains heat better if the ratio of its surface area to its volume is kept as low as possible. This is why people adapted to cold climates tend to be short and stout, while those adapted to hot climates tend to be tall and slim, giving them a greater ratio of surface area to volume to help them get rid of heat through convection and radiation.
Air-cooled engines will have sets of fins around the cylinders to allow the heat to be dissipated. Similarly, inspection of the circuit board for, say, a personal computer, will reveal that the processor chip is usually fitted with a finned plate or "heatsink" to dissipate its heat of operation. A personal computer will also generally have an electric fan to get rid of the warmed air inside of the enclosure and draw in cooled air.
* The behavior of heat in a physical system is described by three rules, known as the "Laws of Thermodynamics". The Laws of Thermodynamics govern the efficiency of engines, and also rule out "perpetual motion machines".
Thermodynamics is an abstract field even by the standards of physics. To understand it, a few formal definitions must be set down first. The most important is that of a "thermodynamic system", which is defined as a domain bounded in space where heat can flow across the boundaries in either direction.
The properties of a thermodynamic system at any one time define its "state". The most important properties are the "thermodynamic variables" of temperature, pressure, and volume, but there are other variables, such as density, specific heat, and the coefficient of thermal expansion.
If the properties of a thermodynamic system do not change over time, and if there are no changes in its configuration and no net transfers of heat across the boundary of the system, the system is said to be in "thermal equilibrium". If the thermodynamic system moves from one state of thermal equilibrium to another, a "thermodynamic process" is said to have taken place.
A thermodynamic process can be "reversible" or "irreversible". A reversible process can be run in one direction, and then be reversed to return to the same state that it started from with no net change in system energy. An irreversible process can be run in one direction, but will require a net input of system energy to reverse it.
* There are four Laws of Thermodynamics. Originally, there were only three, but later physicists decided that a fourth law was required. Since this law was basic to the other three, it was called the "Zeroth Law of Thermodynamics" instead of the "Fourth Law of Thermodynamics". The Zeroth Law is basically a definition of the term "temperature". It states that if two thermodynamic systems are in thermal equilibrium with a third thermodynamic system, then the first two systems are in thermal equilibrium with each other. They will all share the same "temperature".
* That formality out of the way, the "First Law of Thermodynamics" defines heat. The First Law of Thermodynamics stems from the work of Joule, Mayer, and Helmholtz, and amounts simply an extension of the law of conservation of energy to include heat. It states that when a warm object (thermodynamic system) is brought into "thermal contact" with a cooler object, meaning a connection that allows the transfer of heat, a thermodynamic process will take place that eventually brings them to the same temperature. The process transfers "heat energy" between the two objects.
The amount of heat transferred into a system, plus the amount of work done on the system, must result in a corresponding increase in the thermal energy of the system. The First Law further implies that if any work is done by the system, it must drain the internal energy of the system. A system where work can be done without draining the internal energy of the system is referred to as a "perpetual motion machine of the first kind". The First Law rules out such machines.
* The "Second Law Of Thermodynamics", established through the work of the English physicist William Thompson, later Lord Kelvin (1824:1907), for whom the "Kelvin" temperature scale is named, and the German physicist Rudolf Julius Emanuel Clausius (1822:1888). It is by far the most important of the four laws, and some have suggested that it really should have been the First law, not the Second, though in practice the Zeroth and First laws act as necessary steps up to it.
The Second Law requires definition of a property known as "entropy" that was
devised by Clausius. In formal thermodynamic terms, it is defined as:
heat_transfer
entropy = --------------------
absolute_temperature
In much more informal terms, entropy measures the thermodynamic "disorder" of
a system, or roughly speaking the tendency of the system's energy to degrade
into random thermal energy. The greater the entropy, the greater the
disorder.
The significance of the specific form of this definition will be discussed in the next chapter. Entropy can actually be expressed in other ways, and the definition of the term has to be carefully considered for any given scenario.
The Second Law states that the entropy of a "isolated" or "closed" system, meaning one where there is no transfer of energy across its boundaries, can never decrease. It may remain the same, or it may increase. A closed system where the entropy decreases is referred to as a "perpetual motion machine of the second kind". The Second Law rules out such machines even though they do not violate energy conservation -- more on this in the next chapter.
* The "Third Law of Thermodynamics" is a bit of an anticlimax after the other laws. It simply recognizes the existence of the absolute temperature scale, and states that absolute zero can never be attained in practice in any finite number of cooling steps. A cooling system can in principle approach absolute zero by a narrower and narrower margin, but it will never actually reach absolute zero.
* In modern times, as mentioned earlier, the concept of caloric has been replaced by an understanding of heat as the motion of the molecules of a system, using statistical methods based on classical mechanics to determine the average behavior of a large number of molecules. In this view, temperature is a measure of the average kinetic energy, or essentially the motion, of the molecules of a system. A temperature increase means that the average kinetic energy has increased.
Similarly, heat transfer between two thermodynamic systems is caused by collisions between individual molecules of the systems. The collisions continue until on the average the net energy passing across the boundary between the two systems is zero, meaning the two systems have achieved thermodynamic equilibrium.
The First Law of Thermodynamics, then, exactly corresponds to the classical law of conservation of energy. The molecular nature of the Second Law of Thermodynamics is a little more subtle, and basically expresses entropy as the measure of the "probability" of a system. Given a large number of molecules in a thermodynamic system, probability dictates that the molecules will be moving in many different directions (high entropy) than in one direction (low entropy).
* The Second Law sets a direction of time, or "arrow of time", in the direction of increasing entropy. On the microscopic atomic scale, interactions are generally reversible: if we make a video of a collision between two atoms, it will be just as valid if the video is run forward or in reverse. At a macroscopic level, this can be also seen in, say, collisions of billiard balls: if a video is made of two balls colliding, ignoring friction it will be valid if run forward or in reverse.
However, the disorder implicit in entropy ensures that in our macroscopic world time only runs in the forward direction. If we make a video of a sugar cube dissolving, it is completely obvious whether the video is being run forward or in reverse. The same is true of the "break" at the beginning of a billiards game, in which the orderly balls are scattered over the table.
Modern physics would show that this arrow of time had a catch, and also show the solution. Given a system with a low entropy, or low probability, in the present, then it is of course likely to evolve to a state with high entropy, or high probability, in the future. This is what the Second Law basically says. The catch is that, given a system with low entropy / low probability in the present, then the odds would suggest that the system also had high entropy / high probability in the past. Of course, this isn't the way things really work, as a rule, in the past the system had even lower entropy and probability.
It was the discovery of the Big Bang, the primordial explosion that created the Universe we live in, that provided the explanation. The Universe began in a state of high energy and uniformity with low entropy, or in other words the Big Bang provided a "zero reference" for entropy, and the entropy has been increasing ever since. Put simply, the Universe is running thermodynamically downhill from an initial height, and that defines the arrow of time on the macroscale. Some have speculated that if the Universe were to cease to expand and fall back in on itself, then time would reverse.
This is getting well off track. This whole notion of the arrow of time has been considered at length and in great subtlety by modern physics, but in classical physics it is taken as a given that time only flows in the forward direction. From the microscopic physics point of view, there is no reason that the motion of all the molecules in a brick that has been dropped to the ground could not point straight up, causing the brick to fly back spontaneously into the air. However, this configuration is vanishingly improbable. It simply doesn't happen -- it is an irreversible process.
Similarly, there is no reason on the microscopic scale that air molecules could not transfer net heat into a brick, making it warmer, but this is improbable as well. A hot brick will release its heat into cooler air, but cooler air will not warm up a brick. Heat always transfers from a warm object to a cooler one, never the reverse. In fact, this is actually just another way to state the Second Law.
* The probabilistic nature of entropy can be illustrated in a simple mathematical fashion with a system consisting of two containers linked by a tube. One of the containers contains four balls colored red (R), green (G), blue (B), and white (W). The tube is big enough to allow the balls to pass back and forth between the containers, and if the system is shaken for an extended period of time the balls will be distributed between the two containers in a random way.
For the four colors of balls, there are two results in which all four balls
are all either in one container or the other, as represented below:
[ R G B W ] [ - - - - ] [ - - - - ] [ R G B W ]
There are 8 results in which there are three balls in one container and one
ball in the other:
[ - G B W ] [ R - - - ] [ R - - - ] [ - G B W ]
[ R - B W ] [ - G - - ] [ - G - - ] [ R - B W ]
[ R G - W ] [ - - B - ] [ - - B - ] [ R G - W ]
[ R G B - ] [ - - - W ] [ - - - W ] [ R G B - ]
There are 6 results in which there are two balls in one container and two
balls in the other:
[ - - B W ] [ R G - - ] [ R G - - ] [ - - B W ]
[ - G - W ] [ R - B - ] [ R - B - ] [ - G - W ]
[ - G B - ] [ R - - W ] [ R - - W ] [ - G B - ]
There are a total of 2 + 8 + 6 = 16 possible results, which gives the
probabilities:
four balls in one container: 2/16 = 0.13
three balls in one container, one in the other: 8/16 = 0.5
two balls in each container: 6/16 = 0.38
As the number of balls in this system increases, the probability of ending up
with all the balls in the same container approaches zero. With all the balls
in one container, the system has low entropy; with the balls divided between
the two containers, it has high entropy.
As Boltzman pointed out, such a probabilistic approach can also to be used to
construct an alternate definition of entropy:
entropy = constant * natural_log( number_of_available_states )
In the "macroscale", Boltzman's definition of energy gives effectively the
same results as Clausius's definition. (There is also a third definition for
entropy from a branch of theoretical math known as "information theory", but
this definition can be seen as a generalization of the Boltzman definition to
cover phenomena outside of thermodynamics, and it doesn't bring anything new
to the party for thermodynamics as such.)
Like the Maxwell-Boltzman statistics, the Boltzman definition of entropy is of little use for simple physics: it is simply given here as a connection to more detailed studies. No matter how entropy is expressed, the essential point is that it is easy to dissolve, mix, or disperse materials, but it is hard to sort them out or bring them together again.
* The Second Law of thermodynamics is occasionally cited as a disproof of Charles Darwin's theory of evolution, in that evolution seems to imply order rising spontaneously out of disorder.
Actually, the term "evolution" is a bit ambiguous, since by itself all the word means is that there has been a progression of species from common ancestors on the Earth, demonstrating a broad increase in diversity and elaboration. Evolution is about as much a scientific fact as anything in the sciences, since the fossil record shows a clear succession of forms, and the various structural relationships between different life-forms don't make any visible sense without assuming "common descent". The question is how it happened: Darwin proposed that evolution has been driven by spontaneous mutations in new generations of organisms, with the least successful dying out and the most successful surviving to create a next generation.
There's plenty of evidence to suggest that "Darwinism", as evolution by natural selection is sometimes called, is not an unreasonable idea. Certainly human selective breeding has been able to transform creatures all out of recognition, pekinese out of wolves as a vivid example, by doing no more than exploiting natural mutations. Darwin never claimed any more than that natural forces -- the competition for food, tolerance to climate conditions, the ability to escape predators, the ability to find a mate -- would perform a similar selection. None of the individual actions, including reproduction, mutation, and selection by natural forces, violate the Second Law -- in fact, all are easily observed and in themselves entirely provable. If none of the components of the process violate the Second Law, is there any way to show that they do so in combination? In practice, evolution by natural selection is clearly demonstrated by the spontaneous emergence of antibiotic-resistant bacteria, and of bacteria that can digest synthetic materials such as nylon.
Darwinism admittedly does claim that more elaborate life-forms are generated from less elaborate ones by "trial and error" processes. Critics find this idea implausible, and while most are willing to admit that it accounts for the emergence of antibiotic-resistant and nylon-digesting bacteria -- "microevolution" -- they claim that it cannot account for the full elaboration of evolution -- "macroevolution". However, if the Second Law ruled out macroevolution, it's hard to see how microevolution could still work: what the Second Law shows to be absolutely impossible on the large scale should also be absolutely impossible on the small.
Even dismissing microevolution, it's hard to see how the Second Law could rule out the emergence of elaborate multicellular animals from single-celled ancestors -- since by the same logic it would similarly rule out the development of a complete human being from a single-celled egg. The two aren't the same thing, of course, but feature the same spontaneous progression from simple to elaborate. If one is a violation of the Second Law on the face of it, then the other should be as well. Critics claim that it is the "intelligence" of the genetic organization of a single-celled egg that allows it to grow up into a complete human; evolutionists reply that it is the "intelligence" of natural selection that shapes a "signal" out of random mutations into new adaptations of organisms.
In fact, it is very hard to identify any violation of the Second Law by Darwinian evolution on the face of it. The Second Law specifies that entropy for a closed system must increase over time. Entropy, in the classic definition, is strictly a thermal concept: heat transfer divided by absolute temperature. There is no thermodynamic reason that biological systems may not acquire more elaborate levels of organization over time; all the Second Law says is that they will waste a lot of energy doing it, in the same way a baby eats a lot of food to grow up into an adult.
The reality is that the metabolism, growth, and reproduction of organisms, or in other words the basic mechanisms of Darwinian evolutionary processes, are all busily increasing entropy through their consumption of energy -- and there is a line of thermodynamic thought that suggests that thermodynamic "gradients", between low and high entropy, often set up complicated phenomena in the first place. In this view, life -- a particularly complicated phenomenon -- is simply a "trap" between the low-entropy energy provided by sunlight, which drives the food chain through plant photosynthesis, and its ultimate degradation as waste environmental heat.
These are all informal arguments, but that observation cuts both ways: nobody has ever produced a formal mathematical analysis to show that the evolution of life violates the Second Law that has been able to stand up to criticism. Nobody been able to come up with a particularly meaningful way to put a value on the entropy of a biological system to permit a valid thermodynamic analysis; nobody's been able to come up with any particularly useful scheme to quantify in any way the complexity of biological systems, or in other words assign a useful value of the complexity of an amoeba versus a human being. From a genetic and structural point of view, a mouse is about as complicated as a human being: a human is bigger, smarter, and lives far longer, but the genetic codes of the two species are roughly the same size, they feature all the same basic systems, and at a functional level the two organisms operate in much the same way. Whether Darwinism is valid or not, the Second Law neither disproves nor proves it.
* There is, however, another question involved in this matter, the issue of how life got started in the first place. This is not an issue with Darwinism as such, since Darwin never claimed he had any idea of how life arose. His theory of the mechanisms of evolution simply assumed that life arose in some unspecified fashion and he only attempted to describe what happened after that -- in much the same way that a book on computer programming doesn't explain how to build a computer. Darwin thought it possible that the origin of life would always remain a mystery. It still is; researchers have a number of speculative theories, but none are regarded as complete or fully convincing.
The basic idea is that life arose under special conditions that encouraged the creation of ever more elaborate systems of molecules -- with this spontaneous emergence of life from nonlife referred to as "abiogenesis". The Second Law does not inherently rule this idea out. The initial creation of life from the organization of simple molecules into more complicated ones does imply a decrease in entropy, but only for the molecules themselves, and not necessarily for the complete thermodynamic system in which they exist.
Although physical systems do tend in general towards greater disorder, it is still not impossible for order to arise spontaneously from disorder under specific circumstances. Consider the example of a mix of red and white balls given above. Obviously, it can shaken forever and the balls will not resort themselves back into red and white -- except if the red balls are twice as big as the white balls, in which shaking the mix will cause the red balls to rise to the top and the white balls to fall to the bottom. (Why this happens is actually an interesting question in itself, but irrelevant to the current argument and not discussed here.) The mix has resolved itself into order, but the Second Law is not violated, since the process required an input of energy that resulted in a net increase in entropy.
Orderly crystals can easily be created out of dispersed solutions of atoms, and of course water molecules in the air can freeze into very orderly and often elaborate snowflakes; nobody sees such things as miraculous. If the atoms have a selective attraction to each other, they will tend to join together in a crystal, and a thermodynamic analysis will show that entropy increases in the overall system -- the snowflake releases heat during its formation, producing an overall increase in entropy.
A snowflake is referred to as an "emergent system", one that exhibits self-organization. Simple lab experiments have been performed in which complicated molecules have been spontaneously self-assembled out of simple molecules through electric sparks. The Second Law is not violated because energy inputs are used to promote the reactions that created the molecules. It is important to realize, however, that such examples do not constitute the slightest proof of abiogenesis: comparing the self-organization of snowflakes to the architecture of a living system is like comparing the complexity of a set of wind chimes to that of an elaborate music-box -- indeed, such a comparison is a gross understatement.
The origins of life remain mysterious, and it is not unreasonable to retain a little healthy skepticism that the self-assembly of molecules, even over billions of years, could produce living organisms, However, since nobody really knows what happened, it's equally impossible to say that the Second Law -- or for that matter any other law of physics -- either proves or disproves it.
