Heat is work and work’s a curse
And all the heat in the universe
Is going to cool down because it can’t increase
Then there’ll be no more work. And there’ll be perfect peace.
That’s entropy, man.
Flanders and Swann
In the popular mind the second law of thermodynamics is defined as heat loss, or in a slightly more sophisticated version heat loss due to process. It is seen as a refutation of the law of conservation of energy. According to the law of conservation the quantity of matter/energy in the universe is a constant. Matter/Energy can neither be created, nor destroyed. But in the popular mind entropy means that all the energy in the universe is going to just go away. Or in another image to become a uniform grey soup of atoms equally spaced and energized.
Allow me to defend the law of conservation. Without the law of conservation there is no science. All the formulae of physics would be false. Take for example F = ma. This formula does not work if the number for mass is not stable. If some mass can just disappear then the formula has absolutely no use. I hope that a defense of the law of conservation was unnecessary, but it means that whatever expression we use for the Second Law of Thermodynamics it cannot be one in which heat just goes away. If the energy goes away the energy has to go somewhere.
Let us begin with the first law of thermodynamics. Benjamin Thompson was observing the boring of cannon and became fascinated by a curious phenomenon, heat passing from a cooler to a hotter. As the canons were bored they heated up. The filings from the drill were hot enough to boil water. Thus a cool drill produced hot filings. They went from a cooler state to a hotter state simply because of the work that was being done on them. This observation lead to the first law of thermodynamics. Often expressed as “Work is heat and heat is work.” It would be more exact to say that Work at the mechanical level can be translated into heat at the molecular level, because work and heat are both types of energy. (Thermo = heat. Dynamics = work.) Scientists later established that 1 calorie of heat = 4.186 Joules (Newton/meters) of work.
The second law of thermodynamics the first formulation:
You cannot build a heat engine that is 100% efficient.
Or It is impossible to construct a heat engine that, operating in a
cycle produces no other effect than the absorption of heat from a reservoir
and the performance of an equal amount of work.
Or You can not convert heat quantitatively into work with no
other effect.
Thermodynamics is about heat engines. How do you transform molecular energy in the form of heat into mechanical energy in the form of Newton-Meters or Joules? In the discussion of heat engines the first law tells us that you can’t get more energy out in Joules than the amount of heat in calories you put in. The second law tells us that you can’t turn all of the molecular energy into mechanical energy.
At the practical level the operation of the machine will take up energy. A simple example of this is that your car engine has to warm up. Some of the heat that could go to work has to go to heating the engine. The engine is subject to heat loss from friction and conduction. The efficiency of this kind of engine can be measured by the formula Eff = 1- heat loss /heat gain. This is expressed as a percentage.
In this version the second law is simply a practical reality. Any engine is going to involve some conduction and friction. Even as simple a machine as the lever will have some absorption of energy by the lever and the fulcrum. Where does the energy go? It goes into the surrounding environment.
This is a practical consideration, but what about the ideal situation. What if you had an ideal engine that had no friction and no conduction? Is there a limit to how much heat you can transform into work? Carnot proposed the Carnot engine, an ideal four stage engine. As an ideal engine it has no friction or heat absorption by the structure, but it still requires energy to return the system to its original position. Using the ideal gas law, Carnot showed that the thermal efficiency of a Carnot engine can be calculated by the formula Effc = 1 – temperature (in degrees Kelvin) of the hot reservoir/temperature (in degrees Kelvin) of the cool reservoir. If these temperatures are the same, then the efficiency is zero. As the values diverge the efficiency increases. The value could only be 100% if the temperature of the cool reservoir was 0 K or absolute zero. Where does this heat go? It doesn’t go anywhere. It just doesn’t get transformed into mechanical energy. It remains molecular energy. Thus we are no longer talking about the popular idea of heat loss due to process. We are talking about the amount of molecular energy that can be translated into mechanical energy.
If you look at a heat engine as the translation of molecular energy
into mechanical energy then for all of the kinetic energy of the molecules
to be transformed into mechanical energy, the molecules would have to come
to a complete stop i.e. absolute zero. Given Newton’s Second Law that an
object in motion tends to remain in motion this is unlikely to happen.
Thus in the Carnot version the second law is simply a corollary of
the third law. Absolute Zero is unattainable. (Experiments
are currently underway to see if it is attainable.)
Several things are important to note here. 1. Carnot’s portrayal of a heat engine as the flow of heat from a hot reservoir to a cool reservoir becomes standard. 2. Carnot’s substitution of temperature for energy becomes standard. 3. Carnot’s calculations are based on the ideal gas law where the specific heat of an ideal gas is a constant (R the ideal gas constant.). In other words the substitution of temperature for energy only applies to an ideal gas. In the ideal gas law the ratio of work to temperature is a linear relationship. The assumption that this is a universal law will create much confusion.
The second law is a formal constraint on a specific kind of machine. It is not a general law of energy.
The second law of thermodynamics: a second formulation:
Heat won’t pass from a cooler to a hotter.
You can try it if you like, but you’d far better notter.
Flanders & Swann
I throw in the popular expression of this law because it is so clearly wrong. If you don’t have a refridgerator in you kitchen you should have. If you didn’t learn how they work in school you should sue your school board. The refridgerator is only one of a class of machines called heat pumps. With the heat engine, heat passes from a hot reservoir to a cool reservoir and part of that molecular energy is transformed into mechanical energy. A heat pump is exactly the opposite. Mechanical energy is put into the system and heat is transferred from the cool reservoir to the hot reservoir. Heat passes from a cooler to a hotter. The beginning of thermodynamics is the observation of heat passing from a cooler to a hotter in the boring of cannon. The mechanical energy of the drill is transformed into heat. The cool drill creates the hot filings. Thus heat passes from a cooler to a hotter.
The second law of thermodynamics: a revised second formulation.
Heat will not flow spontaneously from a cold object to a hot
object.
Or
Heat does not move spontaneously from a cold body to a hot body with
no other effect.
The addition of the term “spontaneously” is important because it means that we are no longer dealing with a general law, in the sense that the first law is a general law i.e. there are no exceptions. The addition of the term “spontaneously” means that under certain circumstances which may be described as “spontaneous” this holds true. In this context spontaneously may be defined as “where no mechanical, chemical, electrical, solar or atomic energy is being transformed into molecular energy.”
The worry about heat death is a bit more obvious. If work is heat and heat is work, then the steady flow of heat from hotter to cooler will create a universal equilibrium temperature. Since you need a hot reservoir and a cool reservoir to have a heat engine, you would not be able to construct a heat engine. We know from Carnot that if the temperature of the expansion heat reservoir is the same as the temperature of the compression heat reservoir the engine has an efficiency of 0. There would be no more work. Quite terrifying until you realize that although heat is work, heat is not the only form of work. Heat is only work at the molecular level. There is a mechanical universe above the molecular level and an atomic level below it. The purpose of heat engines and heat pumps is to move energy back and forth between the mechanical level and the molecular level. And we are not even talking about moving energy from the atomic level.
This is a tremendously important formulation of the second law. At this point the study of thermodynamics goes off on a tangent. Scientists forget about the central problem of thermodynamics, the transformation of heat energy into mechanical energy, and begin to focus on the transfer of heat as a phenomenon in itself. For thermodynamics the limiting factor is the “difference” in heat energy between one reservoir and another. Using the caloric theory this is transformed into the idea that the limiting factor is that heat flows from a hotter to a cooler. You will note that this formulation still conceives of heat as little packets of caloric energy that can “flow” or “pass” from one object to another.
The second law of thermodynamics: a third formulation.
The entropy of the universe increases in all natural processes.
You will notice that we have introduced another vague word, “natural.” Once again physics lacks an operant definition of “natural.” What would a physicist describe as an “unnatural” process? Thus in this formulation we are no longer talking about a general law, but a special law that operates only under circumstances vaguely referred to as “natural.”
But this formulation implies the measurement of a quantity called entropy. You can only talk about an increase in entropy if you have a way to measure it. The man was Rudolph Clausius. He was looking for a way to measure the capacity for work of a closed system. Since work is heat, the measure could be understood as heat capacity. Heat capacity is measured in joules per degree Kelvin. You can calculate the heat capacity of something by first taking the total number of calories or joules that are in the object. You then divide by the temperature in degrees Kelvin. This gives you the amount of energy relative to temperature, the heat capacity in J/K. Clausius called this Entropy, usually symbolized by a capital S. The three terms are synonymous: heat capacity, capacity to do work, and entropy and the measurement is Joules per degree Kelvin (J/K).
This is related to the concept of specific heat. The amount of total energy in an object is related to its mass. Thus three grams of a substance at the same temperature and pressure as one gram of the substance will have three times as much energy and thus three times as much entropy. The specific heat of a substance is the number of calories or joules that it takes to increase the temperature of one gram of that substance one degree Kelvin. In fact the measurement of heat energy is taken from the specific heat of water. One calorie is the amount of heat that it takes to raise the temperature of one gram of pure water from 15.5C to 16.5C.
Clausius developed a formula:
The change in entropy (S) of a system is equal to the heat (Q) flowing
into the system as the system changes from one state to another divided
by the absolute temperature (T). DS = DQ/T
The problem with this definition is that S is not a measurement. In order for it to be a measurement it would have to be a continuous variable. For example: I have a ruler. It is a nice shiny metal ruler. There is only one problem with this ruler. It has been bent. So it has kinks in it. It can’t be used to measure things any more. It doesn’t even make straight lines. I don’t know why I keep it. We have the same problem with S. If you draw a J/T graph measuring the total energy in an object against the temperature of the object, there are kinks in the line at phase change. It is discontinuous. It takes quantum leaps. If S were a continuous variable it would translate directly into specific heat. But it doesn’t. Not only are there quantum leaps, the slope of the line changes too. The specific heat of ice is .05 cal/g/K. The specific heat of water is 1 cal/g/K and the specific heat of steam is .48. This is one really crooked line. I don’t know why we keep it.
If this measurement worked then it would translate directly into specific heat. We know that the specific heat of an ideal gas is a constant. Specifically it is Boltzmann’s constant. The ideal gas law is that Pressure * volume = number of molecules * the specific heat of an ideal gas (Boltzmann’s constant) * temperature in degrees Kelvin. Thus the entropy of an ideal gas is constant. An ideal gas does not experience a change in entropy.
Where then would you find a change in entropy? At phase change. There is only one small problem. Clausius’ formula does not work. For the formula to work, for the heat to actually be available, the change in entropy would have to translate into changes in specific heat. But it doesn’t. For example it requires 540 calories to turn a gram of water at 373K into a gram of steam at 373K. The change in S calculates out to 540/373 = 1.45. Thus the specific heat of steam should become 1.45 J/K greater. Unfortunately, we know experimentally that the specific heat of steam is only .48 J/K or .52 less than the specific heat of water.
Entropy is really one of those ideas like caloric packets that seems reasonable on first blush but turns out to be wrong in the light of experimental data.
If the measurement is used in this formulation is not a real measurement then the formulation is clearly wrong.
It may also be worth noting that a change in the indicator S indicates a change in phase and an increase in latent heat. Thus an increase in the entropy of the universe would come about through an increase in temperature. The universe would be heating up, not cooling. The universe would be turning into a gas. And that sounds kind of like heat death. But it is clearly nonsense. Remember gravity.
The second law of thermodynamics a fourth formulation:
Isolated systems tend towards disorder and entropy is a measure
of that disorder.
If entropy is not a legitimate measurement then this statement makes no sense. But if we substitute “indicator” for “measure” we can get closer to what Boltzmann was talking about.
What do we mean by order?
Order is acting in unison. Consider a parade square with a company
of militia marching along in good order. They are marching in the
same direction at the same speed. Note that the marching square has
clear boundaries. When the Drill Sargeant tells them to turn they
all turn in unison. This is order. We can see disorder when
the Drill Sargent shouts “Take cover.” They will all disappear off in different
directions at different speeds. You no longer have clear boundaries
either.
The soldiers on the parade ground act like a liquid. They each
move individually, but in unison. However if we took the soldiers
that were marching on the parade ground and a sadistic sargeant put a rope
around them and packed them altogether so that there was little space between
them and their individual movement was quite limited then they would be
more like a solid. I compare the energy sufficient to create a liquid
to orbital energy. The moon has enough energy to orbit the earth,
but it moves in unison with the earth. With an additional amount
of energy the moon could break orbit and explore the universe. (For
all I know NASA may have a plan for it.)
At the beginning of the movie The Apprenticeship of Duddy Kravitz a group of cadets is marching down the street. In front of them a horse drops some manure. The Sargeant walks directly through the manure. The cadets flow around it. The Sargent represents order, he is a solid. The cadets have less order. Their movement is fluid. Duddy breaks away from the group and heads into the crowd. This is disorder. This is a gas.
There is a significant difference between my definition of order and Boltzmann’s. I defined order in physical terms, in terms of motion, motion in unison. Boltzmann describes it purely in statistical terms. By “disorder” Boltzmann means a random distribution, or what we call a curve of normal distribution, or a bell curve. By “order” Boltzmann means a non-random distribution.
Boltzmann is thinking of entropy in Carnot’s terms. A decrease in the temperature difference between two reservoirs results in a decrease in the amount of molecular energy available for transformation into mechanical energy. A difference in temperature between two reservoirs is a non-random situation. The greater the difference in temperature the more non-random it becomes. If you take the combined energy of the molecules in both reservoirs and then calculate the probability that the energies would be distributed in this way, then the probabilities become less as the difference increases.
For example: You have two reservoirs of an ideal gas each with 50 molecules, and the average energy of the hundred molecules is 5 joules. In a normal distribution each reservoir would have 250 joules. The odds are 50/50. The odds of one being 300 joules and the other 200 joules would be 60/40. The odds of one being 400 joules and the other 100 joules would be 80/20. The odds of one being 450 joules and the other 50 joules would be 90/10. As the difference increases the probability decreases. Using the standard efficiency formula the percentage of available energy can be calculated 1-100/400 = 1-20/80 = 75% or 400-100=300 joules of available energy.
Using the ideal gas law (P*V=m*kB*T) and thus dividing by Boltzmann’s constant. (T=P*V/m*kB) you could translate this into temperature and have the same probability structure. This also gives you the source of Boltzmann’s formula S = kB*logW and why he used Boltzmann’s constant. He uses logW because he assumes you have to use calculus to translate numbers for S.
The probabilities of Boltzmann translate directly into the basic calculations of efficiency. Boltzmann is quite correct the constraints of the second law are formal constraints and not energy constraints. There may be large quantities of energy, but if they are not in the right form then they cannot be transformed into mechanical energy. There is no reason to go through the complicated process of trying to turn it into units of “entropy.”
We can then say that what this formulation of the second law of thermodynamics might legitimately say is “Systems not governed by the laws of physics are governed by the laws of probability.” It would, however, still be wrong. You need to add in the laws of information so that it becomes “Systems not governed by the laws of physics or the laws of information are governed by the laws of probability.”
A small final note: Boltzmann’s statistical mechanics has led some moderns to a fifth expression of the second law. “There is one and only one equilibrium point for an object with a given energy and a given number of particles and a set of constraints.” This is a technical version of heat will not pass from a cooler to a hotter. Basically what this says is that two objects in thermal contact will arrive at a common mean temperature. Two objects in thermal equilibrium with each other are at the same temperature. Statistically the molecules of the object are in constant flux, but they will form a curve of normal distribution around a mean temperature. The mean temperature will remain constant. The original thermodynamic idea of molecular energy in the form of heat becoming mechanical energy in the form of joules has been completely lost.
This discussion would not be complete without a reference to the most popular use of entropy. James Lovelock in his book Gaia argues that the evidence for life on other planets would be negative entropy. In purely technical terms this would mean any planet where there are differences in temperature. This is clearly nonsense. He is using Boltzmann’s concept of order. Where there is order, there is evidence of life. Order is not an energy concept. Order is an information concept and is the realm of logic and mathematics. If there is an order not produced by the laws of physics, then that order is produced by a cybernetic system of communication and control. Life transforms information into physical order. Lovelock’s observations are not dependent on the idea of entropy.
We can conclude by observing that thermodynamics has been plagued by
abortive attempts to create universal generalizations from the specific
behaviour of heat engines. The result has been a bogus measurement
and outrageous generalizations.
Robert Johannson 2007/01/20