How It Looks From Here

The Laws of Thermodynamics

2012-02-18T10:07:00.001-07:00

If one looks around the Internet, he or she can find a multitude of pages that state the laws of thermodynamics; so why add one more? I've been writing a lot of posts on thermodynamics, but nowhere have I given a concise statement of the laws; also, I am not always happy with how the laws are described. So this post is my contribution.

The Zeroth Law of Thermodynamics

The zeroth law states:

If two bodies are in thermal equilibrium with a third body, then they are in thermal equilibrium with each other.

For those who have studied mathematics, this law may seem obvious. Perhaps, it should not seem so obvious when one considers all the ways that macroscopic bodies can store energy. At some point I may write a post devoted to this law. Note that this law is essential to defining such a thing as temperature. Two bodies at the same temperature are in thermal equilibrium with each other.

Two bodies are in thermal equilibrium if they have reached the final temperature that they will reach if allowed to exchange heat with one another. At such a point, no net heat is exchanged. In many ways, the zeroth law depends on an understanding of the other laws. One could use the mnemonic, "let's keep score" for the zeroth law, but that phrase is usually used for the third law.

The First Law of Thermodynamics

I have written a post that goes into more depth on the first law of thermodynamics. The first law states:

Heat is not a conserved quantity, and work is not a conserved quantity, but the sum of heat and work is a conserved quantity.

The sum of heat and work is called the internal energy. Conservation of internal energy is a special case of the more general principle of conservation of energy. Conservation of energy can be derived from Noether's Theorem, which is perhaps one of the most profound theorems of physics.

The internal energy is a state function. Its value does not depend on the path taken. Another example of a state function that is closely related to the internal energy is a quantity called enthalpy. A mnemonic for remembering the first law is "you cannot win." You cannot get more energy out of a system than you put in.

The Second Law of Thermodynamics

I have written a series of posts on the second law of thermodynamics. It is my contention that this law is one of the most misunderstood and abused law in the sciences. There are several different ways to state the second law; here I'll stick to one:

There can be no process with the sole result of absorbing heat and completely converting it to work.

A mnemonic for remembering this law is "you cannot break even."

The second law is intimately connected to a state function called entropy. Entropy is often falsely stated to be a measure of disorder. Whereas the entropy is a mathematical relationship between the heat transferred reversibly, and the thermodynamic temperature, it can also be described from a statistical-mechanical viewpoint as a quantity related to the number of ways to arrange a system with a given energy range.

The Third Law of Thermodynamics

I have written a post on the third law of thermodynamics. It states:

It is impossible to reach absolute zero in a finite number of steps.

A mnemonic for this law is "you cannot leave the game." Alternatively, a mnemonic is "let's keep score." This one arises because the third law is the basis for the thermodynamic temperature scale. It is possible, however, to define a temperature scale based upon the zeroth law. The third law is greedy: it gets two mnemonics, whereas the zeroth law gets zero. In the spirit of fair play, I assign one of the third law's mnemonics to the zeroth law.

Mnemonics

0th Law: "Let's keep score."
1st Law: "You cannot win."
2nd Law: "You cannot break even."
3rd Law: "You cannot leave the game."

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Feynman, Richard P., Leighton Robert B., Sands, Matthew, The Feynman Lectures on Physics, Addison-Wesley, Menlo park, CA, 1965

A Few Thoughts On Ender's Game

2012-01-27T15:09:00.003-07:00

On the recommendation of a friend, I recently finished reading Orson Scott Card's book, Ender's Game. This book has a lot to recommend it. So much so that I have heard that it has been used in actual military education classes alongside Sun Tzu and von Clausewitz. I found Card's anticipation of the Internet to be visionary, and there were twists in the plot of the book that I did not expect. There was one element of the story that bothered me, however.

Card is not a scientist, and why should he be? He is a writer of fiction. It is fiction: he is free to create the world as he sees fit. We the readers should suspend our disbelief. Yet, there is one aspect to the story that made it hard for me to suspend my disbelief.

In the book, Card discusses a device called the Ansible that allows instantaneous, faster than light communication. We all know that such a notion as communicating faster than the speed of light violates relativity, but that does not particularly bother me. It is not the only story to violate relativity; so why should such a thing be bothersome?

At several points in the book, Card invokes relativity, specifically the idea of time dilation. It is through time dilation that characters are able to live fewer years in the same time that other characters live more years. Such an idea, though perhaps impracticable is consistent with relativity.

If we admit, the idea of time dilation, however, the concept of simultaneity suffers consequences. Two events that take place simultaneously at different locations, are only simultaneous for a given frame of reference. For other frames of reference, one event occurs before the other event.

An instantaneous message is one that is received simultaneously with its transmission. There is in relativity, however, no privileged frame of reference. Suppose a message is sent from point A to point B instantaneously in a given frame of reference. In other frames of reference, the transmission and reception will not be simultaneous, i.e., the message will not be instantaneous.

Even more problematic. There will be frames of reference, in which the message is received before it is sent. Such an occurrence constitutes causality violation. Such a causality violation would enable one side or the other in the war to go back and change the outcome after the war was over. That would be a weapon more formidable than Dr. Device (which already required me to suspend my disbelief).

Card's story is a great story, and we should endeavor to suspend our disbelief. The fact that I had trouble doing so is not very interesting, but I thought that the reason I had this trouble might help shed some light on why faster-than-light travel is forbidden in relativity without having to introduce a lot of math.

I am of course aware of recent, hard-to-explain results regarding the speed of neutrinos, but that is something that makes that story so interesting. If the finding is correct, what we understand about relativity must be fundamentally flawed in some way. I am skeptical of the idea that the neutrinos were actually faster than light. I suspect that we may be seeing issues related to defining passage in the correct frame of reference with respect to GPS signals used to synchronize the clocks, or some other similar issue. Of course, I could be wrong about that.

The Third Law of Thermodynamics

2012-01-22T08:30:00.000-07:00

The Third Law of Thermodynamics states that it is impossible to reach absolute zero in a finite number of steps. A shortcut way to remember the law is "you cannot leave the game." An alternate shortcut to remember the law is "let's keep score" which is based on the fact that the third law provides the foundation for thermodynamic temperature scales.

Considerations regarding the Third Law

Perfect crystalline solids have zero entropy at absolute zero. Note that materials that are not perfect crystals do not necessarily have zero entropy at absolute zero.

The Third Law provides a zero for temperature that is really a zero. The Fahrenheit and Celsius temperature scales have a zero, but that zero is arbitrarily assigned. In a thermodynamic temperature scale that zero is absolute zero, the coldest temperature possible, or rather a lower bound on the coldest temperature possible.

The Kelvin scale is the International System (SI) standard thermodynamic temperature scale. The unit of the Kelvin scale is the kelvin, not a "degree Kelvin." It is represented by "K."

0 K = -273.15 °C

The equality is exact, as the Celsius scale is now defined in terms of the Kelvin scale. The Rankine scale is a less common thermodynamic temperature scale. See my post Converting Units of Temperature for more discussion.

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Feynman, Richard P., Leighton Robert B., Sands, Matthew, The Feynman Lectures on Physics, Addison-Wesley, Menlo park, CA, 1965

Capitalism

2012-01-02T19:36:00.002-07:00

Yes, I am a capitalist. Actually, I support a mixed-market economy, but my point is that I am not opposed to profiting from my blogging, if is possible. For some time, I have been getting enough page views to make me wonder whether it's worth allowing advertisements.

I realize the pitfalls. I am especially sensitive to the potential for inappropriate ads. If you are a regular reader and have a strong opinion about advertisements on blogs, please let me know. I am going to experiment with it. If you see ads that seem inappropriate, please let me know.

Belzec, Sobibor, and Treblinka, Holocaust Denial and Operation Reinhard

2011-12-30T10:46:00.001-07:00

From the earliest days of their movement, Holocaust deniers have largely centred their arguments on the Auschwitz death camp. Surveying the literature which makes up so-called Holocaust Revisionism, the obsession with Auschwitz is undoubtedly one of its defining features. Since the early 1990s, with the advent of the modern world-wide web, Holocaust deniers have taken to the internet to try and argue their case. Until recently, the ensuing online debates between advocates of Holocaust denial and their critics have likewise focused on Auschwitz.

My friends at the Holocaust Controversies Blog have written an article that focuses on the Operation Reinhard camps, Treblinka, Belzec, and Sobibor, and the attempts to deny the reality of the murder at those camps by Holocaust deniers, Carlo Mattogno, Jürgen Graf, and Thomas Kues. They dedicated this work to the memory of Harry W. Mazal OBE.

The work is entitled, Belzec, Sobibor, and Treblinka, Holocaust Denial and Operation Reinhard: A Critique of the Falsehoods of, Mattogno, Graf, and Kues.

The Heat Death of the Universe

2011-12-27T12:09:00.000-07:00

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled Time's Arrow. The previous post is essential to understanding this post.

In most of the discussion of nonsense in this series, the nonsense stems from a poor understanding of physics. This post introduces some nonsense that must be taken seriously. Perhaps, this nonsense, also stems ultimately from a poor understanding of physics. The people with the poor understanding this time, however, are some of the most brilliant minds in physics.

The School-Book Story

This discussion starts with the school-book story of the heat death of the universe. By calling it the "school-book" story I do not mean to pooh-pooh it too much. In fact, it is most likely the correct story. Much of this post, however, will focus on caveats and complications to the story as it is usually told.

In thermodynamics, the universe is defined as the system and its surroundings. We have seen that the second law requires that for any change the total entropy of the system and the surroundings must increase or stay the same. As time goes by, therefore, the entropy of the universe increases.

The consequence is that the universe will eventually head toward its maximum entropy, and there is no going back. The universe will become a very boring place. All the stored energy that could be used to do useful work will be dispersed as heat, and the universe will become a very cold, well-mixed, dispersed, undifferentiated place. The universe will be at thermodynamic equilibrium. There will not be hot places and cold places.

Criticism of the School-Book Story

As stated above, the school-book story is a good story. In fact, it is probably right, but is the school book story definitely true? To make such an assertion, it is necessary to examine some of the holes in the story. It is also necessary to enter the world of speculative science. First I turn to the obvious hole.

By defining the universe as the system plus the surroundings, I played a neat trick. I essentially defined the universe as being a closed system in the thermodynamic sense (we have to be careful here because cosmologists are talking about something else when they refer to a closed universe).

What if the universe in not a closed system?

When examining a small system it is not hard to see the logic of the idea that the change in energy of the system is equal and opposite to the change in energy of the surroundings. If energy is conserved, such a conclusion must be true, but what about the edges of space and time? What about extreme conditions that might exist somewhere in the universe. Do we really know that the universe is closed?

If the universe were not closed, one of two things must be the case. 1) There must be someplace outside the universe from which energy and/or matter can be transferred to or from the universe, or 2) energy is not really conserved. Let us examine each of these possibilities.

Is the Universe All There Is?

In a semantic sense, it is easy to say yes. The word "uni" means one, meaning there is only one universe composed of everything that is. It is important to understand, however, that this answer is based on a definition.

If there are places outside what we commonly call the universe, we need a new word for what we call the universe. Unfortunately, semantics aside, the common way to discuss this issue is to brush aside the etymology of the word "universe," and allow for the possibility that our universe is not all that there is.

If there are truly places outside our universe that can transfer energy to our universe, then the universe is not a closed thermodynamic system, and we cannot assert that the heat death will take place. Although, see the section below on multiple universes.

On the other hand, consider everything put together, is the universe combined with everything outside the universe a closed system? If so, then there would be an inevitable heat death of this super-universe that truly includes all the stuff. What happens if the collection of all the stuff is actually infinite? What if there are regions that actually obey different physical laws?

Conservation of Energy and Noether's Theorem

One of the most profound theorems in physics is Noether's Theorem. Noether's Theorem is a way of deriving conservation laws from fundamental symmetries.

For example, the conservation of momentum can be related to the symmetry of space. It does not matter where we perform an experiment: we can translate the axes, and the laws of physics still apply. This symmetry can be shown to lead to the conservation of momentum.

In three dimensional space the laws of physics apply no matter which way the axes are oriented; in other words we can rotate the experiment in space without altering the results. This fact implies that angular momentum is a conserved quantity. (I'm proud to be a student of the man who literally wrote the book on angular momentum!).

Conservation of energy is a result of the symmetry of time (note that we are not talking about backward-forwards symmetry as discussed in the previous post is entitled Time's Arrow.). To quote Feynman

In quantum mechanics it turns out that the conservation of energy is very closely related to another important property in the world, things do not depend on the absolute time. We can set up an experiment at a given moment and try it out, and then do the same experiment at a later moment, and it will behave in exactly the same way. Whether this is strictly true or not, we do not know. If we assume that it is true, and add the principles of quantum mechanics, then we can deduce the principle of the conservation of energy. It is a rather subtle and interesting thing, and it is not easy to explain. (Feynman, I-4-7).

So the assumption that energy is conserved depends on the assumption that absolute time does not matter. If energy is not conserved, then the cosmos is not strictly a closed system, and the second law does not require the increase of entropy.

We live in a time in which time seems to maintain such symmetry; thus we experience conservation of energy. Is it possible that at the very beginning of the universe, or at the very end that time may behave differently? Such a notion is probably complete nonsense.

There is no reason to posit that time behaves any differently. There are no observable events that are explained by such a notion. Perhaps, one day such an observation may be made, however.

If energy is conserved, and we consider the whole cosmos of however many "universes," or other entities there may be, then the school-book story of the heat death of the universe seems to hold up. Still, there are some added subtleties that need exploring.

Some of the ideas that I will examine in this post are a little different from normal science because they involve potential models of the cosmos that make predictions that are almost by definition not observable by us. If a model makes a predictions that we cannot observe, it is more speculation than science. Still, it is necessary to consider such ideas, if we want to assert that we know the fate of the universe.

Many Worlds

The many-worlds interpretation of quantum mechanics (QM) was proposed to explain the finding of QM with regard to the collapse of a wavefunction. In Everett's words:

Alternative 5: To assume the universal validity of the quantum description... The general validity of pure wave mechanics, without any statistical assertions, is assumed for all physical systems, including observers and measuring apparata. Observation processes are to be described completely by the state function of the composite system which includes the observer and his object-system, and which at all times obeys the wave equation...

The idea is that instead of probability determining to which alternative a quantum system must collapse, all the alternatives are in fact real.

To properly understand this idea requires an understanding of quantum statistics, in which a system can be in a superposition of two (or more) states before a measurement determines the state. In classical mechanics, we can flip a coin and not know the result, but there is a result. In QM, the result is fundamentally undetermined before the measurement. The wavefunction is 50% heads and 50% tails.

This idea is so bothersome that the idea that both results actually occur but in different worlds has been proposed.

I should state up front that I am not happy with this theory because it does not make any predictions that would have a different result if the theory is true or if it is not. It does not appear to be a falsifiable theory.

Nevertheless, there are people a lot smarter than I am, who take this notion seriously. So, this is a piece of nonsense that we need to admit as possibly true.

To examine the consequences of the many world hypothesis on the heat death of the universe, let us take a shortcut and pretend that we can model QM by a coin toss. There are some subtle differences between coins and particles governed by quantum mechanics. Such particles come in two flavors: 1) fermions, particles that obey Fermi-Dirac statistics, and 2)bosons, particles that obey Bose-Enistein statistics.

If we have two pennies, we can distinguish them from one another. The same is not the case for fermions or bosons. In fact, the wavefunction that describes them must not be able to distinguish between them.

If this idea seems like an insignificant difference, it is worth pointing out that the very existence of covalent bonding in molecules owes to this fact. The universe would look very different if electrons did not obey Fermi-Dirac statistics.

Having said all that, I am now going to ignore it, and treat the collapse of a wavefunction just like a coin toss; except that in the many-worlds interpretations both results of the coin toss take place in different actually existing worlds.

If we flip a coin a large number of times, most of the worlds will look very similar. Consider my post on fluctuations to better understand the statistics involved.

Consider the fact that in one of the worlds, heads comes up every time. In this world, entropy would not increase.

It is important to realize, two facts, however. The probability that the world in which only heads comes up is the world in which we live is vanishingly small. Consider also, that if we consider the cosmos to consist of all of those worlds put together, that the tendency toward maximum entropy will be observed.

The many-worlds scenario does not contradict the idea of the heat-death of the universe, but it certainly complicates the issue.

Multiple Universes

Serious minds have also proposed the idea of multiple universes, perhaps infinitely many universes that exist throughout all time and constantly pop into existence. Perhaps, not all of these universes follow the same physical laws that our universe follows. Consider, however that a subset of them do, or perhaps they all do.

We cannot assert much about a universe that obeys different physical laws; so let us constrain our imagination to consider universes that have similar laws.

What if there are infinitely many universes very similar to our own? What if the number of universes is so large that it dwarfs the number of ways to arrange all of the particles in the universe?

Is it possible that in some of those universes entropy decreases with time?

Even if that is the case we must again consider two facts: 1) the probability that we live in such a universe is so improbable that it is essentially impossible. 2) The collection of all of the universes taken together would still increase their entropy.

Again, this scenario does not exactly contradict the story of the heat death of the universe, but it certainly complicates it.

Gravity and Entropy

To sum up this post the school-book story of the heat death of the universe is most likely a good story, but there are certain seemingly nonsensical, but possible arrangements of the cosmos that make the story a bit simplistic. There is much that I have not discussed. We live in an expanding universe, but what if we did not? What about black holes?

The next post entitled Gravity and Entropy addresses some of these questions.

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Feynman, Richard P., Leighton Robert B., Sands, Matthew, The Feynman Lectures on Physics, Addison-Wesley, Menlo park, CA, 1965
Noether's Theorem in a Nutshell
E. Noether, "Invariante Varlationsprobleme", Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse (1918), 235-257; English translation M. A. Travel, Transport Theory and Statistical Physics 1(3) 1971,183-207.
Byers, Nina, E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws, presented at The Heritage of Emmy Noether in Algebra, Geometry, and Physics, Bar Ilan University, Tel Aviv, Israel, December 2-3, 1996.
Zare, Richard N., Angular Momentum: Understanding Spatial Aspects in Physics and Chemistry, Wiley-Interscience, 1st edition, 1988.
Pauling, Linus, and Wilson, E. Bright, Jr., Introduction to Quantum Mechanics: With Applications to Chemistry, Dover, New York, 1935, 1963
Cohen-Tannoudji, Claude, Diu, Bernard, and Franck, Laloe,Quantum Mechanics, John Wiley & Sons, New York, 1977.
Everett, Hugh, III, The Theory of the Universal Wavefunction
Everett, H., (1957) ‘Relative State Formulation of quantum mechanics’, Review of Modern Physics 29, pp. 454-462; see also ‘The Theory of the Universal Wave Function’, in B. De Witt and N. Graham (eds.), The Many-Worlds Interpretation of Quantum Mechanics, Princeton NJ: Princeton University Press, 1973.
The Many Worlds Interpretation of Quantum Mechanics
Wikipedia: Multiverse
Tegmake, Max, Parallel Universes, Science and Ultimate Reality: From Quantum to Cosmos, honoring John Wheeler's 90th birthday,J.D. Barrow, P.C.W. Davies, & C.L. Harper eds., Cambridge University Press (2003)

Contents

Introduction
What the Second Law Does Not Say
What the Second Law Does Say
Entropy is Not a Measure of Disorder
Reversible Processes
The Carnot Cycle
The Definition of Entropy
Perpetual Motion
The Hydrogen Economy
Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
The Second Law and Swamp Coolers
Entropy and Statistical Thermodynamics
Fluctuations
Partition Functions
Entropy and Information Theory
The Second Law and Creationism
Entropy as Religious, Spiritual, or Self-Help Metaphor
Free Energy
Spontaneous Change and Equilibrium
The Second Law, Radiative Transfer, and Global Warming
The Second Law, Microscopic Reversibility, and Small Systems
The Arrow of Time
The Heat Death of the Universe
Gravity and Entropy
The Second Law and Nietzsche's Eternal Recurrence
Conclusion

Time's Arrow

2011-10-30T11:15:00.000-06:00

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled The Second Law, Microscopic Reversibility, and Small Systems. The previous post is essential to understanding this post.

Why does time move forward instead of backward? In the spatial dimensions, one can move left or right, up or down, backward, or forward.

Time, on the other hand, has a preferred direction. Why is that so? The underlying physics does not seem to have a preferred direction, but time does.

I am not going to answer this question in this post. Rather, the fact that entropy has a preferred direction in time provides an excuse to think about this issue. The previous post helps to explain why entropy increases with time, but it includes an underlying assumption that time moves forward.

Microscopic Physics and Time

The previous post, The Second Law, Microscopic Reversibility, and Small Systems looked at the fact that the underlying microscopic physics is not dependent on the direction that time flows, but that such microscopic interactions combine to produce a flow of increasing entropy with time because of the statistics involved.

In fact, this description was a bit of an oversimplification because it ignored electrical charge and parity.

CPT Symmetry

In the previous post, I discussed the idea that the basic physical laws of the universe were symmetrical with respect to time. In fact, this notion is an oversimplification. The microscopic laws of the universe are not symmetric with time, but under the standard model, they are symmetric under a deeper symmetry called CPT symmetry.

CPT-Symmetry is the invariance with respect to charge, parity, and time simultaneously.

Charge: Particles can have positive charge, negative charge, or no charge. For example an electron has negative charge, whereas a positron has positive charge. Charge is a conserved quantity (provable from gauge invariance).

Parity: Parity, in three three dimension is simply the inversion of all three Cartesian coordinates (x, y, and z) to their opposites (-x, -y, and -x).

Time: Trying to define time could be the subject of a book, several books, or an entire library of books. The purpose of this post is to wonder about entropy and time.

Consider a particle moving forward in time. Now consider reversing its charge, reversing its parity, and propagating it backward in time. Under CPT symmetry, these two situations are indistinguishable.

A violation of CP-symmetry on the other hand, implies a violation of T-symmetry and vice verse.

The universe is not invariant under CP symmetry; in fact, the standard model predicts very small deviations from such symmetry. The standard model in invariant under CPT symmetry, but allows violations of CP symmetry. Such CP asymmetries have been observed for interactions involving the weak force. This asymmetry implies a violation of T-symmetry as well. So in rare cases, time is not actually symmetric for certain interactions.

It is not clear that such small asymmetries can help with understanding why time moves forward instead of backward, but it is a necessary detour to make our understanding a bit more sophisticated. Even if such interactions were more common than they are, they can still be understood as a consequence of the fact that time moves forward. They are not sufficient to explain why time moves forward.

The universe is also not known to be invariant under CPT symmetry. String and quantum gravity theories allow the possibility of CPT violation. There are some very recent experiments looking for CPT-violation as well.

Time

John Wheeler famously said that "time is what stops everything from happening at once." In fact, it is necessary to think a little deeper about what time is. It turns out that our intuitive sense of simultaneity is not something that we can trust. It turns out that that in relativity simultaneity depends on one's frame of reference. Two events that happen simultaneously but in different places in one frame of reference do not happen at the same time in a different frame of reference.

The definitions of time therefore needs to be operational. Time is what we measure with a clock. Of course we want to use a very good clock, but aside from that it is not particularly important what the mechanism of the clock is.

For the purposes of standardization the second has been defined. Time is one of the seven fundamental quantities in International System (SI) units. The unit of time is the second, and it is defined as the time it takes for a specific number of periods (9,192,631,770) of the radiation corresponding to the transition between the two hyperfine levels of cesium 133 in its ground state. It's not important to understand all of that; what is important is to understand that we need to be able to measure time to define it. Because of relativity, we need to be careful how we measure it.

Entropy and Time

Aside from the very infrequently observed violations of CPT symmetry, entropy appears to be the only physical quantity that is dependent upon the direction in which time flows.

It is tempting to want to draw some grand conclusions from such facts. Is it possible that entropy itself is responsible for the fact that time moves forward? Or, is it more reasonable to think that time just does move forward and increasing entropy is a natural consequence?

As scientists, we need to be careful about jumping to conclusions about questions to which we do not know the answer. If we propose an answer, is there a consequence of that proposal that leads to a result that we can actually measure?

One area of research that is of current interest is the relationship between the inflation of the universe, entropy, and time. To understand some of these issues it is necessary to understand the proposed heat death of the universe. These topics are part of the next post.

The next post in this series is entitled, the Second Law, the Universe, and Cosmology.

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Child, M.S., Molecular Collision Theory, Dover, Mineola, NY, 1974, 1984 Reprint.
Tolman, R.C., The Principles of Statistical Mechanics, Oxford University Press, Oxford 1938
Wikipedia: CPT Symmetry
CPT Invariance
SLAC: The Standard Model
CPT Violation and the Standard Model
Feynman, Richard P., Leighton Robert B., Sands, Matthew, The Feynman Lectures on Physics, Addison-Wesley, Menlo park, CA, 1965
Wu, C.S., Ambler, E., Hayward, R. W., Hoppes, D.D., Hudson, R. P., Experimental Test of Parity Conservation in Beta Decay, Physical Review 105 (4) 1413-1415, 1957
Princeton: CPT Violation Experiment (CPT-I and CPT-II)
Kornack, Thomas Whitmore, A test of CPT and Lorentz Symmetry Using a K-3He Co-magnetometer." Dissertation, Princeton University (2005)
http://developeronline.blogspot.com/2008/04/time-is-what-keeps-things-happening-all.html
NIST: SI Units

Contents

Introduction
What the Second Law Does Not Say
What the Second Law Does Say
Entropy is Not a Measure of Disorder
Reversible Processes
The Carnot Cycle
The Definition of Entropy
Perpetual Motion
The Hydrogen Economy
Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
The Second Law and Swamp Coolers
Entropy and Statistical Thermodynamics
Fluctuations
Partition Functions
Entropy and Information Theory
The Second Law and Creationism
Entropy as Religious, Spiritual, or Self-Help Metaphor
Free Energy
Spontaneous Change and Equilibrium
The Second Law, Radiative Transfer, and Global Warming
The Second Law, Microscopic Reversibility, and Small Systems
The Arrow of Time
The Heat Death of the Universe
Gravity and Entropy
The Second Law and Nietzsche's Eternal Recurrence
Conclusion

The Second Law, Microscopic Reversibility, and Small Systems

2011-10-01T15:22:00.001-06:00

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled The Second Law, Radiative Transfer, and Global Warming.

On a small scale, individual physical events are reversible; yet on a macroscopic scale, it is not so. I used to find that confusing. I'd like to try to cut through some of the confusion. In so doing, the underlying mechanism of the second law may become clearer.

Figure Source (Monopoly by Hasbro).

Microscopic Reversibility

The principle of microscopic reversibility is simple. It states that at a microscopic level a reverse reaction takes place by the same mechanism as the forward reaction, only it is reversed.

It is perhaps not so simple to understand without an example. Consider a billiard table, one ball comes from the left and hits another and imparts all its momentum to that ball. The first ball takes the place of the second ball, and the second ball continues to roll.

If we assume a friction-less, drag-less ideal billiard table, we can play the film backwards. The second ball comes from the right, imparts its momentum to the first ball and takes its place. The first ball keeps moving.

Energy is conserved. All of the momentum can be transferred forward or backward. Of course with a macroscopic billiard table, our idealization matters. In real life, friction occurs and we can tell the difference between the forward process and the reverse process even if we play the film backwards.

On a microscopic level, however, processes are, in general, reversible.

Why does taking a process that is reversible microscopically and scaling it up to the macroscopic scale make it suddenly "care" about the forward and backward direction?

If you understand the answer to that question, you understand the basis of the second law of thermodynamics itself.

A Modified Monopoly Game

To understand the this question better, I am going to modify the rules to the game of Monopoly. Ordinarily, in Monopoly, one has to roll doubles to get out of jail. Rolling two dice gives 36 possible combinations, only six of which are doubles. So the probability of rolling doubles on any turn is 1/6.

In ordinary Monopoly, the path by which one ends up in jail is different than the way one gets out of jail. I am going to change the rules to make it symmetrical.

To enter jail, one lands on the "Go to Jail," square. Then one goes to jail, if and only if one rolls doubles. To leave jail, one rolls doubles, and then takes a turn starting from the "Go to Jail" square.

To make it completely symmetrical, one can move around the board either clockwise or counter-clockwise. if one enters jail clockwise, one must leave counter-clockwise and vice-versa.

Community Chest cards and Chance cards that send a player to jail are to be taken out of the deck.

There are 40 squares on the Monopoly board. The probability of landing on a given square, of course, depends on what square a player starts on, and in this modified game, the direction the player is headed. I am going to make a simplifying assumption that we have no information about what square a player was on previously.

For a small number of players (and let N = the number of players), this assumption seems silly, but as N gets large the probabilities converge.

So the chances of landing on any one square in a turn, by assumption, are 1/40.

Note that the path into jail is the reverse of the path out of jail:

Into jail: 1) Land on "Go to Jail." 2) Roll doubles.

Out of jail: 1) Roll doubles. 2) Move from "Go to Jail."

Let's make one more modification: all players start the game together in jail instead of "Go." There is only one possible arrangement of the players on the board with all of them in jail. It can be considered the state of zero entropy.

Consider a 1-player game. Eventually, the player will roll doubles and get out of jail. It is certainly possible that the player will end up back in jail. So, one could consider the system reversible.

Consider N=2. The state where both players are in jail has only one possible arrangement. The state with only one player in jail has 80 possible arrangements (if we don't know which player is in jail). There are 1600 possible states with both players out of jail ("Just Visiting" is a square like any other).

Eventually, both players get out of jail. It is certainly likely that eventually one or the other player will end up in jail. It is not too improbable that from time-to-time both players may wind up in jail at the same time. In fact, if the game goes on long enough, as Monopoly games seem to do, it will probably happen that both players are in jail at the same time again.

Note that as N increases, the number of states with increasing players outside of jail, also increase dramatically. Try calculating for N=3, the number of states with all players in jail, the number with one of the three players out of jail, the number with 2 players out of jail, and the number with 3 players out of jail.

Now imagine that N = 6 x 10²³. Suddenly the game seems different.

The games starts with all players in jail. On the first turn, 1/6 of the players will get out of jail. The fluctuations from that prediction will be so small as to be negligible. See my post on fluctuations to understand why in more detail.

As the game goes on, the players will distribute themselves in a more statistically likely distribution. There will always be players in jail, and there will always be players on every square. The most likely configuration will eventually prevail, and the probability of going to a less likely configuration will be so minuscule as to be negligible.

In other words, the macroscopic "reaction" of players leaving jail is irreversible, even though it is reversible on a microscopic level. This fact is really the basis of the second law of thermodynamics.

Small Systems

The astute reader may be asking about small systems. What if N=2, both players have been freed from jail, and spontaneously end up back in jail. Did they not violate the law of increasing entropy?

The more important point to understand is that such a movement of pieces does not violate the statistical basis of increasing entropy. The probability of decreasing entropy in this small system is large enough that it can happen on occasion.

In real microscopic physical systems, it is possible for the system to spontaneously move from a higher density of states to a lower density of states, as long as such a move is not too statistically improbable.

Thermodynamics is really the study of macroscopic systems. The second law works because macroscopic systems involve such large numbers.

Time

So, on a microscopic scale, a reaction can be reversible. We can run the film backwards on our friction-less billiard table and not be sure whether time is moving forward or backward. When we scale things up to the macroscopic, on the other hand, it is obvious which direction time goes. The broken glass never leaves the floor to become a fully formed glass on top of the table. Entropy seems to have a preferred direction forward in time.

The next post in this series is entitled the Arrow of Time.

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Child, M.S., Molecular Collision Theory, Dover, Mineola, NY, 1974, 1984 Reprint.
Tolman, R.C., The Principles of Statistical Mechanics, Oxford University Press, Oxford 1938
Monopoly by Hasbro

Contents

Introduction
What the Second Law Does Not Say
What the Second Law Does Say
Entropy is Not a Measure of Disorder
Reversible Processes
The Carnot Cycle
The Definition of Entropy
Perpetual Motion
The Hydrogen Economy
Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
The Second Law and Swamp Coolers
Entropy and Statistical Thermodynamics
Fluctuations
Partition Functions
Entropy and Information Theory
The Second Law and Creationism
Entropy as Religious, Spiritual, or Self-Help Metaphor
Free Energy
Spontaneous Change and Equilibrium
The Second Law, Radiative Transfer, and Global Warming
The Second Law, Microscopic Reversibility, and Small Systems
The Arrow of Time
The Heat Death of the Universe
Gravity and Entropy
The Second Law and Nietzsche's Eternal Recurrence
Conclusion

A Religious Test

2011-08-20T13:59:00.001-06:00

This post discusses the startling results from a poll/quiz previously posted on this blog regarding the US Constitution. Surprisingly, only a minority of respondents can correctly identify which statement about God and religion is actually in the Constitution.

Now, of course this poll is not a scientific one, and one should use caution in interpreting the results of such a poll, but nevertheless I suspect it is indicative of profound ignorance of the US Constitution.

Before I reveal the results, let me state the poll one more time to give the reader an opportunity to see which answer he or she would have marked as correct.

The poll asks the reader to finish the statement, "the Constitution states:"

The following options were given as choices:

Men are "endowed by their Creator with certain unalienable Rights"
In God We Trust
There shall be "separation between Church and State"
"no religious test shall ever be required as a qualification to any office or public trust under the United States"
One Nation Under God

The results of the poll are shown in the following bar graph:

Only 39%, a plurality, but not a majority got the correct answer, "no religious test shall ever be required as a qualification to any office or public trust under the United States."

Article VI of the Constitution states:

The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States. (Source)

This phrase is the only mention of God or religion in the body of the Constitution itself (The First Amendment is discussed below. Its text was not one of the choices in the quiz; note also that the date that the Constitution was adopted is stated as "the Seventeenth Day of September in the Year of our Lord one thousand seven hundred and Eighty seven. " ).

So where do these other phrases come from?

Endowed By Their Creator

This phrase comes from the Declaration of Independence, which states:

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. (Source)

It is important to note that the legal meaning of the Declaration of Independence as to its standing in law is somewhat ambiguous:

Although the Declaration of Independence stands with the Constitution as a founding document of the United States of America, its position in U.S. law is much less certain than that of the Constitution. The Declaration has been recognized as the founding act of law establishing the United States as a sovereign and independent nation, and Congress has placed it at the beginning of the U.S. Code, under the heading "The Organic Laws of the United States of America." The Supreme Court, however, has generally not considered it a part of the organic law of the country. For example, although the Declaration mentions a right to rebellion, this right, particularly with regard to violent rebellion, has not been recognized by the Supreme Court and other branches of the federal government. The most notable failure to uphold this right occurred when the Union put down the rebellion by the Southern Confederacy in the Civil War. (Source)

In God We Trust

This phrase appears on US coins. It was directed by an act of Congress in on April 22, 1864. Since that time it has occasionally been left off of coins:

The use of IN GOD WE TRUST has not been uninterrupted. The motto disappeared from the five-cent coin in 1883, and did not reappear until production of the Jefferson nickel began in 1938. Since 1938, all United States coins bear the inscription. Later, the motto was found missing from the new design of the double-eagle gold coin and the eagle gold coin shortly after they appeared in 1907. In response to a general demand, Congress ordered it restored, and the Act of May 18, 1908, made it mandatory on all coins upon which it had previously appeared. IN GOD WE TRUST was not mandatory on the one-cent coin and five-cent coin. It could be placed on them by the Secretary or the Mint Director with the Secretary's approval.

The motto has been in continuous use on the one-cent coin since 1909, and on the ten-cent coin since 1916. It also has appeared on all gold coins and silver dollar coins, half-dollar coins, and quarter-dollar coins struck since July 1, 1908. (Source)

In 1956, at the height of the cold war, the motto was adopted as an official motto of the United States. The phrase now appears on paper currency as well as on coins.

Separation Between Church and State

On January 1, 1802 Thomas Jefferson wrote a letter to the Danbury Baptists in support of the First Amendment's religious clauses. The text of the final letter that he sent is as follows:

The affectionate sentiments of esteem and approbation which you are so good as to express towards me, on behalf of the Danbury Baptist association, give me the highest satisfaction. my duties dictate a faithful and zealous pursuit of the interests of my constituents, & in proportion as they are persuaded of my fidelity to those duties, the discharge of them becomes more and more pleasing.

Believing with you that religion is a matter which lies solely between Man & his God, that he owes account to none other for his faith or his worship, that the legitimate powers of government reach actions only, & not opinions, I contemplate with sovereign reverence that act of the whole American people which declared that their legislature should "make no law respecting an establishment of religion, or prohibiting the free exercise thereof," thus building a wall of separation between Church & State. Adhering to this expression of the supreme will of the nation in behalf of the rights of conscience, I shall see with sincere satisfaction the progress of those sentiments which tend to restore to man all his natural rights, convinced he has no natural right in opposition to his social duties.

I reciprocate your kind prayers for the protection & blessing of the common father and creator of man, and tender you for yourselves & your religious association, assurances of my high respect & esteem. (Source)

In an 1879 decision, Reynolds v. United States, the Supreme Court of the United States found that Jefferson's language of a wall separating Church and State was an authoritative statement on the meaning of the religious clauses of the First Amendment:

Coming as this does from an acknowledged leader of the advocates of the measure, it may be accepted almost as an authoritative declaration of the scope and effect of the amendment thus secured. Congress was deprived of all legislative power over mere opinion, but was left free to reach actions which were in violation of social duties or subversive of good order. (Source)

So although this phrase may be an authoritative interpretation of an amendment to the Constitution, it is not itself to be found within the Constitution.

Under God

The Pledge of Allegiance was written by a socialist named Francis Bellamy in 1892. Originally, he wrote:

I pledge allegiance to my Flag and the Republic for which it stands, one nation, indivisible, with liberty and justice for all. (Source)

In 1923, it was changed to read:

I pledge allegiance to the Flag of the United States of America and to the Republic for which it stands, one nation, indivisible, with liberty and justice for all. (Source)

It was adopted by Congress in this form in 1942. Twelve years later, in 1954, during the cold war, at the urging of President Eisenhower, the phrase "one nation, indivisible" was divided with the phrase "under God." The issue of whether to include the phrase has been divisive one ever since.

The First Amendment

The battle of the role of religion in public life centers around the First Amendment to the Constitution, which reads:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.(Source)

It is necessary to mention that the First Amendment has been incorporated into the Fourteenth Amendment and is therefore binding on the states as well as the Federal Government.

The amendment prohibits the establishment of religion and also protects the free exercise of religion. These parts of the amendment are termed "the establishment clause," and the "free exercise" clause. There is an inherent tension between establishment and free exercise.

A perfect example of that tension is the existence of the Army Chaplain Corps. The Corps exists to guarantee soldiers the right to free exercise of their religion. Yet the Chaplain Corps must carefully walk the line of not violating the establishment clause. A cursory look at court cases involving the Chaplain Corps make evident the difficulty of walking this line.

The Chaplain Corps is perhaps a microcosm of the nation at large that must constantly struggle to find the right balance between "establishment" and "free exercise."

Sources

Petagrams of Carbon

2011-07-09T13:28:00.000-06:00

Sometimes carbon dioxide is referenced in units of ppm, and sometimes it is referenced in petagrams of carbon. What are the meanings of these units and how does one convert between them?

In a previous post I explained how to convert to and from units of ppm. The current post explains the units petagrams of carbon, and how to convert from ppm to petagrams of carbon.

Petagrams

First it is necessary to understand units of petagrams. The SI units of mass are kilograms (kg). A kilogram is 1000 grams. One can construct other units of mass by prepending a prefix to the units grams.

1 kilogram (kg) = 10³ grams (g)
1 megagram (Mg) = 10⁶ g = 10³ kg
1 gigagram (Gg) = 10⁹g = 10⁶kg
1 tertagram (Tg) = 10¹²g = 10⁹kg
1 petagram (Pg) = 10¹⁵g = 10¹²kg

So one petagram is equal to one quadrillion grams, or one trillion kilograms.

The Atmosphere

The mean mass of the atmosphere is 5.1480 x 10¹⁸ kg or 5.1480 x 10²¹g. The molar mass of air is 28.966 g. So the atmosphere contains 5.1480 x 10²¹/28.966 = 1.7773 x 10²⁰ moles of air.

Moles of Carbon Dioxide

Mole fractions of carbon dioxide are expressed in ppm and directly convertible from parts-per-million by volume.

So 1 ppm carbon dioxide = 1.7773 x 10²⁰ /1,000,000 = 1.7773 x 10¹⁴moles carbon dioxide.

In June of 2011, the Mauna Loa site measured 393.69 ppm of carbon dioxide, which equates to

393.69 x 1.7773 x 10¹⁴= 6.9970 x 10¹⁶ moles of carbon dioxide. It is worth noting that carbon dioxide is near its high point in the cycle for 2011, its last low point was about 386 ppm or 6.86 x 10¹⁶ moles.

Mass of Carbon Dioxide

It is now a simple matter to convert from moles of carbon dioxide to mass of carbon dioxide. The molecular mass of carbon dioxide is

12.01 + 16.00 + 16.00 = 44.01 grams/mole.

So 1 ppm carbon dioxide =

1.7773 x 10¹⁴moles x 44.01 grams/mole = 7.822 x 10¹⁵grams or 7.822 petagrams of CO2.

So 393.69 ppm = (393.69 x 7.822) = 3079 Pg Pg CO2
and 386 ppm CO2 3019 Pg CO2.

Petagrams of Carbon

For every mole of carbon dioxide, there is one mole of carbon. The molar mass of carbon is 12.01 g.

So 1 ppm carbon dioxide =

1.7773 x 10¹⁴moles x 12.01 grams/mole = 2.134 Pg of carbon.

So 393.69 ppm CO2 = 840.1 Pg of carbon
and 386 ppm of CO2 = 823.7 Pg carbon

Sources

Trenberth, Kevin E; Smith, Lesley, The Mass of the Atmosphere: A Constrained Global Analysis, J. Climate 18, 864-875 (2005)
John M. Wallace and Peter V. Hobbs, Atmospheric Science, Second Edition, Elsevier, Amsterdam, 2006
Earth System Research Laboratory Global Monitoring Division: Recent Mauna Loa CO2.

Fixing Norton Ghost

2011-07-04T21:27:00.000-06:00

If you are running Symantec's Norton Ghost to backup a computer, you may occasionally run into the following helpful error:

"An interaction with the catalog's database layer failed unexpectedly: sqlite3_step--5: database is locked."

Surprisingly, the Norton online help is no use at all. This problem, however is not difficult to fix. As I have had this same issue with multiple versions of Ghost and also it seems to occur on multiple operating systems, the solution is probably not exclusive to my environment. However, I am running Vista 64 bit and Norton Ghost 14.

To solve this problem it is necessary to stop the Norton backup service. Press Control-Alt-Delete all at the same time and choose "Start Task Manager."

Click on the "Services" tab:

If you like, sort by name; you do not have to do that, but it makes it easy to see which services are running:

In my case the service is called "norton Ghost," but it may also be called "BESR," Backup Exec System Recovery." It may be listed as either a Norton or a Symantec product.

Select it, right click, and choose "stop service."

It may deny you access;

We'll simply have to outsmart it. Click on the services button in the lower right corner. Windows may ask you for permission; grant it, and the following window should appear:

Select the service and click "stop the service" in the upper left:

Norton may complain that it cannot connect to the "agent." Disregard the complaint. Keep the services window open; you will need it again to restart the service. Got to "Computer," or "My Computer" or whatever your operating system calls it and navigate to the C drive.

Double click on Local Disk (C:) and then double click on "Program Data."

Double click on "Symantec."

Double click on "FileBackup"

Delete the file "catalog.dat

It might also be found in C:\Documents and Settings\All Users\Application Data \ file backup

You may have to hunt around in similar locations depending on your OS and Ghost version.

Go back to the services window, choose Norton Ghost and select "Start the service."

Run Ghost. It may take awhile to start as it has to rebuild the database. You may need to redefine your backup jobs, but you probably do not. Click on "status" and see what happens. If you have to redefine the jobs, go ahead and do it, but I did not have to do so.

The Second Law, Radiative Transfer, and Global Warming

2011-07-01T13:38:00.010-06:00

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled Spontaneous Change and Equilibrium.

Does global warming violate the second law of thermodynamics? Such a claim may seem strange. The idea that the vast majority of physical scientists would subscribe to an idea that somehow violates a fundamental law of thermodynamics on its face seems odd. Yet, such a claim is often made by people calling the science behind global warming into question.

I prefer to call such people doubters rather than deniers. I have written a post that explains the reason for this preference. The term denier is too closely associated with Holocaust denial, and it is an unfair brand.

The term skeptic is also inappropriate because doubters rarely behave like true skeptics. In fact, the skeptical view is to follow the evidence. ~~, and the~~ The preponderance of the evidence supports the case for anthropogenic global warming., but this fact will have to be demonstrated.

To understand that evidence, we must bite off one piece at a time, and this essay focuses specifically on the erroneous claim that global warming somehow violates the second law of thermodynamics.

Radiative Transfer

Before it is possible to address this claim it is necessary to understand the context in which it is raised. For those wishing a more in-depth understanding, I have written a primer on infrared spectroscopy and global warming. Radiative transfer involves the absorption, emission, and scattering of electromagnetic radiation.

These processes in the earth's atmosphere are extremely important in understanding the energy balance of the troposphere. Carbon dioxide, in particular, absorbs infrared radiation emitted from the earth.

The carbon dioxide in each layer of the atmosphere also emits infrared radiation. This emission is dependent on the temperature of that layer; it is also dependent on the infrared spectrum of carbon dioxide, which is not a blackbody because it has discrete spectral absorption and emission.

A comment to one of the posts in that series from a person that I assume to be a student in the physical sciences in that series reads as follows:

In my opinion the radiative transfer theory puts in the following inconsistencies when deals with planetary atmosphere.

1) It firstly contradicts the second law of thermodynamics assuming that thermal radiation (heat) flows spontaneously from a molecule at lower temperature to a molecule at higher temperature (feedback radiation).

2) Moreover one estimates the atmospheric CO2 15 microns irradiance adopting, with extreme easiness, the black body radiation laws (Plank, Stefan-Boltzmann, Kirchhoff) as if its thermal radiation is both always and however possible at any temperature.

We know very well that the radiation at 15 microns pertains to a precise vibrational resonant frequency of the CO2 molecule. On the other hand we also know that the CO2 molar heat capacity at constant volume, at atmospheric temperatures, is around 2.5*R (R is the universal gas constant) or rather that such molecules still behave in practice as rigid bodies since the collision intensity with the other molecules, due to the thermal random motion, is absolutely insufficient to start and keep up a meaningful vibrational forced oscillation and to bring it in resonance, that is the necessary condition for photon emission. But then, if there are no forced vibrational oscillations thermically rising, how can the molecules to have a thermal radiation at a vibrational frequency? Why this is possible the CO2 molar heat capacity at constant volume should be at least equal to 3.5*R, value that is reached around a temperature of 700-800 K.
With these two substantial thermodynamic inconsistencies I think that is all wrong, all to start again.

I replied briefly and perhaps a little condescendingly to this comment as follows:

Thanks for your comment. You are a little confused about the second law of thermodynamics. Specifically, you are confused about the difference between kinetics and thermodynamics. Colder bodies do spontaneously radiate to hotter bodies, it's just that the hotter bodies radiate more and the average result is that the hotter body warms up the colder body. You ought to read up on rate equations and equilibrium.

Your argument about heat caopacity would seem to argue that CO2 cannot radiate, but it is easy to observe CO2 radiating. The answer is that at room temperature only about 1 in a million CO2 molecules is in v=1, vibrational motion is quantized. The effect on the heat capacity is small.

The inconsistencies you bring up are not substantial, but indicate that you have some more studying to do. Keep on doing it! But think for yourself!

Other than the condescending tone, there are two things I would change about my reply. The first is that I would omit the word spontaneously. It is confusing to use it in this context. Secondly, instead of referring to v=1, I should have referred to ν2 = 1. The response hits the major points, but deserves some further explanation.

The Second Law

One misunderstanding of the second law of thermodynamics is that it does not allow heat to flow from a cold body to a hot body. The person who made the comment above also misunderstands the second law of thermodynamics, but in a way that takes some explanation to understand.

The second law does prevent heat from flowing from a cold body to a hot body in a cyclic process with no other effect. See my posts on air conditioners and swamp coolers to understand why heat can be transferred from a cold body to a hot body.

Still, that is not the mistake made in this comment.

It is true that under normal (spontaneous) conditions that the net transfer of heat must be from a hot body to a cold body and not the reverse.

This fact does not mean that energy is not or cannot be transferred from the cold body to the hot body. It simply means that more energy must flow from the hot body to the cold body than vice versa.

A world in which energy could not flow from a cold body to a hot body would be very strange indeed.

Consider two gas samples of carbon dioxide at two different temperatures. Consider that the samples are placed so that infrared radiation emitted from one sample impinges on the other and vice versa.

The cold sample will emit radiation that is dependent on its temperature and its infrared spectrum. It does not somehow magically "know" that there is a hotter sample of carbon dioxide near it. It does not know that it should not emit radiation in the direction of the hotter sample.

Infrared photons that approach the hotter sample are of the correct frequencies to be absorbed by the hotter sample. The hotter sample does not magically know that the origin of these photons is a sample of carbon dioxide that is cooler than it is. It will absorb those photons in line with Beer's Law.

Now the fact is that the hotter sample of carbon dioxide radiates more photons than the cold sample. The cold sample absorbs these photons as well. The net effect is that more photons are emitted from the hot sample and absorbed by the cold sample than are emitted from the cold sample and absorbed by the hot sample.

So the net heat transfer is indeed from the hotter body to the colder body. See my post on a multi-layer model of carbon dioxide to understand this situation at a deeper level.

Why does it matter?

Consider the troposphere. In the troposphere, the temperature generally decreases with increased altitude. See my post on the structure of the atmosphere to understand the temperature profile of the atmosphere in more depth.

The plot below shows an idealization of the troposphere with a lapse rate of 6.5 K per 1000 meters.

An important effect of adding carbon dioxide to the troposphere is that more of the infrared radiation that originates from the earth is absorbed in the troposphere.

Therefore the troposphere warms. There are feedback mechanisms involving water that increase the effect of this warming.

The upper troposphere is still much colder than the surface of the earth. So why should a warming troposphere have any effect on surface temperature? After all, we all "know" that heat cannot be transferred from a cold body to a hot body.

As explained above, the net heat does flow from hot to cold, but energy also flows from cold to hot and that energy affects the net heat flow.

Why do we use blankets to keep us warm? The trapped air, after all, is still cooler than our body temperature. In fact that trapped air keeps us warm because there is energy flow both ways.

The net effect is that heat flow is proportional to the gradient of the temperature. In other words, your body loses more heat outside on a cold day than it does on a warm day. There is nothing magical or counter-intuitive going on here. For a deeper understanding see my post Spontaneous Change and Equilibrium.

Vibrational Excitation

The second part of the comment may seem a bit more esoteric for the casual reader, but it does not make it any more accurate. I will try to unpack this comment a bit and explain why it is based upon an incorrect understanding.

The essence of the argument is as follows. The individual makes a back-of-the-envelope calculation of the energy content of carbon dioxide as a function of temperature. At temperatures of interest, let's say 200K - 300K to keep things simple, it is clear that the energy per molecule is not enough to excite the ν2 (~15 micron) band of carbon dioxide.

If this reasoning were correct, the vibration could never be excited at the temperatures of interest, and therefore carbon dioxide could not emit infrared radiation through the 15 micron band.

It is an empirical fact that carbon dioxide at temperatures from 200-300K does emit radiation from the 15 micron band. So where did the reasoning go wrong?

The error is that the individual who made this calculation does not sufficiently understand energy distributions in a collection of molecules. Not every molecule in a sample of carbon dioxide has the average energy of that sample. Some molecules have less energy than the average; some molecules have more energy.

The distribution of energies is well understood and is governed by Boltzmann statistics.

Back-of-the-Envelope Calculation of Average Energy

Because it is the principle that maters, not the absolute numbers, I will follow suit with a back-of-the envelope calculation without quibbling about the details.

If the constant-volume heat capacity of a mole of carbon dioxide is assumed to be about 2.5 R, then the internal energy content is about 2.5 x R x T, where R is the universal gas constant and T is the absolute temperature.

At 200 K, that is equal to about 4160 Joules/mole. At 300 K it is equal to about 6240 Joules/mole.

In a Joule of carbon dioxide, there are Avogadro's number (6.022 x 10²³) of carbon dioxide molecules. So the average energy of a molecule at 200 K is 4200 divided by Avogadro's number, or 6.69 x 10^-21 Joules.

At 300 K that works out to 1.04 x 10^-20 Joules.

The center of the 15-micron band is at about 670 cm^-1. To convert from wavenumbers to microns quickly we take 10⁴ and divide by 670.

Now let's convert to Joules.

1 Joule(J) = 6.242 x 10¹⁸ eV = 5.034 x 10²² cm^-1.

So 670 cm^-1 = 670/5.034 x 10²² = 1.33 x 10^-20 Joules.

So, as stated the average molecule does not have enough energy to excite the vibration.

Now consider what happens if one out of a million molecules is in the excited vibrational state. Out of a mole of molecules that would mean that 6.022 x 10¹⁷ molecules have this added energy. If we sum up all of that energy it is (1.33 x 10^-20 Joules) x (6.022 x 10¹⁷ molecules), which equals 0.00801 Joules or 8.01 mJ.

Recall that a mole of carbon dioxide at 200 K has 4160 Joules and at 300 K, it has 6240 Joules.

If we subtract the energy used to excite the vibrationally excited molecules from the energy available, it has a negligible effect. In other words, the effect of the heat capacity of the vibrational excitation is negligible.

If we are worried about the heat capacity, we are free to approximate a sample of carbon dioxide as if it were made of rigid bodies, but we must not confuse an approximation with reality. Some of those molecules are vibrationally excited, a fact that can be shown by application of Boltzmann statistics.

This calculation contains many simplifications and approximations, but the exact numbers are not important. The key point to understand is that there is a distribution of energies among the molecules in a sample. Bulk properties, such as the heat capacity are not very sensitive to the fact that some of the molecules have enough energy to be in excited vibrational states.

Taking an average and concluding that none of the molecules have enough energy to be in a vibrationally excited state is a misunderstanding of the basic physics involved.

Otherwise, carbon dioxide at temperatures between 200-300 K would not thermally emit radiation via the 15 micron band.

This thermal radiation can be directly observed. If one looks at a cold sky at night, emission from water, carbon dioxide and ozone can be readily observed with passive infrared spectrometry.

A few examples of such observations are listed below.

A brief search of the literature will reveal many other such observations.

The Second Law Meme

I am not sure where the idea that global warming violates the second law of thermodynamics comes from. I have done some searching on Google and have found various repetitions of the claim a few examples are below:

There are a lot more examples. Even the paper by Gerlich and Tscheuschner relies on the same faulty reasoning, and they ought to know better. These arguments are routinely put forth and are rarely challenged by the so-called "skeptic" community.

Skepticism

Using talking points that one does not understand to dismiss science that one does not understand is not skepticism. In my post Glacier Gate and Healthy Skepticism, I discuss this issue:

Healthy skepticism involves thinking for oneself and following the trail of evidence. There is a tendency in a politicized environment to try to lead the evidence rather than follow it. In leading the evidence one takes potshots at the evidence without actually trying to understand it. A true skeptic keeps an open mind, but listens to what the evidence is saying. Claiming that the Theory of Evolution is untrue because one does not wish to believe it, is not skepticism. So it is with global warming.

Neither should a true skeptic take a claim that the Himalayan glaciers will melt by 2035 at face value. A true skeptic should dig deeper and ask what such a claim is based upon. If the claim is based upon good science, one should be willing to accept it. If the IPCC researchers had dug deeper, they would have found that the prediction was premature based upon the science.

A true skeptic would be as interested in refuting such a bogus claim with valid physics as he or she would be interested in challenging the claims of established climate science.

Additionally, one cannot be a skeptic of an argument that one does not understand. The right to call oneself a skeptic is earned, not asserted.

The next post in this series is entitled The Second Law, Microscopic Reversibility and Small Systems.

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Infrared Emission Spectrum of the Atmosphere, Raymond Sloan, John H. Shaw and Dudley Williams, JOSA vol. 45, Issue 6, pp. 455-457 (1955)
Upper Atmospheric Infrared Radiance from CO2 and NO Observed During the Spirit 1 Rocket Experiment, Alber-Golden, S.M.;Matthew, M.W.; Smith D.R., Journal of Geophysical Research, vol 96., Jul1. 1991 pp. 11,319 - 11,329
Rocketborne Cryogenic (10 K) high-resolution interferometer spectrometer flight HIRIS: Auroral and Atmospheric IR Emission Spectra, Stair, A.T, Jr.; Pritchard J.; Coleman, I.; Bohne C.; Williamson, W.; Rogers, J.; Rawlins, W.T., Appl Opt. 1983, Apr 1; 22 (7), pp. 1056-1069
NIST WebBook
A post by Alan Siddons From Climate Realists
An article by Rebecca Terrell at the New American
A Journal article by Gerhard Gerlich and Ralf D. Tscheuschner entitled Falsification of CO2 Greenhouse Effects Within the Frame of Physics

Contents

Introduction
What the Second Law Does Not Say
What the Second Law Does Say
Entropy is Not a Measure of Disorder
Reversible Processes
The Carnot Cycle
The Definition of Entropy
Perpetual Motion
The Hydrogen Economy
Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
The Second Law and Swamp Coolers
Entropy and Statistical Thermodynamics
Fluctuations
Partition Functions
Entropy and Information Theory
The Second Law and Creationism
Entropy as Religious, Spiritual, or Self-Help Metaphor
Free Energy
Spontaneous Change and Equilibrium
The Second Law, Radiative Transfer, and Global Warming
The Second Law, Microscopic Reversibility, and Small Systems
The Arrow of Time
The Heat Death of the Universe
Gravity and Entropy
The Second Law and Nietzsche's Eternal Recurrence
Conclusion

Patanol vs Alaway (Ketotifen Fumarate) Eye Drops

2011-06-11T09:16:00.001-06:00

In July of 2010, I wrote a piece on my experience with Medco getting my prescription for Patanol filled. The medication is a prescription antihistamine that is administered by drops to the eye. It is prescribed for hay fever and similar allergies that cause itchy eyes.

During my struggles with Medco, my local pharmacist recommended an over-the-counter medication that he said was just as effective called ketotifen fumarate, which is sold under the brand name of Alaway but also under other brand names.

After some experience using both medications, I thought I should report back on my experience. First, I need to insert all the necessary caveats. I am not a medical doctor or a pharmacist. If you need medical advice, see your family doctor, or ophthalmologist.

My experience as a patient is that both medications helped to stop my eyes from itching, however, Alaway stings when the drops are added, whereas I do not have this experience with Patanol. If you do not mind the sting, Alaway may be fine, but I prefer the Patanol.

Spontaneous Change and Equilibrium

2011-06-05T15:23:00.006-06:00

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled Free Energy.

Diamond turns spontaneously into graphite; yet we may have to wait longer than the lifetime of the universe to see such a change. Hydrocarbons are spontaneously oxidized into carbon dioxide and water; yet gasoline requires a source of heat before it burns.

Spontaneity

In thermodynamics, the term spontaneity has a very specific meaning. Dictionary definitions include the following.

From Dictionary.Com::

1. coming or resulting from a natural impulse or tendency; without effort or premeditation; natural and unconstrained; unplanned: a spontaneous burst of applause.
2. (of a person) given to acting upon sudden impulses.
3. (of natural phenomena) arising from internal forces or causes; independent of external agencies; self-acting.

From Merriam Webster:

1. proceeding from natural feeling or native tendency without external constraint
2. arising from a momentary impulse
3. controlled and directed internally : self-acting
4. produced without being planted or without human labor : indigenous
5. developing or occurring without apparent external influence, force, cause, or treatment
6. not apparently contrived or manipulated : natural

Notice that even the dictionary definition does not necessarily imply a time-frame, i.e., it does not have to happen right away to be spontaneous, even though in common parlance the word is often used in contexts that imply that an event occurred immediately.

The thermodynamic definition is perhaps closest to the definition that states "arising from internal forces or causes."

A caveat is necessary, however. Some changes that are spontaneous, in a thermodynamics sense, can require an external impetus to initiate them. The change arises from forces inherent in the system, but an external nudge may be necessary to overcome an activation barrier.

Consider a glass sitting on a table. If it falls to the floor and shatters, the occurrence arose from the force of gravity that was always present in the system. Of course, it may require a clumsy person, or a curious cat to provide a nudge to get over an activation barrier, but after that initial nudge, the energy was already stored in the system.

In thermodynamics, a spontaneous change is just such a change, in which the driver for change is internal to the system. We do not care how fast the change happens, or whether something external might help it get going.

A ball spontaneously rolls down a hill; it does not spontaneously roll up a hill. The fact that it may need a nudge to get started is actually a feature of kinetics.

Kinetics focuses on how fast a change will occur, thermodynamics is only concerned with the direction in which the net change occurs.

A change that is not spontaneous can still happen. It is possible to roll a ball up a hill. It is possible to pick a glass up off the floor and put it on a table. It is possible to transfer heat from a cold reservoir to a hot reservoir. All these non-spontaneous changes, require something to drive them.

The driver for a non-spontaneous change, in fact, must be a change that is spontaneous. Spontaneous changes provide the energy that is used to drive a non-spontaneous change.

Of course, we must be careful, when we talk about providing energy. Energy is conserved. Energy cannot strictly be used up.

It is the distribution of energy that changes during a spontaneous change. The useful energy comes from changing the way that energy is distributed in the system and the surroundings.

At constant pressure, a change is spontaneous if the change in Gibbs Free Energy is negative.

It is possible to drive a change, in which the free energy is positive by a bigger change in which the free energy is negative.

Equilibrium

If you were truly at equilibrium, you'd be dead. We live in a world that is constantly in flux. Perhaps, at the heat death of a closed universe, equilibrium will be reached. Equilibrium is a useful concept, but we must understand equilibrium to be both local and temporary.

Consider a glass of ice water. In this glass, I have placed enough water and enough ice that they can come to local equilibrium. That equilibrium is reached at the freezing point of water (32 °F, 0 °C, 273.15 K).

Of course, if we wait long enough the glass will warm to the temperature of the room and all the ice will melt. Let's assume that the glass is well insulated from the room so that heat from the room cannot be transferred to the glass and vice versa. At least for the short term, it is possible to obtain equilibrium in the glass.

At equilibrium, ice is melting to become liquid water and absorbing heat, but liquid water is freezing on the surface of the ice cubes and giving off heat.

These processes are in balance, water becomes ice, and ice becomes water. If the glass is truly at equilibrium, the ratio of ice to water remains the same as these changes occur.

The same occurs in a chemical reaction. Consider a simple reaction, in which two reactants, A and B react to form two products C and D.

A + B → C + D

Suppose that this reaction is spontaneous, i.e., ΔG < 0 for the reaction. Still, it is necessary to consider the reverse reaction, which after all can be driven by energy from somewhere in the system.

C + D → A + B

At equilibrium, a solution of the reactants and products will have all four species. The amount of the constituents is determined by an equilibrium constant, K.

K = [C]*[D]/[A]*[B]

Here the square brackets indicate the concentration. It is simplest to take the concentrations as molar concentrations.

The equilibrium constant K is related to the standard change in Gibbs energy for the reaction.

K = exp (-ΔGrxn/RT )

Exp is the exponential function. ΔG rxn is the standard molar change in Gibbs energy for the reaction. R is the universal gas constant, and T is the thermodynamic temperature.

At equilibrium the forward change is balanced by the backward change. It does not mean there is no change; it means that the change balances.

Energy Flow

The same is true of energy flow from one body to another body. Two bodies in equilibrium with each other are at the same temperature (the so-called zeroth law of thermodynamics). If two bodies are at the same temperature, it is not the case that no energy is flowing between the bodies.

Rather the energy flowing between the two bodies exactly balances. There is no net flow of energy because the flow of energy cancels.

Now consider a hot body and a cooler body. It is important to realize that energy flows in both directions. It flows from the hot body to the cool body and from the cool body to the hot body. It is not a violation of the second law of thermodynamics.

More energy flows from the hot body to the cool body, and the net flow of energy is from the hot body to the cool body.

Consider a black body at 100 K. It radiates according to its characteristic radiation curve.

Figure source

That body at 100K does not know whether it is radiating toward empty space or whether there may be another body there that could be cooler or hotter.

Now imagine that there is another hotter black body at 300 K. It is a black body; so it will absorb any photons that come its way. It does not know the source of the photons is cooler than it is. There is no magic here; the bodies do not somehow know to stop radiating to a hotter body.

However, the body at 300K radiates more than the body at 100 K. More photons, and thus more energy flow from the hot body to the cool body than vice versa. The net flow of energy is from hot to cold, but energy flow both ways.

If allowed to come to equilibrium (and ignoring the influence of other bodies), the hot body will cool as it radiates, the cool body is absorbing radiation from the hot body and it will warm. As the hot body cools, it radiates less. As the cool body warms it radiates more. When they reach the same temperature, the flows of energy will be equal, resulting in no net flow of energy.

If it did not work this way, there would be no use putting blankets over you at night. It is an exercise for the reader to consider why a blanket that is colder than one's body can nevertheless keep one warm.

The next post is entitled The Second Law, Radiative Transfer and Global Warming.

Sources

Dictionary.Com: Spontaneous
Merriam Webster: Spontaneous
Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Anderson, H.C., Stanford University, Lectures on Statistical Thermodynamics, ca. 1990.
http://commons.wikimedia.org/wiki/File:BlackbodySpectrum_loglog_150dpi_en.png

Contents

Introduction
What the Second Law Does Not Say
What the Second Law Does Say
Entropy is Not a Measure of Disorder
Reversible Processes
The Carnot Cycle
The Definition of Entropy
Perpetual Motion
The Hydrogen Economy
Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
The Second Law and Swamp Coolers
Entropy and Statistical Thermodynamics
Fluctuations
Partition Functions
Entropy and Information Theory
The Second Law and Creationism
Entropy as Religious, Spiritual, or Self-Help Metaphor
Free Energy
Spontaneous Change and Equilibrium
The Second Law, Radiative Transfer, and Global Warming
The Second Law, Microscopic Reversibility, and Small Systems
The Arrow of Time
The Heat Death of the Universe
Gravity and Entropy
The Second Law and Nietzsche's Eternal Recurrence
Conclusion

Conclusion

The Holocaust History Project: Pamphlet on David Irving

2011-05-22T12:48:00.004-06:00

The Holocaust History Project Blog of The Holocaust History Project has announced an updated pamphlet on Holocaust denier David Irving.

The British court found that Irving's

misrepresentation of history was

"deliberate in the sense that he was motivated by a

desire borne of his own ideological beliefs to

present Hitler in a favourable light."

Free Energy

2011-05-20T13:33:00.007-06:00

This post is part of a series,Nonsense and the Second Law of Thermodynamics. The previous post is entitled Entropy as Religious, Spiritual or Self-Help Metaphor.

You cannot get something for nothing, and the term "free energy" does not mean energy that has no cost. Rather it refers to energy that is available to do something useful. The use of the word "free" is in the sense of "liberated." Free energy is energy that can be liberated to do something useful.

We know that the second law places limitations on how much energy can be used to do useful work. In most situations, some of the energy must be dissipated as heat that cannot be used to do something useful. The portion of the energy that can do something useful is the free energy.

At constant volume, the free energy is the Helmholtz Free Energy, given the variable A.

The Helmholtz Energy

The Helmholtz energy is equal to the internal energy, U, minus the product of the absolute temperature, T and the entropy S.

A = U - TS

The internal energy is defined in my post on the first law of thermodynamics. T is the temperature in Kelvin, and S is the entropy defined in this series.

As an example, suppose that temperature is kept constant. In such cases:

ΔA = ΔU - TΔS

The delta symbol, Δ, indicates the change in the quantity. By convention:

ΔA = A(final) - A(initial)

The Gibbs Energy

The Gibbs energy is analogous to the the Helmholtz energy except that it is the free energy when pressure is kept constant. The Gibbs energy is equal to the enthalpy, H, minus the product of the absolute temperature, T and the entropy, S.

G = H - TS

The enthalpy is defined in a post on enthalpy as:

H = U + PV

So the Gibbs energy can also be written:

G = A +PV

A constant temperature:

ΔG = ΔH - TΔS

Application

At constant pressure, a system is in equilibrium if

ΔG = 0

On the other hand, if delta G is positive:

ΔG > 0

Energy must be provided to drive the the change.

If a change releases Gibbs free energy, i.e., delta G is negative.

ΔG < 0

The change is spontaneous. Oxidation of hydrocarbons to form carbon dioxide and water is a spontaneous change. It provides energy that can be used to drive other changes. In thermodynamics the term spontaneous is used a little differently than in plain language.

For example diamond changes spontaneously into graphite because ΔG < 0.

The next post in the series is entitled Spontaneous Change and Equilibrium.

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Anderson, H.C., Stanford University, Lectures on Statistical Thermodynamics, ca. 1990.

Contents

Introduction
What the Second Law Does Not Say
What the Second Law Does Say
Entropy is Not a Measure of Disorder
Reversible Processes
The Carnot Cycle
The Definition of Entropy
Perpetual Motion
The Hydrogen Economy
Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
The Second Law and Swamp Coolers
Entropy and Statistical Thermodynamics
Fluctuations
Partition Functions
Entropy and Information Theory
The Second Law and Creationism
Entropy as Religious, Spiritual, or Self-Help Metaphor
Free Energy
Spontaneous Change and Equilibrium
The Second Law, Radiative Transfer, and Global Warming
The Second Law, Microscopic Reversibility, and Small Systems
The Arrow of Time
The Heat Death of the Universe
Gravity and Entropy
The Second Law and Nietzsche's Eternal Recurrence
Conclusion

Enthalpy

2011-05-07T14:00:00.018-06:00

This post continues a tangent from my series on the second law of thermodynamics. It discusses another quantity in thermodynamics, but it is necessary before I can get to the next post in the series, which is on free energy.

This post discusses the term enthalpy.

At constant pressure the change in enthalpy is the heat transferred to a system.

  ΔH = q at constant pressure.

Heat is not a state function, but enthalpy is.

The change in enthalpy only depends on initial and final values, it does not depend upon the path taken. Heat on the other hand does depend on the path taken.

Relationship to Internal Energy

In a previous post on the first law of thermodynamics, I defined a state function called the internal energy, U. Recall that:

ΔU = q +w

If the only work being done is PV work:

ΔU = q - PΔV

Also recall that at constant volume:

ΔU = q

Under conditions of constant pressure:

ΔH = q

Definition of Enthalpy

Therefore:

  ΔH = ΔU + PΔV

In integral form:

H = U + PV

This final equation is the definition of enthalpy.

Examples

One example of the use of enthalpy is the conversion of diamond to graphite. Assume this transition occurs at the thermodynamic standard conditions of a pressure of 1 barr (1 barr = 100,000 Pa), and a temperature of 298.15 K (25 °C ).

C (diamond) → C (graphite)

This reaction is endothermic, i.e., heat is released. For one mole of carbon:

ΔH = -1.895 kJ

The reverse reaction is endothermic:

C (graphite) → C (diamond)    ΔH = 1.895 kJ/mol

The latter reaction is the reaction for the enthalpy of formation for diamond. Graphite is the standard state of carbon and its enthalpy of formation is zero by definition. Another example is the enthalpy of formation of carbon dioxide:

C (graphite) + O₂ (gas) → CO₂ (gas)   ΔH = - 393.5 kJ/mol

This reaction is exothermic; burning carbon gives off heat. In fact most of the energy we depend upon comes from burning hydrocarbons to form carbon dioxide and and water. This topic will be discussed in numerous future posts.

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donald A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
UC Davis
Chemwiki: Enthalpy

Site Redesign

2011-04-30T14:21:00.002-06:00

Frequent visitors may have noticed that I've redesigned the site. An unfortunate consequence is that two of the polls were lost; I've recreated them but lost the votes to date.

Second Law Poll

2011-04-30T10:15:00.002-06:00

The purpose of this post is to document the final results of a poll on the site. I do not intend to comment on the results until I conclude my series on the second law of thermodynamics. It is worth noting at this point that there is no clear consensus on the correct answer from respondents.

I think that this fact is reflective of the state of confusion that exists in public discourse regarding the second law of thermodynamics.

The poll asked the respondents to pick an option to complete the statement: "The second law of thermodynamics states:"

The option were:

The disorder of the universe must increase.
You can't win.
The entropy of an isolated system must stay the same or increase.
The entropy of a closed system must stay the same or increase.
Heat cannot be transferred from a cold body to a hot body.
The entropy of the world must increase.
Let's keep score.

The results are shown in the following figure with truncated responses:

The First Law of Thermodynamics

2011-04-24T12:07:00.006-06:00

I need to take a tangent from my series on the second law of thermodynamics and discuss the first law of thermodynamics.

Heat is not a conserved quantity. Work is not a conserved quantity, but the sum of heat and work is a conserved quantity. The first law is related to the law of conservation of energy; in fact it is one case of that law.

Heat

Heat is a form of energy, often represented by the variable q. Heat, however is not a state function. Its value is dependent on the path taken. It is possible to convert heat into work, and it is also possible to convert heat into work. At some point I need to write a post describing the relationship between heat and temperature, but for now, understand that they are not the same thing.

Work

Work is also a form of energy. It is often represented by the variable w. Work can be defined as the integral of force over distance. Note that force has units of Newtons, distance has units of meters, and that the product is a Joule, a unit of energy.

One type of work that is often of interest in thermodynamics is called PV-work. PV-work is the integral of pressure over volume. It is negative because the convention is that work done on a system is positive. Sometimes engineers use the opposite convention.

Note that pressure has units of Pascals (Pa). A Pascal is equal to one Newton per meter squared. Volume has units of meters cubed. Pressure times volume has units of Newtons times meters, otherwise known as Joules.

Work is not a conserved quantity. It is not a state function. Its value is dependent on the path taken. Work can be converted into heat, and heat can be converted into work.

Internal Energy

The sum of heat and work is called the internal energy. The internal energy is often represented by the variable U; sometimes it is also represented by the variable E.

ΔU = w + q

The internal energy is a state function. It is a conserved quantity. It can neither be created nor destroyed. The internal energy is independent of the path taken.

Work can be converted to heat, and heat can be converted to work, but the sum of heat and work is a conserved quantity. This statement is the first law of thermodynamics. In shorthand the first law is sometimes stated as "You can't win."

For a continuing cyclic process, you cannot get more energy out than you put in.

Conservation of Energy

The first law of thermodynamics is a special case of the law of conservation of energy. Conservation of energy itself is a consequence of Noether's theorem. As long as a physical system is symmetric with respect to time, energy must be conserved.

Noether's theorem itself is more general, and the various conservation laws can be derived from Noether's theorem.

More on Internal Energy

If the only work done, PV work, then:

ΔU = q - PΔV

Notice that under conditions of constant volume:

ΔU = q

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donald A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
UC Davis Chemwiki: http://chemwiki.ucdavis.edu/Physical_Chemistry/Thermodynamics/State_Functions
Wikipedia: Noether's theorem.

Entropy as a Religious, Spiritual or Self-Help Metaphor

2011-04-01T12:40:00.021-06:00

This post is part of a series,Nonsense and the Second Law of Thermodynamics. The previous post is entitled The Second Law and Creationism.

While browsing around the Internet for misuses of the term "entropy," I found some examples of the use of entropy as a metaphor. For example:

In human development and performance, entropy is somehow equated with limitations. It should be noted that if we go on to accept that people have limitations and a capacity for sin, then the natural pattern of human performance is not towards excellence but mediocrity. I say this because there are challenges, adversities, and even suffering, which are essential for healthy growth, although we don't normally seek or invite them. Overcoming these challenges, help us to see limitations as mere imaginations. Since entropy is very difficult to keep at bay, why must we continue to struggle against it in life? (Source)

I think that this paragraph has to be read as a somewhat confusing metaphor. It is not easy to characterize as a correct or incorrect understanding of entropy, but I suggest it is a bad metaphor. Entropy is not something that we can struggle against in the long term.

We do struggle against it in a sense when we build more efficient gasoline engines. We try to be as efficient as we can, the best we can be. The second law tell us that the best we can be involves dissipating heat that cannot be used for useful work. We can build more efficient engines, but ultimately, the efficiency that can be achieved is a place of total limitation.

Trying to frame our human struggle as war against entropy strikes me as a bad idea. It is framing the discussion in terms of a war we can never win. Better realistically to assert that creating some entropy in the universe is ok. We are here to use some of the energy available to us and transform it into less useful forms of energy.

If our goal were to fight entropy, then we should not do anything interesting. If you clean your room, you increase the entropy of the universe.

If you do not want to contribute to the entropy of the universe, do not clean your room. (Caveat: in the popular imagination entropy is incorrectly equated with disorder. See Entropy is Not a Measure of Disorder. Whether a messy room has more entropy than a clean one is debatable - it probably does, but only by a minuscule amount. What is not debatable is the fact that any entropy decrease in the room as a result of housekeeping increases the entropy of the room and its surroundings taken as a whole.)

Physics cannot tell us how we should live our lives, but if I wanted to use the second law as an inspiring metaphor, I would start with the idea that it means we are privileged to be here. The metaphorical lesson is that I am here instead of someone else; the useful work that I use for my benefit could have been used for someone else's benefit. Let us use it to do truly useful work, to love, and to make the lives of people around us a little bit better.

Even this metaphor is a bit weak, however. Realistically, there is so much energy in the universe (from the sun, from others stars, from nuclear energy) that there is plenty of energy left over for others to use, if only we knew how to harvest it safely. See, however, Isaac Asimov's short story The Last Question, for a fun view of this question.

The next post is entitled Free Energy.

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Anderson, H.C., Stanford University, Lectures on Statistical Thermodynamics, ca. 1990.
Laird, Brian B. Entropy, Disorder, and Freezing
Vitus Ejiogu, Entropy - A Place of Total Limitation
Asimov, Isaac, The Last Question, Multivax Website. Thanks to Ruth Shear for pointing this story out to me.

Contents

Introduction
What the Second Law Does Not Say
What the Second Law Does Say
Entropy is Not a Measure of Disorder
Reversible Processes
The Carnot Cycle
The Definition of Entropy
Perpetual Motion
The Hydrogen Economy
Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
The Second Law and Swamp Coolers
Entropy and Statistical Thermodynamics
Fluctuations
Partition Functions
Entropy and Information Theory
The Second Law and Creationism
Entropy as Religious, Spiritual, or Self-Help Metaphor
Free Energy
Spontaneous Change and Equilibrium
The Second Law, Radiative Transfer, and Global Warming
The Second Law, Microscopic Reversibility, and Small Systems
The Arrow of Time
The Heat Death of the Universe
Gravity and Entropy
The Second Law and Nietzsche's Eternal Recurrence
Conclusion

Converting Units of Temperature

2011-03-27T13:05:00.000-06:00

In the course of writing articles for this blog, on occasion I write a post on the conversion of units. Units can be confusing for some, and if I can add some clarity, I think it is worth doing. This post addresses conversion between units of temperature.

The most common temperature conversion that one wants to do is to convert degrees Celsius to degrees Fahrenheit or vice versa. I hope to explain how to do this conversion in such a way that one need not memorize a formula, even to get the exact answer.

First, it is worthwhile to mention a quick and dirty approximation that almost anyone can do in his or her head. It is not exact, but it is useful, if you are traveling to a country that uses a scale that is different than the one you use to think about ambient temperatures.

Approximate Conversion Between Celsius and Fahrenheit

The approximate method of conversion is very simple. To convert from degrees Celsius to degrees Fahrenheit, take the temperature in Celsius, double it, and add 30.

For example, suppose you live in the US but are visiting France. Someone tells you that the temperature outside is 20 degrees, and you want to know what the temperature is in Fahrenheit. Doubling twenty yields 40 and adding 30 yields 70.

The exact answer is 68; so you are off by 2 degrees Fahrenheit or about 1 degree Celsius.

The reverse calculation can be carried out as well. Suppose someone gives you a temperature of 50 degrees Fahrenheit and you want to know the temperature in Celsius. Subtract 30 and divide by 2. 20 divided by 2 is 10.

The exact answer in this case is also 10. This approximation is only exactly correct at 10 degrees Celsius or 50 degrees Fahrenheit. As the temperature gets colder or hotter, the approximation gets worse.

Between 5 and 95 degrees Fahrenheit, the error does not exceed 5 degrees Fahrenheit. Between 32 degrees Fahrenheit and 68 degrees Fahrenheit, the error does not exceed 2 degrees Fahrenheit.

The approximation works pretty well for most ambient temperatures. One may desire to be more exact, however, or one may need to calculate a temperature that is outside the range where the approximation works well. In either case, it is not difficult to calculate the answer as precisely as one desires.

Exact Conversion Between Celsius and Fahrenheit

First a caveat: the exact conversion is only exact if one does not round the answer one gets from the mathematical operations. In some cases that math works out to provide an exact conversion. For example 32 degrees Fahrenheit is exactly 0 degrees Celsius.

In other cases, one will round the result to the appropriate number of decimal places. If one wishes to keep uncertainty to a minimum, one should perform the calculation with enough accuracy that the conversion itself is not the source of uncertainty in the temperature measurement. Rather, the uncertainty in the original measurement should be the source of uncertainty.

Some people like to memorize formulas and that is fine, but others find it easier to reason their way to an answer. To convert between the Fahrenheit and Celsius scales, there is no need to memorize the formula.

It is sufficient to remember simple facts that you probably already know. Water freezes at 0 degrees Celsius, which is 32 degrees Fahrenheit. Water boils at 100 degrees Celsius, which is 212 degrees Fahrenheit.

These numbers are the only number you need to remember.

0 degrees Celsius is 32 degrees Fahrenheit. This fact gives us an offset between the two scales of 32. consider that fact and then set it aside.

Between 0 and 100 degrees Celsius, how many Fahrenheit degrees are there? Well, we know that the Fahrenheit scale goes from 32 to 212 in that same span. 212 - 32 = 180.

So for every difference of 100 degrees Celsius, there is a difference of 180 degrees Fahrenheit.

Suppose we want to convert 20 degrees Celsius to Fahrenheit. How far does the answer differ from 0 degrees Celsius or 32 degrees Fahrenheit?

In Celsius, the answer differs by 20 degrees. Every difference of 100 degrees Celsius is a difference of 180 degrees Fahrenheit. 20 degrees is one-fifth of 100 degrees. Our answer must differ from the freezing point by one fifth of 180. One fifth of 180 is 36. The freezing point in Fahrenheit is 32. So the answer must be 68 Fahrenheit.

We did not make any approximations in our math; so this answer is exact, assuming our input was exactly 20 degrees Celsius.

This method effectively derives the formula for converting between the Celsius and Fahrenheit scales. 180 divided by 100 is equal to 9/5.

To convert from Celsius to Fahrenheit, one multiplies by 9/5 and adds 32.

9/5 multiplied by 20 is 36, and 36 + 32 is 68.

If one wishes to convert from Fahrenheit to Celsius, simply reverse the logic.

To convert from Fahrenheit to Celsius, subtract 32 and multiply by 5/9.

Note again that there is no need to memorize these numbers if you know the temperatures at which water freezes and boils.

The Kelvin Scale

The third law of thermodynamics provides an absolute zero for temperature, and tells us that it is impossible to get there in a finite number of steps. The Kelvin scale is defined so that 0 kelvin (K) is absolute zero.

Note that in the Kelvin scale, one does not refer to "degrees Kelvin." Because the scale is an absolute scale it is recognized by making the unit of temperature be the kelvin (K). Unlike degrees Fahrenheit (ºF), there is no degree (º) sign. The unit is the kelvin (K).

Zero degrees Celsius is defined to be exactly 273.15 K. Conversion between the Kelvin and Celsius scales is easy.

To convert from kelvin to degrees Celsius, simply subtract 273.15.

To convert from degrees Celsius to kelvin, add 273.15. So, 25.00 ºC = 298.15 K.

The conversion is exact; so one may maintain as many significant figures as are available in the original measurement.

300.19478 K = 27.04478 ºC. One may keep all of the significant figures.

If one wishes to convert from the Kelvin to the Fahrenheit scale, the easiest method to remember is to convert from Kelvin to Celsius first and then to Fahrenheit. The reverse is also true. To convert from the Fahrenheit scale to the Kelvin scale, first convert to Celsius, and then convert from Celsius to Kelvin.

The Rankine Scale

Just as the Kelvin scale is an absolute temperature scale in which a difference of one kelvin is a difference of one degree Celsius, there is an absolute temperature scale that is offset from the Fahrenheit scale such that one degree Fahrenheit is one unit on that scale.

The Rankine scale is offset from the Fahrenheit scale by (9/5)*273.15 - 32 = 459.67.

Zero degrees Fahrenheit is defined to be exactly 459.67 degrees Rankine (ºR). Inconsistent with the designation of the kelvin (K) as a unit, the Rankine scale uses degrees Rankine (ºR).

To convert from degrees Fahrenheit to degrees Rankine, add 459.67.

To convert from degrees Rankine to degrees Fahrenheit, subtract 459.67.

Again, the conversion is exact.

Sources

NIST: International System of Units

The Second Law and Creationism

2011-03-25T13:36:00.015-06:00

This post is part of a series,Nonsense and the Second Law of Thermodynamics. The previous post is entitled Entropy and Information Theory.

Creationists and other people confused about the second law of thermodynamics often bring up some variant of the idea that life is somehow a counter-example the second law of thermodynamics. If entropy were disorder (which it is not), is it not obvious that life is highly ordered?

Therefore they seem to conclude that life is an example of decreasing entropy. The flaw in their thinking is not so much the confusion between disorder and entropy; the argument could be made with an accurate description of entropy. It would still be incorrect.

An Open System

The second law does not state that a system's entropy cannot decrease. It says that a closed system's entropy cannot decrease. Life, as we know it, is not a closed system. Life exists in an environment with energy streaming in and out from the surroundings.

The most obvious problem is the sun. The sun is streaming energy to the earth. What we earthlings do with that energy is negligible compared to what happens on the sun.

Even if we could somehow exist without the sun, our energy has to come from somewhere. When we eat we get energy, but we must emit waste products. The energy that we living things consume is not all used for useful work; much of it is given off as heat.

The second law regulates the upper limit on useful work that can be done with the amount of energy used.

Living things create entropy every time they take action. Any local decrease in entropy must be paid for by an increase in the entropy of the environment. In fact, it must be paid with interest.

Creationism and Entropy

A primary example of a Creationist using such a flawed argument is Henry M. Morris, Ph.D. in his article: The Scientific Case Against Evolution:

The main scientific reason why there is no evidence for evolution in either the present or the past (except in the creative imagination of evolutionary scientists) is because one of the most fundamental laws of nature precludes it. The law of increasing entropy -- also known as the second law of thermodynamics -- stipulates that all systems in the real world tend to go "downhill," as it were, toward disorganization and decreased complexity.

Note that the author in this case tries to argue with the obvious flaws in his argument:

This naive response to the entropy law is typical of evolutionary dissimulation. While it is true that local order can increase in an open system if certain conditions are met, the fact is that evolution does not meet those conditions. Simply saying that the earth is open to the energy from the sun says nothing about how that raw solar heat is converted into increased complexity in any system, open or closed.

The fact is that the best known and most fundamental equation of thermodynamics says that the influx of heat into an open system will increase the entropy of that system, not decrease it. All known cases of decreased entropy (or increased organization) in open systems involve a guiding program of some sort and one or more energy conversion mechanisms.

The Missing Link

The argument here is a beautiful piece sophistry. To prove that evolution does not violate the second law of thermodynamics, those of us who believe in Evolution must now prove every link in the chain.

We must show how plants convert sunlight into stored energy in the form of reduced forms of carbon. We must show that this process is not 100% efficient, but that some of that some of the energy must be discarded as heat. We must trace the use of this reduced carbon through the food chain and show that each time it is oxidized to do useful work that some energy must be dissipated as heat.

All of the processes by which life converts energy to do something useful follow the second law of thermodynamics. The second law is not some vague pronouncement about order and disorder. It is a precise thermodynamic quantity that can be measured.

Morris's argument is beautiful because it moves the missing link paradox to the realm of thermodynamics. The missing-link paradox is that Creationists claim that intermediate forms between other primates and humans have never been found.

When somebody find a species N that is intermediate between species B and species W, the Creationist can respond that no one has found a species intermediate between species B and species N. The Creationist can play this game forever, no matter how many intermediate species are found, even if we exhaust the letters of the alphabet.

Source: http://en.wikipedia.org/wiki/File:Archaeopteryx_lithographica_paris.JPG

Anyone who takes up Morris's argument must now do the same thing for each process in which a living organism converts energy to useful work. In other words, it is necessary to understand everything about every possible biological process.

Until we understand everything, Morris can gleefully state that Evolution is "unproved." He does not need to compare the explanatory power of Evolution to his alternative hypothesis.

Once life exists, the mechanisms of evolution and life involve genetics, genetic mutation. and natural selection. These processes are, of course, constrained by thermodynamics. The process of reproduction requires energy, not all of that energy can be used for useful work. Living things must produce waste products and heat to reproduce.

The Origin of Life

The question of the origin of life is conceptually different from the question of evolution, but it is still an important subject for thought, and it is a related question.

The origination of the first living cell is a conceptually difficult topic. We do not know precisely how life came to be despite decades of research from the days of A. I. Oparin. The event could have been a singular event, i.e., all life evolved from a single occurrence of a cell becoming alive. Alternatively, multiple events could have occurred under some optimal condition.

It is not unreasonable to suppose that whatever happened was statistically unlikely.

Such an even is statistically unlikely enough that it has never been observed directly. It is possible that under certain conditions it may be more probable. It is possible that this event was singular that it only happened once, but we do not know that.

That statistical unlikelihood does not make it a violation of the second law of thermodynamics!

The nascent cell, like all other thermodynamic processes, must have required energy. Some of that energy was used to do useful work, but not all of it. The entropy of the system, i.e., the nascent cell, and the surroundings, i.e., everything else increased according to the second law.

There is no thermodynamic problem for the origin of life. Do not confuse statistical arguments for the unlikelihood of life's origin with statistical thermodynamic arguments about entropy. The occurrence of a statistically unlikely event is not inconsistent with the second law of thermodynamics.

That is not to say that we know everything there is to know about how life came to be. Not knowing how something happened does not make it a violation of the second law. The next post is entitled Entropy as a Religious, Spiritual or Self-Help Metaphor.

Sources

Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
McQuarrie, Donal d A., Statistical Thermodynamics, University Science Books, Mill Valley, CA, 1973
Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
Anderson, H.C., Stanford University, Lectures on Statistical Thermodynamics, ca. 1990.
Henry M. Morris, The Scientific Case Against Evolution
A. I. Oparin, The Origin of Life
Wikipedia: Arcaeopteryx Lithographica Paris

Contents

Introduction
What the Second Law Does Not Say
What the Second Law Does Say
Entropy is Not a Measure of Disorder
Reversible Processes
The Carnot Cycle
The Definition of Entropy
Perpetual Motion
The Hydrogen Economy
Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
The Second Law and Swamp Coolers
Entropy and Statistical Thermodynamics
Fluctuations
Partition Functions
Entropy and Information Theory
The Second Law and Creationism
Entropy as Religious, Spiritual, or Self-Help Metaphor
Free Energy
Spontaneous Change and Equilibrium
The Second Law, Radiative Transfer, and Global Warming
The Second Law, Microscopic Reversibility, and Small Systems
The Arrow of Time
The Heat Death of the Universe
Gravity and Entropy
The Second Law and Nietzsche's Eternal Recurrence
Conclusion

Entropy and Information Theory

2011-03-04T14:48:00.011-07:00

This post is part of a series,Nonsense and the Second Law of Thermodynamics The previous post is entitled Partition Functions.

The posts in this series are primarily about the second law of thermodynamics, the concept of entropy, and the use and abuse of these ideas. I would be remiss, however, not to mention information theory and the role that entropy plays. This post is not intended to be a comprehensive introduction to information theory. Readers especially interested in this topic will want to read other sources in addition to this post.

Information Theory

Information theory is concerned with modeling communication systems. These communication systems could be understood, for example, to be radio frequency communication, but the results of the theory are more general than a specific system of encoding information and some of the consequences of the theory can be broad in scope and significance.

Ash states:

It is possible to make a case for the statement that information theory is essentially the study of one theorem, ... which states that "it is possible to transmit information through a noisy channel at any rate less than channel capacity with an arbitrarily small probability of error."

Information and Uncertainty

Information can be understood as the reduction of uncertainty. If I say that I have a message, I have not provided much information about the content of the message.

On the other hand, if I say that the the message contains twelve words in the English language, I have reduced the uncertainty of the message content and provided information. If I say the first word is "the," I have further reduced the uncertainty, again conveying information.

Bits

It is common to reduce information to a simple form: from ones and zeros. A bit can be either one or zero, and by looking at a fundamental bit of information, it is possible to understand the mathematical laws that govern information.

As human beings, we may rebel against the notion that information can be conveyed as ones and zeros. Computers are based on binary systems and we think of ourselves as being complex beings whose messages cannot be reduced to ones and zeros.

Consider, however, that the character set that I am using to write this blog is, in fact, coded in ones and zeros. All of the great poetry of the world, such as the works of Shakespeare, Whitman, Homer, Sophocles, and Baudelaire can be encoded in ones and zeros. If the ones and zeros are not transmitted correctly, some of the information may be lost.

-P log P

Suppose we want to convey one of sixteen possible messages that are equally probable. It turns out that this can be done with 4 bits of data. Each bit can have two possible values and 2 x 2 x 2 x 2 = 16.

The probability of any message is 1/16, and it requires -log(1/16) = 4 bits of information to convey, if the log is taken to be log base two for convenience.

Here we have assumed that that messages are equally probably, but if they are not, they have to be weighted by their probability.

The uncertainty of the information is the sum over all messages of the quantity - P log P. This uncertainty is also called the entropy by analogy with thermodynamics.

S = - ΣP log P

Thermodynamic Entropy

Recall that an expression for the thermodynamic entropy is:

S = -k ΣP*ln(P)

If we define the redefine the thermodynamic entropy as S/k, recalling that k is merely a constant we can make the equations the same and eliminate the units. The difference between the log used is inconsequential as we can multiply by a constant and redefine the entropy appropriately.

One should be careful in taking the analogy too far. Remember that in thermodynamics that the numbers involved are extremely large. In most cases, information theory does not involve such large numbers.

Information theory can become quite complicated, and this discussion is only intended to be a cursory mention of the topic, not a definitive comparison.

The next post discusses the Second Law and Creationism.

Sources

Ash, Robert B., Information Theory, Dover Publications Inc., New York, 1965
Touretzky, David S. Basics of Information Theory, http://http://www.cs.cmu.edu/~dst/Tutorials/Info-Theory

Contents

Introduction
What the Second Law Does Not Say
What the Second Law Does Say
Entropy is Not a Measure of Disorder
Reversible Processes
The Carnot Cycle
The Definition of Entropy
Perpetual Motion
The Hydrogen Economy
Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
The Second Law and Swamp Coolers
Entropy and Statistical Thermodynamics
Fluctuations
Partition Functions
Entropy and Information Theory
The Second Law and Creationism
Entropy as Religious, Spiritual, or Self-Help Metaphor
Free Energy
Spontaneous Change and Equilibrium
The Second Law, Radiative Transfer, and Global Warming
The Second Law, Microscopic Reversibility, and Small Systems
The Arrow of Time
The Heat Death of the Universe
Gravity and Entropy
The Second Law and Nietzsche's Eternal Recurrence
Conclusion

Temperature Anomalies and Graphing Data

2011-01-14T15:27:00.008-07:00

Globally Averaged Temperature Anomaly

One statistic that is used to understand climate is the annual globally averaged temperature anomaly. It is not the only measure of global warming; there are a great many others, but it is one that the media tend to focus on because it is a convenient way of explaining what is happening to surface temperatures as a function of time.

The data here are taken from Global Land-Ocean Temperature Index (C) (Anomaly with Base: 1951-1980), which includes data from 1882-2007. There are updated numbers available for more recent years, but I am using these data to respond to an argument made by a friend.

The data come from the National Aeronautics and Space Administration's (NASA's) Goddard Institute of Space Studies (GISS). The base period for calculating the anomaly used by GISS is 1951-1980. This choice is arbitrary.

The Hadley Centre, for example, prefer to use the base period 1961-1990. The data are presented relative to the average of the base period. That means that a temperature anomaly of 0.2 is 0.2 K hotter than the average temperature for the base period.

Polemics or Science?

A friend of mine (who shall remain anonymous unless he chooses to out himself) recently posted a plot on Facebook. He plotted the temperature in the Kelvin scale vs. year from the same source I used above.

The kelvin (K) is a unit a thermodynamic temperature scale, the Kelvin scale. The third law of thermodynamics states the it is impossible to reach absolute zero in a finite number of steps, or alternatively that the entropy of a system at absolute zero is zero.

This law provides a natural zero-point for temperature. The Kelvin scale is simply the Celsius scale adjusted so that zero is absolute zero (-273.15 ºC). Because it is an absolute scale, the unit is referred to as the kelvin, rather than degrees Kelvin.

If one were to assume that the mean temperature of the earth from 1951 to 1980 were 287 K, one would obtain the results my friend plotted.

I found three discrepancies in numbers that he provided me. It is not really pertinent to the argument, but I include my changes for completeness. For 1944, he had 287.07 K instead of 287.19 K; for 1973, he had 286.86 K instead of 287.14 K, and for 1977, he had 286.88 K instead of 287.12 K. All other values agreed with my calculation. I used my calculated values throughout.

The argument seems to be that on a Kelvin scale, the warming trend does not look very dramatic. My friend implies that the usage of temperature anomaly is based upon polemics rather than science, and if only the data were plotted in kelvin, it would be clear that a trend of 0.16 K/decade is not something to worry about.

Before addressing this argument directly, I would like to use an analogy.

An Analogy: The Dow Jones Industrial Average

Suppose that I want to track how the the Dow Jones Industrial Average (DJIA) changes over a period of ten calendar days from Jan. 3, 2011 to Jan. 13, 2011. Note that there are only data points for days when the stock market is open, but that should not hamper the analogy.

One way that we could track this data would be to plot the gains or losses relative to that first day as a function of date.

Note that I have scaled this graph using the data limits so that the reader can gain information from this graph. Of course this graph does not inform us of the actual value of the DJIA on a given day; so it might be nice to plot the value as a function of date.

Note again that I chose to plot the data using the data limits. Despite the fact that the DJIA is in the 11,000 range, I have preserved the information in the graph by using the data limits. If for some reason I wanted to obscure that information, I could plot it differently.

Note that these are the same data that I plotted above, but that now I have obscured the information. The only information that this graph gives us is that the DJIA remained between 11,000 and 12,000 during the time period in question.

There is no reason to present this information in a graph. I am obscuring any trend that might be in the data.

Note that it makes no difference whatsoever if I choose to use the price or the gains/losses. It is the data range that obscures the information, not the choice of units. I could do the same thing with gains and losses.

This plot obscures the information as well. About the only information to be gained is that the DJIA did not fluctuate more than +/- 100 points during the week in question. If I had used the data limits, I could surmise that the DJIA fluctuated from losses of about 32 points to gains of about 84 points relative to the first day of the period in question.

Nowhere have I changed the data. It is all the same data plotted differently. If I am going to bother to plot data, I should use a scale that preserves the maximum amount of information. It does not matter what units I use. In fact, I can plot the data all on the same graph.

Here I have plotted gains and losses against the left axis an the price against the right axis. Note that the data are superimposed upon each other because both are plotted on a scale related to the data limits.

Examples

Note major news sources plot index prices plots it in a manner related to the data limits.

There is no reason to plot the information if one does not preserve the information by using appropriate data limits.

Thermodynamic Temperature vs Temperature Anomaly

Of course the difference between my friend's plot and the plot of GISS temperature anomalies has nothing to do with the fact that my friend chose to report data as an absolute temperature. Here is the same data plotted as absolute temperature in Kelvin, but with appropriate data limits.

I can also plot the temperature anomaly in a way that obscures the data.

It is absurd to plot the information this way as the plot does not display the information content of the data, but it is possible to do. Note whether one chooses an appropriate data range or not has nothing to do with the units chosen to display the data.

I can display the data in Celsius.

I can display the data in Fahrenheit.

Heck, I can display the data in Rankine.

It does not matter what units I choose; it is the same data. With any units I can choose to plot it according to the data limits, or I can obscure the information content by plotting it on a larger scale. This fact is true for any data set whatsoever!

I can plot the absolute temperature and the temperature anomaly on the same scale.

The plots lie on top of each other if plotted using the data limits. It's the same data; so it should not be a surprise.

I can plot, temperature in Kelvin, Celsius, Fahrenheit, and Rankine along with the temperature anomaly in Kelvin all on the same plot.

So Why Use Temperature Anomaly?

There is no polemical reason to use temperature anomaly. If I were only interested in polemics, the effect is just as obvious using the absolute temperature scale with proper data limits.

If that is the case, why bother to use anomaly at all. Perhaps, one should consider the arguments of the scientists who make such a choice. GISS explain as follows:

Our analysis concerns only temperature anomalies, not absolute temperature. Temperature anomalies are computed relative to the base period 1951-1980. The reason to work with anomalies, rather than absolute temperature is that absolute temperature varies markedly in short distances, while monthly or annual temperature anomalies are representative of a much larger region. Indeed, we have shown (Hansen and Lebedeff, 1987) that temperature anomalies are strongly correlated out to distances of the order of 1000 km. For a more detailed discussion, see The Elusive Absolute Surface Air Temperature.

Consider the complexity of the problem of assigning a surface air temperature to the earth. There is a well-known proverbs that goes:

A man who has a thermometer knows what the temperature is. A man who has two thermometers isn't sure.

Temperature varies locally and globally, not only by latitude and longitude, but also by altitude. There is no agreement about how to measure the surface air temperature (SAT) for a given location, but there is general agreement about how to measure an anomaly. From The Elusive Absolute Surface Air Temperature:

Q. What exactly do we mean by SAT ?

A. I doubt that there is a general agreement how to answer this question. Even at the same location, the temperature near the ground may be very different from the temperature 5 ft above the ground and different again from 10 ft or 50 ft above the ground. Particularly in the presence of vegetation (say in a rain forest), the temperature above the vegetation may be very different from the temperature below the top of the vegetation. A reasonable suggestion might be to use the average temperature of the first 50 ft of air either above ground or above the top of the vegetation. To measure SAT we have to agree on what it is and, as far as I know, no such standard has been suggested or generally adopted. Even if the 50 ft standard were adopted, I cannot imagine that a weather station would build a 50 ft stack of thermometers to be able to find the true SAT at its location.

Over short distances, weather stations may measure very different temperatures for a given day, but temperature anomalies often agree for much larger distances.

The Hadley Centre does provide one way to calculate surface air temperatures from their data, but the calculation comes with a lot of caveats.

Q. How are the daily mean temperatures calculated? And in case of there being several ways, how can you be sure that those ways are equivalent?

A. Because we use temperature anomalies from a station climatology, it doesn't matter how the average temperature is calculated as long as it is always done in the same way (the differences will cancel in the climatology and the monthly values). For UK data we still use the average of the Max and Min temperatures. This gives us homogenous long-term series from a station. Some other countries calculate the average in other ways. So long as they don't change the method of calculation, the results will be consistent. If the calculation method is changed we apply corrections to the reported values. In some instances it is possible that the method was changed, but no record was made. The uncertainties associated with such inhomogeneities are discussed in Brohan et al. 2006.

Q. Why do you use anomalies?

A. Anomalies vary slowly from one place to another - if it is warmer than average in London, it is likely to be warmer than average in Paris too - but actual temperatures can vary greatly from one weather station to its nearest neighbour. The average anomaly for, say, Europe is likely to be representative of a large area, the average absolute temperature will be representative of only a very limited one.

Q. How do you obtain a global annual average temperature from the monthly data?

A. First the monthly anomalies in each grid box are averaged together to give an annual average anomaly for that grid box. The area-weighted averages of these annual average grid-box anomalies are then calculated for the northern hemisphere and for the southern hemisphere. The global average temperature is the arithmetic mean of the northern hemisphere average and the southern hemisphere average. The last step avoids biasing the global average to the more densely observed northern hemisphere. There are, of course, other ways to calculate the global average and each will give a slightly different answer.

Q. The HadCRUT3 data are expressed as anomalies, but I want actual temperatures.

A. HadCRUT3 is an anomalies dataset, and all the uncertainties apply to the anomalies. If you are interested in year-to-year changes it's best to use the anomalies if you can. So before you start using the actuals, think hard to check you can't use the anomalies instead. We can make actuals - we merge the SST climatology from the HadSST2 dataset and the land climatology from CRU high resolution dataset (New et al 2002 - see http://www.cru.uea.ac.uk/cru/data/hrg.htm), to make a climatology for HadCRUT3 and add this to the anomalies dataset. But this has two problems: it adds an additional source of uncertainties which we don't allow for, so our uncertainty analysis is no longer valid; also land surface actuals vary over short distances because of large changes in altitude: so the actual range of temperatures in a 5 degree grid box can be large, and the mean value is not always useful. The absolute global-average annual temperature and the absolute hemisphere-average annual temperatures for 1961-1990 were calculate by Jones et al. (1999). They are: Globe 61-90 average = 14.0°C Northern Hemisphere 61-90 average = 14.6°C Southern Hemisphere 61-90 average = 13.4°C.

It is possible to use such conventions and then to present a mean temperature as a function of time. If the data were presented that way, those who wished to attack the data would find other reasons for doing so. They'd make the very same arguments that climate scientists make for why surface air temperatures are elusive.

To argue that data are presented as anomalies for polemical purposes is absurd. There is no polemical benefit to doing so as I have shown.

Ultimately, as long as one is consistent with conventions and careful on applying uncertainties, one could establish and use an SAT convention and report temperature changes over time. There may in fact be polemical advantages to such an approach, but there are some drawbacks in making the data generally usable.

Climate scientists have established a convention of using temperature anomalies for scientific purposes. I cannot see any scientific objections to this convention vs. any other convention. If someone has a better convention, that person should propose it as a standard; maybe it will be accepted, but that standard will not change the data, nor will it change the way that the data look on a plot.

Sources

Anonymous Friend, Facebook Communication
Global Land-Ocean Temperature Index (C) (Anomaly with Base: 1951-1980)
The Claim of a Warming Pause
Hadley Institute Surface Temperature Anomalies
Anonymous Friend, Facebook communication
Yahoo: DJI Historical Prices
GISS Surface Temperature Analysis (GISTEMP)
The Elusive Absolute Surface Air Temperature
Structure of the Atmosphere
Met Office Hadley Centre FAQ