Bremmer’s land

Mute swan (Cygnus olor).

Bremmer’s land (“Landje van Bremmer”) is a small pasture in Oegstgeest at the Leiden border close to where I live. It is adjacent to a park belonging to castle “Endegeest”, and next to a eco plant nursery: “De Groene Cirkel” (The Green Cirkel). So far I was able to get a decent picture of more than 20 species of bird I found in this area.

In this post I show a few of those pictures. Click on them for a larger view, or an album.

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Apparent Paradox?

We demand rigidly defined areas of doubt and uncertainty.

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In 2001 Bartell published a paper [1] in which he posed, among others, the following question:


Suppose \(N\) photons of frequency \(\nu\) are emitted by a laser aimed at a stationary black body. The black body absorbs the energy \(nh\nu\) and converts it to thermal energy (heat). Compare this with the energy \(N h\nu’\) absorbed if the black body is moving away from the source. By the Doppler effect, \(\nu’\) is less than \(\nu\) and consequently the absorber sees and absorbs photons of lower energy than emitted by the laser. Where did the extra energy go? The answer to this simple question eludes many physical chemistry professors. It does, however, yield some important results

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Gibbs’ Paradox

J. Willard Gibbs

How wonderful that we have met with a paradox. Now we have some hope of making progress.

Niels Bohr


The Gibbs Paradox is one of the topics I discussed in my Finland Lectures as an example of problems that never went away. The paradox was, around 1875, discovered by Gibbs himself, who also proposed a solution. The latest paper I know of claiming a resolution was published in 2014[1] Read more

Another Carnot Mystery

On two occasions I have been asked, “Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?” I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Charles Babbage.

I learned about the Carnot cycle and its consequences teaching thermodynamics. Conscientiously following the textbooks, mainly because as teachers we are no longer allowed to deviate from them in order not to confuse the students. After a number of years I thought I had finally figured out the use and beauty of this cycle to introduce concepts like efficiency and entropy. Almost all textbooks follow the same ritual: introduce the cycle, “prove” how all reversible cycles must have the same efficiency regardless of working substance, and calculate the efficiency using the ideal classical gas isotherms and adiabats. Then finish the exercise by showing that if you integrate heat divided by temperature over the path of the cycle you get zero, and conclude that there is a state quantity, to be called entropy. Then make some remarks about irreversibility and Clausius, and continue on to the next topic. And I like to think that after a few years I could follow that ritual fairly well. Not that I was impressed by the level of student understanding come exam time, but hell, it is thermodynamics, nobody understands that. Fortunately the powers that be, and a lack of others wanting to teach something as arcane and useless as thermodynamics, left me in a position to yearly try to increase my knowledge and understanding of the field.

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Carnot’s Reversible Engine

In bodies employed to realize the motive power of heat there should not occur any change of temperature that may not be due to a change of volume.

Sadi Carnot

Steam engines were very complicated machines by the time Carnot started thinking about their efficiency. More than a century of technological advance had already gone by. He was able to abstract the essentials, and with outdated knowledge, even in his time, derived an expression for the maximum efficiency that we still teach and use today. In the mean time he also laid the ground work for the concept of entropy and the second law of thermodynamics. It is somewhat of a mystery to me how that was possible with all the mistakes he made, and misconceptions he had.

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Debunking Jennings

S happens

Bumper sticker

In 2002 Jennings and coworkers wrote a paper [1] in which they claimed that photosynthesis is more efficient than a Carnot heat engine running between the same temperatures. In fact, their final sentence reads: “Thus, \(1-T/T_r\) represents a kind of efficiency horizon beyond which negative entropy is produced and the second law is not obeyed. As this is impossible for a heat machine, it serves to underline the difference between photosynthetic photochemistry and a heat machine.”

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