Thursday, December 29, 2011

Liquids and Gases and Van der Waals

Johannes Diderik van der Waals
It seems strange to be including a Nobel Laureate in this blog that is supposed to be about lesser known scientists, but I decided to include Van der Waals because I, for one, didn't even realize that he had won a Nobel Prize for his work until I started writing this. Van der Waals won the 1910 Nobel Prize in Physics "for his work on the equation of state for gases and liquids." His name generally appears in general chemistry courses in two places related to those very topics: Van der Waals forces, the intermolecular forces that hold all molecules together, and the Van der Waals equation, which is a correction of the Ideal Gas Law which accounts for the size of and the intermolecular interactions between gas molecules.

Van der Waals was born in Leyden, the Netherlands, in 1837 and became a schoolteacher. He did not know Latin or Greek, and thus was prohibited from taking academic examinations and going to a university. How times have changed! He could still attended classes at Leyden University, however, and he earned teaching certificates in math and physics.  When the laws requiring classical languages were changed, Van der Waals sat for examinations and in 1873 earned his doctorate at the age of 36 with a thesis entitled "Over de Continuiteit van den Gas en Vloeistoftoestand" (On the Continuity of the Gas and Liquid State). Four years later, he became the first professor of physics at the University of Amsterdam.

Here, I think, it is important to realize what Van der Waals contribution actually was and what the other discoveries of his day were.  Rudolf Clausius (1822-1888) had suggested that heat is a measure of motion only in 1850, and published his first work on entropy in 1865.  His work also relied on the Maxwell-Boltzmann distribution, which was developed in the mid-19th century by James Clerk Maxwell (1831-1879) and Ludwig Boltzmann (1844-1906). Van der Waals tried to explain the phenomenon of a "critical temperature" for gases, and determined that it was due to the fact that atoms and molecules have a finite size and interact with each other, both facts that are ignored in the ideal gas law.  After this, he developed the Law of Corresponding States, which provided the theoretical background for the experiments leading to the liquefaction of hydrogen and helium in 1898 and 1908 respectively.

Even with his later work, much of what Van der Waals is known for was in or came directly from his doctoral thesis. So to all of you graduate students out there, you may be working on Nobel Prize winning material!

If you are interested in a better understanding of Van der Waals work, his Nobel Prize acceptance speech is a good summary of his work and also of his interactions with other scientists: Van der Waals Nobel Lecture.

For more information about Van der Waals, I would highly recommend the resources provided on the Nobel Prize website, which has been the main source of information for this post:  J. D. van der Waals.

Tuesday, December 20, 2011

The Poisson Distribution

The Poisson distribution began life with unusual applications.  It is a statistical function that is used to determine how many events of low probability will occur in a given time frame.  It is useful because only the average number of occurrences needs to be known.  From this, one can calculate the probability that the event will happen zero times in a given time interval, or fifty times.  λ is the expected number of occurrences in a time frame, and the probability that there will be exactly k occurrences of the phenomenon in a given time frame is given by the function on the right.

Poisson Distribution for various values of λ
It's discoverer, Siméon-Denis Poisson (1781-1840), was a well-known French mathematician who worked in electrostatics and celestial mechanics.  He even got his name on the Eiffel Tower.  He had studied mathematics at the École Polytechnique in Paris starting in 1798 and became a professor there in 1802.  He had studied with the mathematicians Pierre-Simon Laplace (known for the Laplacian) and Joseph-Louis Lagrange (of Lagrange multipliers), and was a professor at the same time as André-Marie Ampère, known for his work in electromagnetism. His treatise introducing the Poisson distribution, however, bears the particularly unscientific title of Research on the Probability of Criminal and Civil Verdicts.1 It was published in 1837, just three years before Poisson's death, and did not have a great impact at the time.

His distribution was brought to greater prominence by Ladislaus Josephovich Bortkiewicz (1868-1931), a Polish statistician.  In 1898 he published a book entitled, in English translation, The Law of Small Numbers, in which he showed that the Poisson distribution held for events even when the probabilities varied.  His examples were the number of men killed by horse kicks in the Prussian army over a 20 year period and the number of children who committed suicide in Prussia.  His work brought the Poisson distribution to a wider audience, and it is often called the Law of Small Numbers today.  As you can see, the Poisson distribution can be applied to many different situations, and some modern applications include the number of calls that a cell tower receives and the number of beds that an emergency room needs to have available.

[1] For a more detailed discussion of Poisson's work, see "A Study of Poisson's Models for Jury Verdicts in Criminal and Civil Trials" by Alan E. Gelfand and Herbert Solomon in the Journal of the American Statistical Association , Vol. 68, No. 342 (Jun., 1973), pp. 271-278.

Tuesday, December 13, 2011

Introduction to SciHistory

Now that I've written my first post, I want to provide some explanation for this blog and also open up the discussion about what you, my readers, would like to see here.

This blog stems from my curiosity about the people whose names are bandied about in science and math classes around the world attached to other things, like Raoult's Law, Schrodinger's Equation, the Boltzmann Constant, and the Bohr model of the atom.  Teachers and professors rarely stop to explain who (and when) these people were, how they interrelate, and what led to the discoveries that have their names attached to them.  The history of science is not a linear progression of ideas, but something full of arguments, dead ends, discussions, and eureka moments, and these discoveries were made by real people like you and me.

I would like, in this blog, to focus on exploring the lives of the people whose names are familiar in science classes, but to whom most people couldn't even attach a first name.  This means that I will not include, at least for now, people like Newton, Archimedes, and Einstein in favor of lesser known but equally mentioned of scientists, mathematicians, and natural philosophers like Hertz, Fourier, Arrhenius and Doppler.

The biggest problem for me is how to organize SciHistory, and that is where I would like some input from you, my hopefully-to-be-faithful readers. I have had a bunch of ideas about how to start. For now, I am going randomly, but am trying to give variety in field of notoriety, type of discovery, and time period. (For instance, in picking a second person to write on, I thought of Rydberg, who has a constant, but discarded that since he, like Ångström, was Swedish and lived in the 19th century; he will have to wait.) Other ideas include

  • Picking names off the Wikipedia list of craters on the Moon 
  • Starting with units named after people and then moving on to constants and finally equations
  • Writing on whoever a professor mentions first after I finish a blog post
  • Reader suggestions
The first two still need some sort of algorithm unless I go alphabetically, and the third has the disadvantage of only working while I am taking classes.  So the fourth is by far my favorite idea, but it will depend on you, dear readers. Let the suggestions begin!

Anders Jonas Ångström and the ångström

Anders Jonas Ångström
As any good chemist knows, an ångström is equal to 1x10-10 meters and is designated by the symbol Å, complete with the little circle on top.  I’ve often wondered where the circle came from,1 but surprisingly until starting this blog had not wondered if the angstrom was named after a person, especially since it does not follow the metric system prefixes.  Anders Jonas Ångström was born in Sweden in 1814 and studied physics at the University of Upsala.  He didn't leave except for brief sojourns to further science.  He was interested in astronomical work, and studied at the Stockholm Observatory before becoming the observer at the Upsala Observatory.  The Stockholm Academy of Sciences gave him the job of analyzing the magnetic data obtained by the “Eugéne,” a ship which had travelled around the world from 1851-1853.  In 1858 he became chair of the department of physics at Upsala University (I told you he didn’t leave).

Ångström is most known, when he is thought of at all, for his work in optics which led him to be considered one of the founders of spectroscopy.  His work was primarily with things that gave off light, such as electric sparks. His greatest work was with the solar spectrum, and it is from this work that his fame as the namesake of the ångström comes.  By studying the wavelengths of light emitted by the sun using diffraction gratings, he determined in 1862 that the sun’s atmosphere contained hydrogen.  In 1868 he published a book containing a map of the entire visible solar spectrum, consisting of 1000 lines: Recherches sur le spectre solaire.  Others doing similar work used arbitrary units, but Ångström used units of ten-billionths of a meter, or at least he thought he did.  He discovered that the meter from which he measured the gratings was too short, and thus all of his calculations were off.2  He began the work to correct it, but died in 1874, before it was finished, leaving his assistant Thalén to finish the job.  A contemporary said that his “work [was] characterized by such accuracy and completeness as to render it worthy of the highest admiration, to be regarded as a pattern to all investigators.”3  Although others had attempted to make such maps, Ångström’s was the most complete and accurate, and so those who came after him used his units to describe future measurements, calling them at first Ångström units and then ångströms.

Ångströms, however, are not the best units for describing visible light, since they result in numbers in the thousands, such as 6534.  The wavelengths Ångström described are commonly now referred to in nanometers, so the values are only in the hundreds.  Chemists, however, have a great affinity for the ångström, since it is the perfect unit for describing the length of chemical bonds, which are on the order of an ångström.  So, although the angstrom is no longer used in the field in which it originated and has been relegated to the status of a “non-SI unit,” it still finds its uses in chemistry although Ångström and his work have been long forgotten.

[1] Upon further investigation, the ring is not a diacritical mark, but an integral part of the letter in Sweden.
[2] Some idea of the complication of the creation and recalculation of this map can be gathered by looking at the exercises in Robert Alexander Houstoun's  A Treatise on Light, p. 254.
[3] Heinrich Schellen, Spectrum analysis in its application to terrestrial substances, and the physical constitution of the heavenly bodies (Longmans, 1872), p. 237.  The figure is also from this book.
As a reward for reading the footnotes, here is a link to an argument that Ångström had with a fellow physicist in the Philosophical Magazine: "Observations on Certain Lines of the Solar Spectrum".  Note that the fellow who argues against Ångström, Pierre Janssen, doesn't have a unit named after him, but they both gave their names to (separate) lunar craters, and only Janssen has a crater on Mars.