Friday, October 23, 2015

Avogadro's Number Part 1: A History of the Mole with Very Little about Avogadro

Happy Mole Day! I thought that it would be appropriate to recognize the day with a post about Avogadro's Number and the associated Mole.  Avogadro's Number, which denotes the number of objects in a mole (6.022*1023), has a certain fascination for people, particularly, in my experience, among chemistry students.  There is even a website (and foundation) dedicated to Mole Day and the mole. While I have been trying to make these posts about people, to get to the bottom of Avogadro's number, one has to dig a lot deeper than Avogadro.  He does, however, make a good starting place.

Amedeo Avogadro (1776-1856)
Lorenzo Romano Amedeo Carlo Avogadro of Quaregna and Cerreto (phew, that's long) was by training a lawyer. He became interested in math and physics around 1800 and started working with electricity and metallic salts. Despite his lack of formal training, he did end up with a post as the chair of theoretical physics at the University of Turin. One of his projects was determining the electronegativity of various elements. He is best known for, and has the mole name after him because of, his "molecular hypothesis", first suggested in an essay in 1811. He suggested that molecules in a gas are scattered such that the average distance is constant when the temperature and pressure are constant. That is, that at the same temperature and pressure, there are the same number of molecules in a set volume of one gas as of another gas. By using this hypothesis, he was able to calculate the molecular weight of gases by using their densities.This, however, put Avogadro in opposition to John Dalton (1766-1844), who had rejected this idea (see page 555 onwards of A New System of Chemistry, 1810). This was in part because Dalton and others believed that all gases contained only one atom of an element--for instance, that a molecule of water was HO rather than H2O. Since Avogadro kept to himself and tended to cite himself, his hypothesis did not gain much credibility, though André-Marie Ampère (1775-1836) came to the same conclusion in 1814. This additional support doesn't seem to have helped any, so it was a while before his hypothesis was accepted.

Stanislao Cannizzaro (1826-1910)
By the middle of the 19th century, scientists were still debating about the nature of atoms, molecules, and their divisibility, and to make matters worse, they were starting to write formulas, diagrams, and calculate masses without a standard system. They could not agree on notations and other conventions, such as how to write chemical formulas or what the standard weight for describing an atom would be. In order to deal with these problems, August Kekulé (1829-1896) organized what came to be known as the Karlsruhe Congress, which met in September 1860 in Karlsruhe, Germany. The hero of the day was Stanislao Cannizzaro, who had written an article in 1858 that was based on Avogadro's hypothesis. This paper was circulated at the conference, and in it Cannizzaro suggested that the weight of hydrogen be taken as 1.0. He also suggested that oxygen be assumed to be diatomic as a gas, such that the formula for water would be H2O and the mass is 16.0. Cannizzaro clearly stated a theory in which atoms, molecules, and multiple identical atoms in the same molecule are distinguished. His argument and explanation influenced Lothar Meyer and Dimitri Mendeleev to both accept Avogadro's hypothesis, and after the conference it gained a wider popularity.

Few scientists, however, were concerned with how many molecules were actually in that volume of gas. They were more concerned with the hypothesis itself and what that meant for being able to determine other properties of matter, such as how big atoms and molecules actually are. The next step in determining Avogadro's number takes us from Avogadro and Cannizzaro in Italy to Austria, where Johann Josef Loschmidt (1821-1895) was working.
Johann Josef Loschmidt (1821-1895)
(image from Wikimedia commons)

Loschmidt wanted to find out what the actual size of a molecule of a gas was, and used current theories of gases to determine this. Rudolph Clausius had derived in 1859 the mean free path of molecules in a gas in terms of the cross-sectional area, and James Maxwell derived his own expression the following year. Loschmidt then calculated what fraction of the gas was occupied by the molecules themselves based on the mean free path and then assumed that when the gas is liquefied, the volume is only slightly larger than that of the molecules themselves. The problem then was the air had not been liquefied, so he used the work of Hermann Kopp to estimate the density of liquid air. He determined that the size of a nitrogen molecule was 9.69*10-10 m, or about three times too big. But not bad. However, despite knowing how big molecules were, no one seemed particularly concerned about how many molecules were in a region of space. A following paper in 1865, ostensibly by Loschmidt, states that a cubic millimeter of gas contains 866 billion molecules, but that wasn't the point of the article.

Studies of gasses over the course of the rest of the century and into the 20th century would help to elucidate the question of how many molecules are in a given volume, but I will leave that for another day! Check back soon (or subscribe to the emails) for more information about the definition and calculation of Avogadro's number and where the name mole came from.

Original Papers in order of publication

Saturday, October 4, 2014

Friedrich Bessel and the Stars

Friedrich Bessel (1784-1846)
from Wikimedia Commons
This post was going to be about math, but, as usual, these scientists surprise me! The subject of this week's post is Friedrich Bessel, and if you have ever taken a course on differential equations, you have probably heard his name in reference to Bessel functions. Unlike most of the scientists featured here, Bessel appears to have had no higher education after being apprenticed at the age of 14 or 15 to work in an import-export firm. In spite of that, he made significant contributions to the fields of mathematics and astronomy. During his apprenticeship, he self-taught many things, including navigation, astronomy, and foreign languages. In 1804 he wrote a paper on Halley's comet based on observations that had been made in the 1607 and showed it to Wilhelm Olbers, a noted German astronomer, who had it published. He was appointed in 1810 to be the director of the Konigsberg Observatory (which wasn't completed until 1813). He was also given an honorary doctorate, which was important for his position as a professor!

Bessel was the first to measure the distance of a star by parallax. This is the same phenomenon that you can see if you hold you finger up relative to something in the background and watch the position of your finger change as you look at it with each eye. If you know the distance between your eyes and the angle of the shift, you can calculate the distance of the object.The only catch is that as the object gets farther away, the angle of the shift gets smaller and smaller, but scientists hoped that a shift could be observed as the Earth moves around the sun. Bessel was particularly interested because he considered his duty as an astronomer to explain why the celestial bodies moved as they did.

Tycho Brahe tried in the late 1500s, but was not able to observe parallax. Robert Hooke tried again in the 1600s and claimed to have measured the distance to Gamma Draconis, but no one believed him. He was also wrong: he calculated that it was only 0.1 light years away, when it is actually around 154. James Bradley also tried to measure this distance, but was also unable to, although as a result of his measurements he discovered the aberration of starlight--that you need to account for the movement of the earth and the speed of light in a telescope. William Herschel (who later discovered Uranus) also set about measuring stellar parallax, and tried to find a combination of a close and far star so that he could measure the slight changes in position. Instead, he discovered actual pairs of stars, which are no good since they are about the same distance.

Bessel, as the director of the new observatory in Konigsberg in Prussia, had the use of a telescope made by Joseph Fraunhofer, a maker of telescopes with a precision never seen before. Such an instrument was also in the possession of Struve, astronomer at the Dorpat Observatory in Estonia, and he and Bessel began a race. Struve published a value of the paralax of Vega, but with only 16 measurements. Bessel had been interested in the double star 61 Cygni for many years, having published a paper in 1812 on the subject, and proposed that by observing how they moved in relation to each other, the total mass of the two could be determined. Since it is one of the fastest moving stars in the sky, it was assumed to be one of the closest, and it was observable from his observatory for most of the year. Bessel presented his calculations in 1838, giving a distance of 10.3 light years, which is not too far from the current value of 11.4. Because of his careful measurements, the scientific community, including Struve, accepted his accomplishment as the first.

Bessel's careful measurements of the stars enabled him to make a new discovery as well. He observed that Sirius and Procyon, both bright stars, moved oddly, as though something was influencing them, and corroborated this with historic data as well. He posited that there must be other stars that had not been observed, and indeed, the companion star of Sirius was discovered in 1862 and was recognized as the double star that Bessel had predicted, while Procyon B was not discovered until 1896.

It was also his desire for precise astronomy that led to the Bessel functions, which are solutions to a particular differential equation. Special cases of the functions had been studied before by several Bernoullis, Euler, and Lagrange, among others, but Bessel is considered to have been the person to systematize the equations, and as such, they have been named after him. They appear often in cases involving circles and cylinders, and as such, Bessel found them useful in his studies of the stars, though exactly how I do not know. If it were not for this, people might have never heard his name, but I wonder which accomplishments he would most want to be known for? (And hopefully whatever it was isn't one of the ones that I left out of this short summary.)

Selected Works by Bessel
  • "Über den Doppelsterne Nro. 61 Cygni", Monatliche Correspondenz 181226, 148-63.
  • "On the parallax of 61 Cygni", Monthly Notices of the Royal Astronomical Society 1838, 4, 152-161. The beginning is worth a read, as he discusses some of the difficulties that he encountered both with the measurements and the calculations associated with them.
  • "Bestimmung der Entfernung des 61sten Sterns des Schwans", Astronomische Nachrichten 183916, 65-96. doi: 10.1002/asna.18390160502. German paper on the same topic.
  • "On the variations of the proper motions of Procyon and Sirius" Monthly Notices of the Royal Astronomical Society 1844, 6, 136-141. doi: 10.1093/mnras/6.11.136a. This is also a very readable article. Of particular note is the last paragraph, where he deals with the issue of positing the existence of something that can't (or hasn't) been seen.
  • "Ueber Veränderlichkeit der eigenen Bewegungen der Fixsterne", Astronomische Nachrichten 1845, 22 (10), 145-160. doi: 10.1002/asna.18450221002. Again, a German article on the same topic.

Saturday, August 23, 2014

Wilhelm Ostwald

Wilhelm Ostwald (1853-1932)
 Licensed under Public domain
 via Wikimedia Commons
One of my favorite things about writing this blog is discovering the interesting interests that the scientists
that I write about had, in addition to whatever work they are remembered for. So as I started to look for information on Wilhelm Ostwald, I was fascinated to discover that there were articles discussing his involvement with color and the Bauhaus (an art school in Germany in the 20s). Ostwald ripening, which is where I had heard his name before, seems to be one of the more minor contributions that he made: Ostwald won the 1909 Nobel Prize in Chemistry primarily for his work with catalysis, wrote 45 books and about 1000 publications, and is considered one of the founders of physical chemistry.

Ostwald was born in 1853 in Riga, Russia, and became a professor at Riga Polytechnic in 1882. One of his duties was to expand the laboratory, so he toured laboratories in Germany and had the opportunity to meet many German scientists. One of the things that he did at Riga was writing a two volume textbook. Another was translating Gibb’s work on chemical thermodynamics into German, which enabled it to be more widely read in Europe than the original English. He founded, with van’t Hoff, the first journal in physical chemistry,  Zeitschrift für Physikalische Chemie. It wasn’t the first journal he founded—he later started Annalen der Naturphilosophie.

In 1887 he then moved on to a position in Leipzig as the chair of physical chemistry—the only chair in physical chemistry in Germany. That same year he first recognized that catalysis was a kinetic process when he was studying the oxidation of hydrogen iodide by bromic acid, leading to the idea that a catalyst is something that modifies the rate of a reaction without being changed its self. Later, he tried to apply this to the problem of nitrogen fixation, but his method by catalysis with iron did not work. He did, however, patent a process for making nitric acid out of ammonia using a platinum catalyst in 1902, which is still the most commonly used process for making nitric acid today.

Svante Arrhenius published a paper in 1887 on electrolytic dissociation, the first in the field of electrochemistry. Ostwald had studied salt decomposition in obtaining a degree, and returned to it, developing his dilution law. He also wrote Elektrochemie from 1893-1896, a text on the subject, in which he also included the history of the developments and biographies of the scientists involved. He also formed an electrochemical society in 1894.

After about 1900, Ostwald turned more of his attentions to ideas of natural philosophy. In one of his works, Grosse Männer,  he divided scientists into two categories, classicists or romantics. There is actually a very interesting article by Robert Deltete and David Thorsell that compares the working styles of Josiah Gibbs and Wilhelm Ostwald as examples of these two styles. Ostwald is the romantic, jumping from one idea to the next, making connections with many people and not being afraid of publishing work before it is finished. Gibbs, on the other hand, led a fairly reclusive life, taught few students, and did not publish his work on thermodynamics until it was completed to his satisfaction.

Ostwald retired in 1906, but continued being involved in many different projects, societies, and researches. One of these was Brücke (The Bridge), an organization with the goal of organizing science to make it more efficient. Ostwald had been interested in energy for quite some time, and was actually one of the last well-renowned scientists to reject atomism. He favored an energetic explanation, and as such, conserving energy in many forms was one of his interests. He even named his retirement home, which he moved to in 1906, Haus Energie.[1] Some of the things that the Bridge did were to promote Esperanto as a universal language for science, reducing the need for translations of works, and to standardize publication formats for scientific publications. Several other scientists of note were involved in the Bridge, including Svante Arrhenius, Ernest Solvay, Ernest Rutherford, and Henri Poincaré. Ostwald also became involved in the German Monistic Alliance, which also had as its aim the unification of science, but included the reorganization of society as well. Ostwald was involved in both until Brücke closed in 1914 due to lack of funding and the difficulties of unifying science during the Great War.

After World War I, Ostwald he turned his attentions to color theory. He had been a painter since at least 1884, and now turned his scientific energies to color and developing a color theory. One of his most important contributions was giving value to the color grey. He even opened a pigment factory from 1920-1923. Walter Gropius invited him to speak at the Bauhaus and he even became a trustee. Ostwald considered his contributions to color theory some of his greatest work and nominated himself for a Nobel Prize for it (while, having won a Nobel Prize, he could nominate people, one can’t nominate ones self).

Now I don’t know how many of you may be materials scientists or others interested in Ostwald ripening, but I realize I haven’t mentioned it since the first paragraph. It is because very few of the articles I read even mentioned that work, and for a little while I was convinced that I had gotten the wrong Ostwald. But it turns out that this is the right guy. Ostwald ripening, which is a thermodynamic process observed in solutions, either solid or liquid, where larger crystals grow and smaller crystals shrink, comes from his work in the late 1890s.

[1] The house has been preserved, and was recently purchased by Gerda und Klaus Tschira Stiftung and it will serve as a location for scientific meetings. (Ertl, 2009)

Selected Works by Ostwald

Tuesday, May 20, 2014

The Way the Atom Splits: Lise Meitner, fission, and weighty elements

Lise Meitner (1878-1968)
(photo from Wikimedia Commons)
So far, I’ve written about people whose names are memorialized in scientific equipment, equations, and units, but not yet in that most permanent place, an element name. And to rectify that omission, today’s subject is Lise Meitner, the eponym of element 109, meitnerium (Mt).

Lise Meitner was born in Vienna in 1878 and although Protestant, had Jewish roots that will be important later. She attended school and after was tutored in mathematics and physics to enable her to pass the Matura exam, which was required before one could study in a university. She went to the University of Vienna, and enjoyed the lectures by, among others, Ludwig Boltzmann. Her thesis was on “Heat Conduction in Inhomogeneous Materials”, where she showed experimentally one of Maxwell’s formulas related to conduction. Here second paper was on Fresnel’s reflection formulae. Both of these had very little to do with the subjects of her later work.

After Boltzmann’s death in 1906, Meitner began helping Stefan Meyer with his work in radioactivity, measuring alpha and beta radiation. Around the same time, Max Planck visited the University, and she decided that she wanted to go to Berlin for a few semesters to learn more about physics. So in 1907, Meitner went off to Berlin for what turned out to be much longer than a few semesters. She attended lectures, but also went to the head of the institute of experimental physics, Heinrich Rubens, to ask if she could work in his lab. He suggested instead that she work with Otto Hahn, who was looking for a physicist who knew something about radioactivity. This work led to two very fruitful collaborations. Meitner also served as Planck’s assistant from 1912-1915.

Her first notable discovery was made around WWI. In 1913, Hahn and Meitner moved their lab from the University of Berlin to the Kaiser Wilhelm Institute für Chemie. This proved to be very useful for their studies of radioactivity, because the new lab was not contaminated by radiation, so they could do more sensitive experiments. One experiment that they were particularly interested in was looking for an element which produced actinium (element 89). Actinium’s place in the periodic table had been determined in 1913, and according to the displacement laws developed by Frederic Soddy and Kasimir Fajans, also in 1913, actinium could be produced by beta emission from radium or alpha emission by an unknown element 91. Fajans and Oswald Göhring discovered a new element when observing the beta-decay of Thorium-234. It had a very short half-life, and, claiming discoverer’s privilege, they named it brevium. It was a beta emitter, however, and could not produce actinium, though they were close--they had discovered element 91, just not the isotope that that would decay into actinium.

Meitner and Hahn worked to improve the technique Fajans had developed to separate brevium, but still wanted to find an isotope that produced actinium. When WWI started, Hahn was conscripted to serve in the special gas warfare unit that was run by Fritz Haber. This, along with the fact that most of the lab assistants were also serving in the war, meant that Meitner did most of the lab work by herself for the next several years, though Hahn was able to come back a few times and consult. Meitner did go off from 1915-1916 to serve as a nurse in Austria, but chafed at all of the time she was not busy and went back to the lab. She worked on substances derived from pitchblende and monitored them alpha emissions to try to discover the isotope that would lead to actinium. She needed more pitchblende, which, during the war, proved difficult. After trying several times, she was finally able to obtain a sufficient amount to determine the half-life of the alpha emitter that she had found. She and Hahn submitted a paper in March of 1918 entitled “The Mother Substance of Actinium, a New Radioactive Element of Long Half-Life”. They called the element protoactinium, which was shortened in 1949 to protactinium. Although they were not the first to discover element 91, Fajans and Göhring agreed that brevium was a silly name for an element that had isotopes with such long half lives, and agreed to the name. Soddy had also been working on the problem, and published their results in June of 1918, but acknowledged themselves beaten, and everyone agreed to the name.

Meitner obtained her own lab in 1917 as part of the department for radioactivity, and she and Hahn ceased their collaboration in 1920. From 1920-1934 she worked with alpha, beta, and gamma radiation and various nuclear processes. She used a Wilson cloud chamber and was the first to observe electron-positron pair formation by gamma radiation. Meitner followed work being done in other labs, and was intrigued by Enrico Fermi’s work in 1934 where he bombarded elements with neutrons and discovered that this could cause nuclear reactions. Irène Joliot-Curie bombarded uranium with atomic particles and found elements similar to lanthanum and barium, which was a very strange result considering their positions on the periodic table. Meitner was very interested, and so she and Hahn resumed working together to look into it. They also invited Fritz Strassmann, who was skilled in chemical analysis, to join the team in 1935. Hahn didn’t believe the results that Joliot-Curie obtained, but repeated the experiments and found the same thing. It was Meitner who convinced Hahn that the lanthanum and barium-like elements they observed were actually those elements and that what he had done was split the atomic nucleus itself.

At the same time, Jews in Germany were beginning to feel the Nazi persecution, and although Meitner was Protestant, she had Jewish heritage. This was not a problem for a while, since Meitner was Austrian, not German. However, when Germany annexed Austria in 1938, everything changed. She requested permission to leave, but it was denied. Then began a chain of scientists all trying to help Meitner. Hahn and Paul Rosbaud arranged for her to leave Austria illegally. Peter Debye (in Berlin) contacted Dirk Coster (in Gronigen) who was able to obtain her entry into Holland and Coster, along with Adriaan Fokker, helped to get her from Berlin into Holland. She left July 13, 1938. From the Netherlands, Meitner went to Sweden with further help from Debye and Bohr. Niels Bohr had begun working around 1932 to find persecuted Jewish scientists positions in foreign institutions, and it was through his efforts that she was given a position at the Nobel Institute for Experimental Physics in Stockholm, although she complained about having a lack of equipment there.

Meitner continued to advise the research on fission, but had to do it from afar. From December 1938 to early 1939 she worked with her nephew Otto Frisch to develop a theoretical interpretation of the fission that Hahn and Joliot-Curie had observed. Meitner and Frisch published the paper together, but because of the political situation, Hahn working in Germany did not want to publish the paper on fission with Meitner, and her name was left off. Hahn received the 1944 Nobel Prize for the discovery of fission, and Meitner was recognized as a collaborator in the presentation speech. Her paper with Frisch, however, had shown that fission could actually occur and that there was enough energy to split the atom, rather than just break off a piece, and that when it did split, it released huge amounts of energy.

In 1943 she was offered a post with the British scientists going to Los Alamos and in 1947 Fritz Straßmann invited her to join him at the Kaiser-Wilhelm-Institut, but she refused both offers. She retired to Cambridge in 1960, joining Otto Frisch and other relatives there. In 1966 she, Hahn, and Straßmann shared the Enrico Fermi Award for the discovery of the fission of Uranium. Meitner died in 1968 and her tombstone bears the epitaph “A physicist who never lost her humanity.”

Now, as I mentioned at first, Lise Meitner has an element named after her, but if you were paying attention, I haven't talked about its discovery. This would not come until 1982 when element 109 was discovered by a group in Darmstadt. Elements 104-109 were all discovered between 1964 and 1982, some of them by several labs, each of which named the elements, and meitnerium got caught up in these disputes, even though the discovery and name were not disputed. The groups at Lawrence Berkley and Dubna both claimed to have discovered elements 104, 106, and 107 first, which made distributing naming rights difficult. The United States proposed a list, but it gave preference to the claims of the scientists from Lawrence Berkley. In 1986, IUPAC and IUPAP set up the Transfermium Working Group to try to settle who discovered the elements first. This, of course, caused responses from the scientists involved, which were published in 1993. Then the Commission on Nomenclature of Inorganic Chemistry met in 1994 and chose delegates to discuss names for the elements 101-109, which were chosen from names submitted by the various labs involved in the disputes—Lawrence Berkeley, Joint Institute for Nuclear Research in Dubna, Russia, and Gesellschaft für Schwerionen Forschung in Darmstadt, Germany. The United States then complained that Seaborgium had been removed from consideration and the American Chemical Society Committee on Nomenclature rejected the recommendations of IUPAC. Finally, in 1997, IUPAC took the issue to the general assembly and proposed a new list, which was finally accepted and element 109 was officially meitnerium.

Selected Works by Meitner

Wednesday, July 31, 2013

Lord Kelvin: Beyond Degrees

William Thomson, Baron
Kelvin of Largs (1824-1907)
Painted by Hubert von Herkomer
It has been a little while since I wrote on someone who gave his name to a unit, so up this week is the eponym of the Kelvin, William Thomson, who is more often referred to as Lord Kelvin, because, for the last years of his life, he was Baron Kelvin of Largs. He did not inherit the title, but was actually the first scientist to be elevated to the House of Lords, and spent most of his life as William Thomson, although he was knighted in 1866, becoming Sir William Thomson. In addition to being the first Baron Kelvin, he was also the last, as he had no heir to succeed to the title. In just a bit of reading about him, I found out that Thomson/Kelvin was interested in all sorts of things, far more than just thermodynamics. These other interests include telegraph signals, navigational aids, and the age of the earth. I have become quite caught up in his and others work on the telegraph, so I initially intended to write this only on his telegraph work, but I have become so excited about telegraphy that I decided to give it a post all of its own. So this post will focus on Kelvin's work except for telegraphy, but look forward next week (hopefully) to a post all about the telegraph, and perhaps another post on Thomson.

So that I can skip on his scientific and engineering work, I'll give only a brief summary of William Thomson's life. He was born in Belfast and moved to Glasgow as a child. He studied at the Universities of Glasgow and Cambridge (where he was on the rowing team). At the young age of 22 he took a professorship at the University of Glasgow and never left, despite other offers. He gave many lectures, including a series at Johns Hopkins University. He also owned a yacht, the Lalla Rookh, built for him in the late 1860s, and several of his inventions were for improving navigation. He published more than 661 papers/communications and took out 70 patents. After his death in 1907, he was buried in Westminster Abbey next to Isaac Newton.

The Lalla Rookh
From The Life of William Thomson by Silvanus Thompson
The Lalla Rookh was a yacht of 126 tons. Thomson was known to take it out for much of the time between the semesters at the University of Glasgow, and sailed around Scotland and even farther afield, such as Lisbon. He also used it to entertain. In 1871 he planned a cruise to the Hebrides and West Highlands with Hermann von Helmholtz, Thomas Huxley, John Tyndall and James Maxwell, though it seems only Helmholtz was actually able to make it. Having the ship inspired twenty five different patented inventions. One of these was a device for correcting compasses when the ship had a metal hull. Another was an improved sounding (depth finding) device, using a wire rather than a rope such that measurements could be taken at speed.

The purchase of the ship may also have lead to his increased interest in fluid mechanics, which occurred in 1867, and was one of the subjects he discussed with Helmholtz on their journey in 1871. But his interest in fluid mechanics had a little remembered result as well--the idea of vortex atoms. Remember, the structure of the atom was not known in the mid-19th century. A common standard for atomic weights was not determined until 1860, and Mendeleev published the first periodic table in 1869. Even then, people didn't know what made up an atom. J. J. Thomson discovered the electron in 1897, and Rutherford's famous experiment which proved the existence of an atomic nucleus was not until 1909, after Kelvin's death, so the question of the nature of the atom was wide open. William Thomson spent considerable time developing the idea of vortex atoms, based on the descriptions of fluid motion made by Helmholtz, who had expanded descriptions of fluid motion to include more irrotational motion. This theory suggested that atoms are vortexes (such as smoke rings) in the ether that makes up space. By around 1883, Thomson began to feel that his theory was not sufficient to explain matter, but he had gotten other scientists thinking about the idea and furthered the field of hydrodynamics.

There is certainly more to be said, and I had hoped to say it, but hopefully that whets your appetite and you will go looking for more information on your own. The references should be quite helpful. And perhaps I will return to Kelvin at some point to talk about absolute zero, electricity, and the age of the earth. But if I try to cover them now, this post will never be finished.

Miscellaneous Works by Thomson

Thursday, June 27, 2013

Erudite Euler

Leonhard Euler (1707-1783)
Up this week is Euler, the great mathematician. It is impossible for me to do him justice in this post, because of the shear volume of his work and my lack of knowledge about mathematics. If you would like to read more about him, please look at some of the articles in the references. These ones are particularly good.

I mostly know of Euler from two things-Euler's method, which was an approximation method I used in calculus, and Euler's formula, which could actually mean many things, but in this case I refer to his formula that relates exponential and trigonometric functions, eix = cos(x) + isin(x), and is perhaps seen more commonly in less scientific circles in the particular form where x = π, where it simplifies to e = 1. These are both definitely important, and I used the latter in about every other homework that I did this year, but Euler considered many more applied problems than this small sample size would suggest.

Leonhard Euler was the son of a minister and originally intended to study theology. Fields of study were different in the 18th century, though, and at the conclusion of his master's degree he gave a lecture comparing the natural philosophy of Newton and Descartes. At the University of Basel he studied, among other things, mathematics under the tutelage of Johann Bernoulli, who encouraged him to study mathematics more pointedly and was of much help in later years as well. As a young man, Euler competed for the prize question of the Paris Academy of Sciences, a competition open to the greatest scientific minds in Europe, and came in second. Not bad. In later years, he came in first twelve times. He applied to a position in physics at the University of Basel and failed, but then was invited to the Academy of Sciences in St. Petersburg, where he was devoted mainly to mathematics. He stayed in St. Petersburg from 1727-1741. He was then invited to help found the Academy of Sciences in Berlin, Prussia. He did not get along well with Frederick II, though, and when rebuffed from the position of president of the Academy, returned to St. Petersburg in 1766 at the invitation of Catherine II, and stayed there until his death in 1783.

In setting down to consider the accomplishments of Euler, I found it interesting to note who had gone before. Nicholas Fuss, one of Euler's students, in his eulogy on Euler, said:
At the time when Mr. Euler entered into mathematics, nothing could be more discouraging. A mediocre talent simply could not expect to make a name for it and it was best to choose another career or to distinguish one brilliantly. The memory of the recently deceased great men that had been part of the past century and the beginning of ours was still particularly fresh in our minds. Hardly had Newton and Leibniz altered the face of geometry when they died and we had not yet forgotten the important services that the discoveries of Huyghens, Bernoulli, Moivre, Tschirnhausen, Taylor, Fermat and so many other mathematicians had provided to all the branches of mathematics.
Euler clearly chose the second option: "to distinguish one brilliantly". Rather than consider that mathematics had been exhausted, as one might think, or even that certain areas had, he pursued many different avenues and pushed mathematics in many new and old directions.

He tackled the field of mechanics in two volumes, introducing to it integral and differential calculus. He was also very interested in sound, and had written a thesis on it when applying for the position in Basel. But he returned to the subject in St. Petersburg, and extended his writings to include the emotions that sounds can evoke. There he also developed the Γ function (which gives factorials for positive integers, but can be applied to non- and negative integers as well), and the constant γ, called Euler's constant. He also developed the concept of the fuction, and the notation still used today of f(x). While writing on complex and novel mathematical ideas, Euler also wrote works on more basic subjects, like textbooks on arithematic for use in Russian schools, and Théorie complete de la construction et de la manœuvre des vaisseaux, a text for sailors on navigation. Other problems that he dealt with included optimal profiles for the teeth on gears, why disks (think pennies being spun) seem to spin faster as they fall down, the critical load for a rod to buckle, and the number of vertices, edges, and faces for polyhedra. In investigating these, he also often returned to a topic for many years after he first looked at it.

Something that undoubtedly helped his work was his prodigious memory. He was reported to be able to recite the entirety of Virgil's Aeneid (which, having read, I can assure you is no mean feat). It helps, of course, that he could read and write Latin. He was also very good at doing calculations in his head and remembering the results afterwards. This was vital to his work, especially in later years, since he lost the sight in one eye in 1735 and suffered from cataracts in the other, eventually losing his sight almost completely. That did not, however, stop his productivity, and he had his sons and others take dictation, or copy large letters from a slate.

For one with such remarkable skills, he also seems to have been quite humble and well liked, and passed up opportunities to quibble over who had discovered things first. He married twice and had 13 children, five of whom survived to adulthood. He had a fit of apoplexy while playing with one of his grandsons and drinking a cup of tea, and died a few hours later. Fuss spoke glowingly about him: how he dropped calculations for ordinary conversation, explained concepts at the level of the listener, did not hold
grudges, fought injustice where he saw it, and many other praiseworthy qualities. It has been exciting to see how a man could be at the top of his field, clearly pursuing topics that piqued his interest, and yet still  be praised as a humane, relateable, Christian man.

  • Marquis de Condorcet, "Eulogy to Mr. Euler", History of the Royal Academy of Sciences, 1783, Paris 1786, p. 37-68. A nice summary of Euler's life and work, free of equations and with some nice anecdotes.
  • Nicolas Fuss, "Eulogy of Leonhard Euler", read at the Imperial Academy of Sciences of Saint Petersburg, October 23, 1783. Fuss was a student of Euler and a grandson-in-law. His eulogy is longer than Condorcet's, but more personal. If you are interested in what Euler was like as a person, skip to the end. Fuss paints a wonderful picture of a caring, Christian family man.
  • Walter Gautschi, "Leonhard Euler: His Life, the Man, and His Works", SIAM Review 50, no. 1 (2008), 3-33. DOI: 10.1137/070702710. A relatively short summary of his life, providing both an outline of life events and some of his mathematical accomplishments. For a quick summary of some of his math, this is a good place to start, as this one actually includes some diagrams and equations.

Thursday, June 20, 2013

The Elusive Wulff

Wulff construction
A Wulff plot (the surface energies are given in red)
Drawing by Michael Schmid
and used under the GNU Free Documentation License 
I first ran into the name Wulff last year in my thermodynamics class. He gave his name to a method for constructing the shape that a single crystal will take based on the surface energies of different crystallographic directions. It is a clever construction, and that particular homework problem was probably my favorite of the whole year. So I decided to see what I could find out about Wulff, and I have found him very difficult to track down, so I put this post on the shelf. Then in winter quarter I ran into the name again in an x-ray diffraction class where we used Wulff nets, and I decided to track him down again. It proved no easier, but I did get farther! It does seem strange, though, for a man who's name is so often used, that there is so little information on him. He doesn't even have an English Wikipedia page! The first reason for confusion about Wulff is that he was Ukrainian, so there are different transliterations of his name, but worse than that, he went by two names! Georg Wulff was the name he used in German-language publications, and thus is the name that we are more familiar with, but his name in Russian (transliterated, of course) was Yuri Viktorovich. I did, at last, find a nice article on him in the Complete Dictionary of Scientific Biography, and some information in one of my x-ray diffraction textbooks.

Image of a projection from Wulff's 1902 paper
Georg Wulff was born in the Ukraine in 1863 and studied at Warsaw University. In 1907, or 1908, or 1911, he became a professor (or teacher) of crystallography at Moscow University. Nobody seems to be able to agree. What is important, I think, is that between defending his dissertation and his death, he taught in various capacities at universities in Russia and the U.S.S.R.

He published two important papers in 1901 and 1902 regarding crystal structures and stereographic projections. They are both in German, so I don't know exactly what they are getting at. Hammond says that Wulff proposed the Wulff net in 1909, but there seems to be an image of part of a Wulff net in his 1902 paper. His 1901 paper introduced the principles of the Wulff construction, which is a graphical method for determining the faces of a crystal that are expressed based on the surface energy of the different crystallographic directions. This idea built on Josiah Gibbs' proposal that materials want to minimize total surface energy. Wulff himself did not prove mathematically why his construction worked, but it was proved by Conyers Herring (1914-2009) in the 1950s.
Crystal diagram from Wulff's 1901 paper

Wulff was also in communication with William Henry Bragg and his son, William Lawrence Bragg, English crystallographers. Wulff derived an equation for x-ray diffraction in 1913 that was equivalent to the one proposed the year before by the Braggs, and so some people at the time called what is now known as Braggs' Law the Bragg-Wulff Law. Wulff appears to have lost out on the name because he published second, and, more importantly, he did not follow up with as many advancements on the topic as the Braggs. After that, he appears not to have taken on any new areas of study and faded into obscurity, though not without leaving his name for students of thermodynamics and crystallography to stumble upon.