symmetry

 

This site contains material for crystallographers, thermodynamicists, historians of science and technology, and mathematicians interested in group theory, potential theory, and probability.  The owner of this site is Dr Michael Jewess (PHOTO), author or co-author of the work referred to below.

 

(For Michael Jewess’s work on intellectual property, follow the link RESEARCH IN IP.)

 

Sections I to III below deal with “orientational disorder” of ammonium, tetramethylammonium, nitrate, and carbonate ions;  with “positional disorder” of copper ions;  with phthalocyanine and adiponitrile copper complexes;  specifically with ammonium chloride, tetramethylammonium trichloromanganate(II) (“TMMC”), scawtite, bis(adiponitrile)copper(I) nitrate, and copper(II) phthalocyanine;  and with heat capacity anomalies.

 

Section IV below deals with “band groups” (the seven groups of symmetry operations for a frieze, offering an understanding by analogy of the 230 space groups).

 

Section V below deals with the application of modern thermodynamic data to understand a nineteenth-century commercial chemical process, Julius Thomsen’s process for production of washing soda in which cryolite was used as the source of the sodium needed.   

 

Section VI below deals with the thermodynamics of graphane, structurally related to the more famous graphene, and possibilities for synthesising it on a bulk scale.

 

Section VII below deals with optimisation problems relating to the gravity of a planet.

 

Section VIII below deals with two problems in probability:  the “two girls” problem and the Monty Hall game show problem.

 

Section IX below deals the apprenticeship of Michael Faraday in Marylebone, London from 1805 to 1812 and his subsequent recruitment to the Royal Institution by Humphry Davy, and the “blue plaque” on the place of his apprenticeship.

 

Section X below deals with the apprenticehip of Humphry Davy in Penzance, Cornwall from 1795 to 1798, whence he progressed to a career in scientific research, and the presentation of a blue plaque on the place of his apprenticeship on 11 September 2015.

 

Section XI below deals with the achievements of H G J Moseley, who would almost certainly have received a Nobel Prize for his work on atomic numbers had he not been killed in Turkey in 1915.

 

Section XII below deals with the possibility of present-day electrocatalytic nitrogen fixation, including by reference to the work of Humphry Davy.      

 

 

SECTION I: DISORDER OF POLYATOMIC IONS IN CRYSTALS 

 

There follows a link to the full text of an Acta Cryst. paper on the treatment of “orientational disorder” for crystallographers: ORIENTATIONAL DISORDER   (The corresponding doi link is http://dx.doi.org/10.1107/S0567740882006062.)  The key conclusion of this paper is that it is often wrong to locate the central atom of a disordered ion at the centre of symmetry of the site in which the ion is located.  While, with suitably chosen positions among which the ion is disordered, this allows the average symmetry of the ion to be made to match that of the site, it may be that the precise positions (differing only in the orientation of the ions) are energetically impossible because they cannot correspond to potential energy wells.  X-ray crystal determinations in which this mistake has been made are shown to have resulted in structures that are materially questionable.   

 

There follows a link to a J. Chem.Thermodynamics  paper on the heat capacity of tetramethylammonium trichloromanganate(II) (often known as tetramethylammonium manganese chloride, “TMMC”) :  TMMC.  (The corresponding doi link is http://dx.doi.org/10.1016/0021-9614(83)90064-2.) This is one of the crystals referred to in the paper disorder.pdf linked above;  tmmc.pdf reports a heat capacity anomaly associated with disordering of the tetramethylammonium ions.  

 

There follows a link to a J. Chem. Soc. paper on the heat capacity of bis(adiponitrile)copper(I) nitrate:  CU(ADIP)2NO3;this reports a heat capacity anomaly associated with disordering of the nitrate ions in this crystal, more fully discussed in the Acta Cryst. paper linked above.  (The corresponding doi link is http://dx.doi.org/10.1039/F29807600803.)

 

The Acta Cryst. paper discusses also the structures of ammonium chloride and scawtite, which at room temperature have disordered ammonium and carbonate ions respectively.

 

SECTION II: POSITIONAL DISORDER OF THE COPPER ION IN β -MODIFICATION COPPER(II) PHTHALOCYANINE

 

There follows a link to a J. Chem. Soc. paper which reports the heat capacity of β-modification copper(II), nickel(II), and hydrogen phthalocyanines:   PHTHALOCYANINE.  (The corresponding doi link is http://dx.doi.org/10.1039/F29817701757.) This suggests that, on account of the rigidity of the essentially square-planar phthalocyanine divalent anion, the copper (II) ion does not have a minimum energy when located at its centre.  Instead, there may be a local maximum, with four equivalent positions of minimum energy in a square around the centre.  The height of the maximum is small and the effect is observed only at very low temperatures, in a particularly large heat capacity anomaly.  Above the temperature of the anomaly, the ions are apparently disordered among the four positions with equal probability.

 

SECTION III:  HEAT CAPACITY ANOMALIES OFFER A DIRECT APPROACH TO THE NUMBER OF POSITIONS OR ORIENTATIONS AMONG WHICH THERE IS DISORDER

 

Heat capacity anomalies offer a direct approach to the number of positions or orientations among which an ion is disordered in a crystal.  When such a crystal is cooled, there is frequently a transition to a form in which the ions are ordered. If the specific heat is measured across the temperature range in which the transition occurs, a heat capacity anomaly is observed.  The entropy change (readily calculated from the measured heat capacities) includes a component R ln n  per mole of ions where n is the number of positions among which the ion is disordered in the high- temperature form.  Such anomalies are interpreted to advantage in the papers linked above.  Generally, the lower the temperature, the easier it is to estimate the R ln n component, because the vibrational entropy changes that occur over the temperature range of the transition are smaller.

 

The published papers linked in Section I report  R ln n as closest to R ln 2 in the case of bis(adiponitrile)copper(I) nitrate, and to R ln 3 in the case of TMMC.  The published paper linked in Section II reports  R ln n as closest to R ln 4 in the case of

β-modification copper(II) phthalocyanine.

 

It should be noted that heat capacity anomalies are associated also with other types of order-disorder transitions in crystals.  A long-known example of an anomaly is that arising from positional disordering in β-brass (CuZn).  In addition, anomalies are associated with magnetic transitions, transitions from libration to rotation of ions, and electronic transitions (the anomalies in the last case being known as “Schottky anomalies”).       

 

SECTION IV:  THE “BAND GROUPS” (“FRIEZE GROUPS”)

 

The student of crystallography finds it reasonably possible to satisfy himself of the existence of 32 point groups in three dimensions.  The existence of 230 space groups is usually taken on trust rather than checked personally.  A student can be given confidence in this if he considers the seven “band groups” (or “frieze groups”) that can occur in a repetitive two-dimensional pattern repeating only in one dimension, ie a frieze.  Once he is satisfied that these are correct, then his confidence in the space groups is based on analogy rather than trust. 

 

Adolf Pabst of UC Berkeley (1890-1990) is said to have recommended construction of the seven band groups as an exercise for the student.  However, this is not a trivial task, and the following links to such an exercise:  BANDGROUPS.      

 

SECTION V:  JULIUS THOMSEN’S COMMERCIAL PROCESS FOR MANUFACTURING WASHING SODA

 

There follows a link to an essay in the Royal Society of Chemistry’s Historical Group Newsletter and summary of papers: THOMSEN.  This paper relates to the commercial activities of the Danish chemist, Julius Thomsen (1826-1909), best known for his work on the periodic table and on thermochemistry.  He made his fortune by a process which manufactured washing soda from cryolite (Na3AlF6) mined in Greenland, a Danish possession. The paper applies modern thermodynamic knowledge to understand how his process works.  It raises the intriguing possibility that one or more compounds believed to have been first discovered in the 1980s were in fact produced a century before, on an industrial scale, as an intermediate in Thomsen’s process.        

 

SECTION VI:  THERMODYNAMICS OF GRAPHANE

 

There follows a link to a report on graphane:  GRAPHANE.  Graphene is a celebrated material consisting of a single layer of carbon.  Graphane consists of such a layer hydrogenated on both sides (empirical formula CH), and has been formed in small quantities from graphene and atomic hydrogen.  The report estimates its thermodynamic properties and considers which chemical reactions not involving such exotic reactants might be used to synthesise it in bulk.

 

SECTION VII:  OPTIMISING THE ACCELERATION DUE TO GRAVITY ON A PLANET’S SURFACE

 

There follows a link to a Mathematical Gazette paper with the above title:  GRAVITY.  (The corresponding doi link is https://doi.org/10.1017/S002555720000646X.)Two problems are addressed, each concerning how a hypothetical planet-builder can make best use of a given amount of incompressible uniform material if he wishes to obtain, if only at one or two points on the surface, the maximum acceleration due to gravity.  The planet-builder is also granted planet rigidity; he can form the planet into the shape he chooses, without worrying that the planet will under its own gravity revert to being a sphere.  The first optimisation problem constrains the planet-builder to form the material into a spheroid (of any degree of oblateness);  the second imposes no constraint on the form, and therefore is a “calculus of variations” problem.  The solution to the first problem is a surprisingly oblate spheroid, with the acceleration due to gravity maximised at the two poles.  The solution to the second problem is a particular ovoid of revolution, also surprisingly flattened, with the acceleration due to gravity highest at the pole at the blunt end.        

 

SECTION VIII:  THE “TWO GIRLS” PROBLEM AND THE MONTY HALL PROBLEM COMPARED

 

There follows a link to a Mathematical Gazette contribution on the “two girls” problem:  TWO GIRLS.  In a randomly chosen family with two children, if boys and girls are equally likely, there is a ¼ chance that there are two girls.  This problem is concerned with what more accurate statistical inference can be made from the additional, true statement that, in the family, “at least one of the two children is a girl who was born on a Tuesday”.  It is shown that what one infers depends on the process that generated the additional, true statement.  This is analogous to the famous Monty Hall game show, where what one infers from the facts that one of three doors is open showing there is no prize behind it and that one of the other two conceals a prize depends on the process by which the first door came to be opened.        

 

SECTION IX:  APPRENTICESHIP OF MICHAEL FARADAY

 

There follows a link to an essay in the Royal Society of Chemistry’s Historical Group Newsletter and summary of papers: FARADAY.  This paper relates to Faraday’s apprenticeship with George Riebau, bookbinder and bookseller, from 1805 to 1812.  The shop (now 48 Blandford St in the Marylebone area of London) bears a splendid plaque erected in 1875-1876 under the scheme which was the predecessor of the blue plaque scheme.  The paper describes how Riebau’s help and encouragement allowed Faraday to make the leap to scientific work at the Royal Institution in 1813, and contains modern and contemporary illustrations.

 

SECTION X:  APPRENTICESHIP OF HUMPHRY DAVY

 

There follow links to articles describing the installation of a Royal Society of Chemistry Chemical Landmark plaque in the newsletter of the Society for the History of Alchemy and Chemistry, Chemical Intelligence, http://www.ambix.org/wp-content/uploads/2016/01/ChemIntelJan2016.pdf (see pages 30-31), and in the Royal Society of Chemistry’s Historical Group Newsletter and summary of papers:  DAVY).  The author acted as master of ceremonies at the event to present the plaque on 11 September 2015.  Davy was apprenticed to an apothecary, John Bingham Borlase from 1795 to 1798 at No 1 Market Place, Penzance, Cornwall.  Like Faraday, a decade and more later, Davy benefited from a generous master, who in his case released him early from his indentures so that he could pursue a career in scientific research.

 

SECTION XI:  H J G MOSELEY

 

H J G (“Harry”) Moseley was killed at the battle of Chanuk Bair in Turkey in August 1915.  Had he not been killed, he almost certainly would have won a Nobel Prize for his work in determining atomic numbers, ie the charge on the nucleus of atoms.  These in turn determine the number of elecrons in the electrically neutral atom, and thereby the chemical properties of elements.  The modern periodic table is in order of atomic number.  The following link is to a radio interview with given by Michael Jewess: http://www.bbc.co.uk/programmes/p03hxbx0.

 

SECTION XII:  ELECTROCATALYTIC FIXATION OF NITROGEN

 

Fixation of nitrogen from the air by electrolytic means is thermodynamically allowed with low applied potentials.  Reports of such fixation go back to Sir Humphry Davy.  With molecular catalysis, such fixation might be practical today.  There follows a link to an American Chemical Society paper on the possibility of nitrogen fixation by electrolysis in the presence of suitable catalysts:  N2FIXATION.  (The corresponding doi link is http://dx.doi.org/10.1021/acssuschemeng.6b01473.)    

 

CONTACT

 

E-mail michaeljewess[AT]researchinip.com.  This sub-site was last updated on 17 September 2017.

 

 

ADDITIONAL KEYWORDS:  specific heat, history of technology, Isaac Newton, London street names, Marylebone street names, RSA, Royal Society of Arts, memorial tablets, Outhgill, Humphry Davy, Henry Moseley, Harry Moseley.