Contraharmonic mean - Revision history

en>Dbfirs: /* Properties */ say which average

2014-03-06T19:09:41Z

Properties: say which average

← Older revision		Revision as of 21:09, 6 March 2014
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	~~In [[chemistry]] and [[crystallography]], the '''coordination number''' of a central atom in a [[molecule]] or [[crystal]]~~ is ~~the number of its nearest neighbours. This number is determined somewhat differently for molecules than for crystals.~~		Andrew Berryhill is what his wife loves to contact him and he completely digs that name. Her family lives in Alaska but her husband desires them to move. It's not a common factor but what she likes performing is to perform domino but she doesn't have the time recently. Invoicing is my profession.<br><br>My homepage ... tarot readings ([http://help.ksu.edu.sa/node/65129 click the up coming article])

	~~In chemistry, the emphasis is on bonding structure in molecules or ions~~ and the coordination number of an atom is determined by simply counting the other atoms to which it is bonded (by either single or multiple bonds). For example, [Cr(NH<sub>3</sub>)<sub>2</sub>Cl<sub>2</sub>Br<sub>2</sub>]<sup>1-</sup> has Cr<sup>3+</sup> as its central cation, which has a coordination number of 6.

	However the solid-state structures of crystals often have less clearly defined [[chemical bond\|bonds]], so a simpler model is used, in which the atoms are represented by touching spheres. In this model the coordination number of an atom is the number of other atoms that ~~it touches~~. ~~For an atom~~ in the interior of a [[crystal lattice]], the number of atoms touching the given atom is the '''bulk coordination number'''; for an atom at a surface of a crystal, this is the '''surface coordination number'''.

	~~== Chemistry usage ==~~
	~~In chemistry, '''coordination number''' (c~~.~~n.), defined originally in 1893 by [[Alfred Werner]], is the total number of neighbours of a central atom in a molecule or ion.<ref>A.K. De (2003) ''A Text Book of Inorganic Chemistry'~~', p. 88. New Age International Publishers, ISBN 8122413846</ref><ref>{{GoldBookRef \| title = coordination number \| file = C01331}}</ref> Although a carbon atom has four [[chemical bonds]] in most stable molecules, the coordination number of each [[carbon]] is four in [[methane]] (CH<sub>4</sub>), three in [[ethylene]] (H<sub>2</sub>C=CH<sub>2</sub>, each C is bonded to 2H + 1C = 3 atoms), and two in [[acetylene]]. In effect we count the first bond (or [[sigma bond]]) to each neighbouring atom, but not the other bonds ([[pi bond]]s).

	~~In [[inorganic chemistry]] also, only the first or sigma bond between each [[ligand]] and the central atom counts, but~~ not any pi bonds to the same ligands. In [[tungsten hexacarbonyl]], W(CO)<sub>6</sub>, the coordination number of tungsten (W) is counted as six although pi as well as sigma bonding is important in such [[metal carbonyl]]s.

	Examples of high coordination number complexes are the ions formed by [[uranium]] and [[thorium]] with [[bidentate]] [[nitrate]] ion ligands, U(NO<sub>3</sub>)<sub>6</sub><sup>2−</sup> and Th(NO<sub>3</sub>)<sub>6</sub><sup>2−</sup>. Here each nitrate ligand is bound to the metal by two oxygen atoms, so that the total coordination number of the U or Th atom is 12.

	When the surrounding ligands are smaller than the central atom, even higher coordination numbers may be possible. One [[computational chemistry]] study predicted a particularly stable PbHe<sub>15</sub><sup>2+</sup> ion composed of a central [[lead]] ion coordinated with no fewer than 15 helium atoms.<ref>{{cite journal\|title=The Search for the Species with the Highest Coordination Number\|author= Andreas Hermann, Matthias Lein, and Peter Schwerdtfeger\|doi=10.1002/anie.200604148\|year=2007\|journal=Angewandte Chemie International Edition\|volume=46\|issue=14\|pages=2444}}</ref>

	~~== Crystallography usage ==~~
	~~[[Image:Lattice body centered cubic.svg\|thumb\|right\|BCC structure of iron. For greater visibility the atoms are indicated by points rather than touching spheres.]]~~

	~~In [[materials science]], the '''bulk coordination number''' of a given atom in the interior of~~ a [[crystal lattice]] is the number of atoms touching the given atom. Iron at 20 °C has a [[Cubic crystal system\|body-centered cubic (BCC) crystal]] in which each interior iron atom occupies the centre of a cube formed by eight neighbouring iron atoms. The bulk coordination number for this structure is therefore 8.

	The highest bulk coordination number is 12, found in both [[close packing\|hexagonal close-packed]] (HCP) and cubic close-packed (CCP) (also known as [[Cubic crystal system\|face-centered cubic]] or FCC) structures. This value of 12 corresponds to the theoretical limit of the [[kissing number problem]] when all spheres are identical.

	~~[[Image:Graphite-sheet-3D-balls.png\|thumb\|left\|One layer of a graphite crystal (graphene) with carbon atoms and C-C bonds shown in black.]]~~
	~~The two most~~ common ~~[[allotropes]] of carbon have different coordination numbers. In [[diamond]], each carbon atom~~ is at the centre of a [[tetrahedron]] formed by four other carbon atoms, so the coordination number is four, as for methane. [[Graphite]] is made of two-dimensional layers in which each carbon is covalently bonded to ~~three other carbons. Atoms in other layers are much further away and are not nearest neighbours, so the coordination number of a carbon atom in graphite is 3 as in ethylene.~~

	[[Image:Fluorite-unit-cell-3D-ionic.png\|thumb\|right\|Calcium fluoride [[unit cell]]. In this image the Ca<sup>2+</sup>(gray) and F<sup>-</sup>(green) ions are shown as touching spheres]] Simple [[ionic crystal\|ionic]] structures are described by two coordination numbers, one for each type of ion. [[Calcium fluoride]] (CaF<sub>2</sub>) is an (8, 4) structure, meaning that each [[cation]] Ca<sup>2+</sup> is surrounded by eight F<sup>−</sup> [[anion]] neighbors, and each anion F<sup>−</sup> by four Ca<sup>2+</sup>. For [[sodium chloride]] (NaCl), the numbers of cations and anions are equal, and both coordination numbers are six so that the structure is (6, 6).

	~~For an atom at a surface of a crystal, the~~ '''surface coordination number''' is always less than the bulk coordination number. The surface coordination number is dependent on the [[Miller index\|Miller indices]] of the surface. In a [[Cubic crystal system\|body-centered cubic (BCC) crystal]], the bulk coordination number is 8, whereas, for the (100) surface, the surface coordination number is 4.

	~~== Usage in quasicrystal, liquid and other disordered systems ==~~
	~~[[Image:First coordination number of Lennard-Jones fluid.png\|thumb\|First coordination number of [[Lennard-Jones]] fluid]]~~
	~~[[Image:Second coordination number of Lennard-Jones fluid.png\|thumb\|Second coordination number of Lennard-Jones fluid]]~~

	~~The coordination number of systems with disorder cannot be precisely defined.~~

	~~The '''first coordination number''' can be defined using~~ the ~~[[radial distribution function]] <math>g(r)</math> <ref>Yoshio Waseda, ''The Structure of Non-crystalline Materials – Liquids and Amorphous Solids'', McGraw-Hill, New York, 1980, pp~~. 48–51. </ref><ref>{{cite journal\|title=X-ray diffraction study of liquid sulfur\|author= K S Vahvaselkä and J M Mangs \|doi=10.1088/0031-8949/38/5/017\|year=1988\|journal=Physica Scripta\|volume=38\|issue=5\|pages=737}}</ref>

	~~<math>n_1 = 4 \pi \int_{r_0}^{r_1} r^2 g(r) \rho \, dr, </math>~~

	~~where <math>r_0</math> is the rightmost position starting from <math>r=0</math> whereon <math>g(r)</math> is approximately zero, <math>r_1</math>~~ is ~~the first minimum~~. ~~Therefore, it is the area under the first peak of~~ <~~math~~>~~g(r)~~<~~/math~~>.

	~~The '''second coordination number''' is defined similarly:~~

	~~<math>n_2 = 4 \pi \int_{r_1}^{r_2} r^2 g(r) \rho \, dr~~. ~~</math>~~

	~~Alternative definitions for the coordination number can be found in literature, but in essence the main idea is the same~~. ~~One of those definition are as follows: Denote <math>r_p</math> as the position of the first peak,~~

	~~<math>n_1' = 4 \pi \times 2 \int_{r_0}^{r_p} r^2 g~~(~~r) \rho \, dr. </math>~~

	~~The '''first coordination shell''' is the spherical shell with radius between <math>r_0</math> to <math>r_1</math> around the central particle under investigation.~~

	~~==References==~~
	~~{{reflist}}~~

	~~==External links==~~
	* [http://~~www4~~.~~nau~~.edu~~/meteorite/Meteorite/Book-GlossaryC~~.~~html Meteorite Book-Glossary C]~~
	* [http://~~www.d.umn.edu/~pkiprof/ChemWebV2/Coordination/CN1.html A website on Coordination Numbers]~~

	~~[[Category:Chemical bonding]]~~
	~~[[Category:Materials science]]~~
	~~[[Category:Coordination chemistry]~~]

en>John of Reading: Typo/general fixing, replaced: has been has been → has been using AWB

2013-11-06T12:32:43Z

Typo/general fixing, replaced: has been has been → has been using AWB

← Older revision		Revision as of 14:32, 6 November 2013
Line 1:		Line 1:
	The ~~writer~~ is known as ~~Wilber Pegues~~. One of the ~~issues she loves most~~ is ~~canoeing~~ and ~~she's been doing it for quite~~ a ~~while~~. ~~Since he was 18 he's been operating~~ as an ~~info officer but he ideas~~ on ~~changing~~ it. ~~Her family life~~ in ~~Ohio~~ but ~~her spouse wants them to move~~.<br><br>~~Feel free~~ to ~~surf to my blog post~~ ... [http://~~Kard~~.dk/~~?p=24252 real psychics~~]		In [[chemistry]] and [[crystallography]], the '''coordination number''' of a central atom in a [[molecule]] or [[crystal]] is the number of its nearest neighbours. This number is determined somewhat differently for molecules than for crystals.

			In chemistry, the emphasis is on bonding structure in molecules or ions and the coordination number of an atom is determined by simply counting the other atoms to which it is bonded (by either single or multiple bonds). For example, [Cr(NH<sub>3</sub>)<sub>2</sub>Cl<sub>2</sub>Br<sub>2</sub>]<sup>1-</sup> has Cr<sup>3+</sup> as its central cation, which has a coordination number of 6.

			However the solid-state structures of crystals often have less clearly defined [[chemical bond\|bonds]], so a simpler model is used, in which the atoms are represented by touching spheres. In this model the coordination number of an atom is the number of other atoms that it touches. For an atom in the interior of a [[crystal lattice]], the number of atoms touching the given atom is the '''bulk coordination number'''; for an atom at a surface of a crystal, this is the '''surface coordination number'''.

			== Chemistry usage ==
			In chemistry, '''coordination number''' (c.n.), defined originally in 1893 by [[Alfred Werner]], is the total number of neighbours of a central atom in a molecule or ion.<ref>A.K. De (2003) ''A Text Book of Inorganic Chemistry'', p. 88. New Age International Publishers, ISBN 8122413846</ref><ref>{{GoldBookRef \| title = coordination number \| file = C01331}}</ref> Although a carbon atom has four [[chemical bonds]] in most stable molecules, the coordination number of each [[carbon]] is four in [[methane]] (CH<sub>4</sub>), three in [[ethylene]] (H<sub>2</sub>C=CH<sub>2</sub>, each C is bonded to 2H + 1C = 3 atoms), and two in [[acetylene]]. In effect we count the first bond (or [[sigma bond]]) to each neighbouring atom, but not the other bonds ([[pi bond]]s).

			In [[inorganic chemistry]] also, only the first or sigma bond between each [[ligand]] and the central atom counts, but not any pi bonds to the same ligands. In [[tungsten hexacarbonyl]], W(CO)<sub>6</sub>, the coordination number of tungsten (W) is counted as six although pi as well as sigma bonding is important in such [[metal carbonyl]]s.

			Examples of high coordination number complexes are the ions formed by [[uranium]] and [[thorium]] with [[bidentate]] [[nitrate]] ion ligands, U(NO<sub>3</sub>)<sub>6</sub><sup>2−</sup> and Th(NO<sub>3</sub>)<sub>6</sub><sup>2−</sup>. Here each nitrate ligand is bound to the metal by two oxygen atoms, so that the total coordination number of the U or Th atom is 12.

			When the surrounding ligands are smaller than the central atom, even higher coordination numbers may be possible. One [[computational chemistry]] study predicted a particularly stable PbHe<sub>15</sub><sup>2+</sup> ion composed of a central [[lead]] ion coordinated with no fewer than 15 helium atoms.<ref>{{cite journal\|title=The Search for the Species with the Highest Coordination Number\|author= Andreas Hermann, Matthias Lein, and Peter Schwerdtfeger\|doi=10.1002/anie.200604148\|year=2007\|journal=Angewandte Chemie International Edition\|volume=46\|issue=14\|pages=2444}}</ref>

			== Crystallography usage ==
			[[Image:Lattice body centered cubic.svg\|thumb\|right\|BCC structure of iron. For greater visibility the atoms are indicated by points rather than touching spheres.]]

			In [[materials science]], the '''bulk coordination number''' of a given atom in the interior of a [[crystal lattice]] is the number of atoms touching the given atom. Iron at 20 °C has a [[Cubic crystal system\|body-centered cubic (BCC) crystal]] in which each interior iron atom occupies the centre of a cube formed by eight neighbouring iron atoms. The bulk coordination number for this structure is therefore 8.

			The highest bulk coordination number is 12, found in both [[close packing\|hexagonal close-packed]] (HCP) and cubic close-packed (CCP) (also known as [[Cubic crystal system\|face-centered cubic]] or FCC) structures. This value of 12 corresponds to the theoretical limit of the [[kissing number problem]] when all spheres are identical.

			[[Image:Graphite-sheet-3D-balls.png\|thumb\|left\|One layer of a graphite crystal (graphene) with carbon atoms and C-C bonds shown in black.]]
			The two most common [[allotropes]] of carbon have different coordination numbers. In [[diamond]], each carbon atom is at the centre of a [[tetrahedron]] formed by four other carbon atoms, so the coordination number is four, as for methane. [[Graphite]] is made of two-dimensional layers in which each carbon is covalently bonded to three other carbons. Atoms in other layers are much further away and are not nearest neighbours, so the coordination number of a carbon atom in graphite is 3 as in ethylene.

			[[Image:Fluorite-unit-cell-3D-ionic.png\|thumb\|right\|Calcium fluoride [[unit cell]]. In this image the Ca<sup>2+</sup>(gray) and F<sup>-</sup>(green) ions are shown as touching spheres]] Simple [[ionic crystal\|ionic]] structures are described by two coordination numbers, one for each type of ion. [[Calcium fluoride]] (CaF<sub>2</sub>) is an (8, 4) structure, meaning that each [[cation]] Ca<sup>2+</sup> is surrounded by eight F<sup>−</sup> [[anion]] neighbors, and each anion F<sup>−</sup> by four Ca<sup>2+</sup>. For [[sodium chloride]] (NaCl), the numbers of cations and anions are equal, and both coordination numbers are six so that the structure is (6, 6).

			For an atom at a surface of a crystal, the '''surface coordination number''' is always less than the bulk coordination number. The surface coordination number is dependent on the [[Miller index\|Miller indices]] of the surface. In a [[Cubic crystal system\|body-centered cubic (BCC) crystal]], the bulk coordination number is 8, whereas, for the (100) surface, the surface coordination number is 4.

			== Usage in quasicrystal, liquid and other disordered systems ==
			[[Image:First coordination number of Lennard-Jones fluid.png\|thumb\|First coordination number of [[Lennard-Jones]] fluid]]
			[[Image:Second coordination number of Lennard-Jones fluid.png\|thumb\|Second coordination number of Lennard-Jones fluid]]

			The coordination number of systems with disorder cannot be precisely defined.

			The '''first coordination number''' can be defined using the [[radial distribution function]] <math>g(r)</math> <ref>Yoshio Waseda, ''The Structure of Non-crystalline Materials – Liquids and Amorphous Solids'', McGraw-Hill, New York, 1980, pp. 48–51. </ref><ref>{{cite journal\|title=X-ray diffraction study of liquid sulfur\|author= K S Vahvaselkä and J M Mangs \|doi=10.1088/0031-8949/38/5/017\|year=1988\|journal=Physica Scripta\|volume=38\|issue=5\|pages=737}}</ref>

			<math>n_1 = 4 \pi \int_{r_0}^{r_1} r^2 g(r) \rho \, dr, </math>

			where <math>r_0</math> is the rightmost position starting from <math>r=0</math> whereon <math>g(r)</math> is approximately zero, <math>r_1</math> is the first minimum. Therefore, it is the area under the first peak of <math>g(r)</math>.

			The '''second coordination number''' is defined similarly:

			<math>n_2 = 4 \pi \int_{r_1}^{r_2} r^2 g(r) \rho \, dr. </math>

			Alternative definitions for the coordination number can be found in literature, but in essence the main idea is the same. One of those definition are as follows: Denote <math>r_p</math> as the position of the first peak,

			<math>n_1' = 4 \pi \times 2 \int_{r_0}^{r_p} r^2 g(r) \rho \, dr. </math>

			The '''first coordination shell''' is the spherical shell with radius between <math>r_0</math> to <math>r_1</math> around the central particle under investigation.

			==References==
			{{reflist}}

			==External links==
			* [http://www4.nau.edu/meteorite/Meteorite/Book-GlossaryC.html Meteorite Book-Glossary C]
			* [http://www.d.umn.edu/~pkiprof/ChemWebV2/Coordination/CN1.html A website on Coordination Numbers]

			[[Category:Chemical bonding]]
			[[Category:Materials science]]
			[[Category:Coordination chemistry]]

en>JoDu987: Rewrite inequality list P. in general form

2012-03-20T16:32:25Z

Rewrite inequality list P. in general form

New page

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