Strong prime - Revision history

en>Tamfang: /* Definition in cryptography */ numbers here are meaningless clutter

2014-10-11T19:11:37Z

Definition in cryptography: numbers here are meaningless clutter

← Older revision		Revision as of 21:11, 11 October 2014
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	~~{{cleanup\|reason=wrong information, see discussion page\|date=January 2013}}~~		Alyson Meagher is the title her parents gave her but she doesn't like when people use her complete name. To climb is some thing I really enjoy performing. My day job is an invoicing officer but I've currently applied for an additional one. Mississippi is the only location I've been residing in but I will have to move in a yr or two.<br><br>my weblog ... [http://203.250.78.160/zbxe/?document_srl=1792908 psychic readers]
	~~[[File:Sodium-chloride-3D-ionic.png\|thumb\|right\|Sodium chloride crystal lattice]]~~

	~~The '''lattice energy''' of a crystalline [[solid]]~~ is ~~usually defined as~~ the [[internal energy\|energy]] of formation of the crystal from infinitely-separated [[ion]]s, [[molecule]]s, or [[atom]]s, and as such is invariably negative. The concept of lattice energy was originally developed for [[rocksalt]]-structured and [[sphalerite]]-structured compounds like ~~NaCl and ZnS, where the ions occupy high-symmetry crystal lattice sites~~. ~~In the case of NaCl, the lattice energy~~ is ~~the energy released by the reaction~~

	~~: Na<sup>+</sup> (g) + Cl<sup>−</sup> (g) → NaCl (s)~~

	~~which would amount to -786 kJ/mol~~.~~<ref name="Johnson"/>~~

	Some older textbooks define lattice energy with the opposite sign,<ref>{{cite book\|last=Zumdahl\|first=Steven S.\|title=Chemistry\|year=1997\|publisher=Houghton Mifflin\|location=Boston\|isbn=0-669-41794-7\|pages=357–358\|edition=4th ed.}}</ref> i.e. the energy required to convert the crystal into infinitely separated gaseous ions, atoms, or molecules in [[vacuum]], an [[endothermic]] process. Following this convention, the lattice energy of NaCl would be +787 kJ/mol. The lattice energy for ionic crystals such as sodium chloride, metals such as iron, or covalently linked materials such as diamond is considerably greater in magnitude than for ~~solids such as sugar or iodine, whose neutral molecules interact only by weaker [[Dipole-dipole force\|dipole-dipole]] or [[van der Waals force]]s.~~

	~~The precise value of the lattice energy may not be determined experimentally, because of the impossibility of preparing~~ an adequate amount of gaseous ions or atoms and measuring the energy released during their condensation to form the solid. However, the value of the lattice energy may either be derived theoretically from [[electrostatics]] or from a thermodynamic cycling reaction, the so-called [[Born–Haber cycle]].

	~~The relationship between the molar lattice energy and the molar lattice enthalpy~~ is ~~given by~~ the ~~following equation:~~
	:<math>\Delta _G U=\Delta _G H -p\Delta V_m</math>, where <math>\Delta _G U</math> is the molar lattice energy, <math>\Delta _G H</math> the molar lattice enthalpy and <math>\Delta V_m</math> the change of the volume per mol. Therefore the lattice enthalpy further takes into account that work has to be performed against an outer pressure <math>p</math>.

	~~==Theoretical treatments==~~

	~~===Born–Landé equation===~~
	~~{{main\|Born–Landé equation}}~~
	~~In 1918<ref>~~I~~.D. Brown,~~ '~~'The chemical Bond~~ in ~~Inorganic Chemistry'', IUCr monographs~~ in crystallography, Oxford University Press, 2002, ISBN 0-19-850870-0</ref> [[Max Born\|Born]] and [[Alfred Landé\|Landé]] proposed that the lattice energy could be derived from the [[electric potential]] of the ionic lattice and a repulsive [[potential energy]] term.<ref name = "Johnson">David Arthur Johnson, ''Metals and Chemical Change'',Open University, Royal Society of Chemistry, 2002,ISBN 0-85404-665-8</ref>

	~~:<math>E = -\frac{N_AMz^+z^- e^2 }{4 \pi \varepsilon_0 r_0}\left(1-\frac{1}{n}\right),</math>~~
	~~where~~

	~~:''N''<sub>A</sub> is the [[Avogadro constant]];~~
	~~:''M'' is the [[Madelung constant]], relating to the geometry of the crystal;~~
	~~:''z''<sup>+</sup> is the charge number of cation;~~
	~~:''z''<sup>−</sup> is the charge number of anion;~~
	~~:''e'' is the [[elementary charge]], equal to {{val\|1.6022\|e=-19\|u=C}};~~
	~~:''ε''<sub>0</sub> is the [[permittivity of free space]], equal to {{val\|8.854\|e=-12\|u=C<sup>2</sup> J<sup>−1</sup> m<sup>−1</sup>}};~~
	~~:''r''<sub>0</sub> is the distance to closest ion; and~~
	~~:''n'' is the Born exponent,~~ a ~~number between 5 and 12, determined experimentally by measuring the [[compressibility]] of the solid,~~ or ~~derived theoretically~~.<~~ref~~>~~Cotton, F. Albert; Wilkinson, Geoffrey; (1966). Advanced Inorganic Chemistry (2d Edn.) New York:Wiley-Interscience.~~<~~/ref~~>

	~~The [[Born–Landé equation]] gives a reasonable fit to the lattice energy~~.~~<ref name = "Johnson"/>~~

	~~{\| class="wikitable" border="1"~~
	\|-
	~~! Compound~~
	~~! Calculated Lattice Energy~~
	~~! Experimental Lattice Energy~~
	\|-
	~~\| NaCl~~
	~~\| −756 kJ/mol~~
	~~\| −787 kJ/mol~~
	\|-
	~~\| LiF~~
	~~\| −1007 kJ/mol~~
	~~\| −1046 kJ/mol~~
	\|-
	~~\| CaCl<sub>2</sub>~~
	~~\| −2170 kJ/mol~~
	~~\| −2255 kJ/mol~~
	\|}

	~~From the [[Born–Landé equation]] it can be seen that the lattice energy of a compound is dependent on a number of factors~~

	* as the charges on the ions increase the lattice energy increases (becomes more negative),
	* when ions are closer together the lattice energy increases (becomes more negative)
	Barium oxide (BaO), for instance, which has the NaCl structure and therefore the same Madelung constant, has a bond radius of 275 picometers and a lattice energy of -3054 kJ/mol, while sodium chloride (NaCl) has a bond radius of 283 picometers and a lattice energy of -786 kJ/mol.

	~~===Kapustinskii equation===~~
	~~The [[Kapustinskii equation]] can be used as a simpler way of deriving lattice energies where high precision is not required~~.~~<ref name = "Johnson"/>~~

	~~=== Effect of polarisation ===~~

	~~For ionic compounds with ions occupying lattice sites with~~ [[crystallographic point groups]] ''C''<sub>1</sub>, ''C''<sub>1</sub>''<sub>h</sub>'', ''C<sub>n</sub>'' or ''C<sub>nv</sub>'' (''n'' = 2, 3, 4 or 6) the concept of the lattice energy and the Born–Haber cycle has to be extended.<ref name= ZPB1995a>{{cite journal \| author = M. Birkholz \| title = Crystal-field induced dipoles in heteropolar crystals – I. concept \| journal = Z. Phys. B \| volume = 96 \| pages = 325–332 \| year = 1995 \| doi = 10.1007/BF01313054 \|bibcode = 1995ZPhyB..96..325B \| url=http://www.mariobirkholz.de/ZPB1995a.pdf}}</ref> In these cases the [[Ionic polarization\|polarization]] energy ''E<sub>pol</sub>'' associated with ions on polar lattice sites has to be included in the Born–Haber cycle and the solid formation reaction has to start from the already polarized species. As an example, one may consider the case of [[pyrite\|iron-pyrite]] FeS<sub>2</sub>, where sulfur ions occupy lattice site of point symmetry group ''C''<sub>3</sub>. The lattice energy defining reaction then reads

	~~: Fe<sup>2+</sup> (g) + 2 pol S<sup>−</sup> (g) → FeS<sub>2</sub> (s)~~

	where pol S<sup>−</sup> stands for the polarized, gaseous sulfur ion. It has been shown that the neglection of the effect led to 15% difference between theoretical and experimental thermodynamic cycle energy of FeS<sub>2</sub> that reduced to only 2%, when the sulfur polarization effects were included.<ref name= BJPC1992>{{cite journal \| author = M. Birkholz \| title = The crystal energy of pyrite \| journal = J. Phys.: Condens. Matt. \| volume = 4 \| pages = 6227 \| year = 1992 \| doi = 10.1088/0953-8984/4/29/007 \|bibcode = 1992JPCM....~~4.6227B \| url=http://www.mariobirkholz.de~~/~~JPhysC1992.pdf}}<~~/~~ref>~~

	~~== See also ==~~
	* [[Bond energy]]
	* [[Born–Haber cycle]]
	* [[Chemical bond]]
	* [[Madelung constant]]
	* [[Ionic conductivity]]

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

	~~{{chemical solutions}}~~

	~~[[Category:Crystallography]]~~
	~~[[Category:Solid-state chemistry]~~]

74.216.92.75: /* Definition in number theory */

2014-01-13T07:13:09Z

Definition in number theory

← Older revision		Revision as of 09:13, 13 January 2014
Line 1:		Line 1:
	The title of the ~~author~~ is ~~Jayson~~. I'~~ve usually loved living~~ in ~~Alaska~~. ~~To play domino~~ is ~~some thing I truly enjoy doing~~. ~~Since he was 18 he~~'~~s been working~~ as ~~an information officer but he ideas~~ on ~~changing it~~.<br><br>~~Have a look at my blog~~ - [~~http~~://~~fashionlinked~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~13453~~/~~info~~/ ~~are psychics real~~]		{{cleanup\|reason=wrong information, see discussion page\|date=January 2013}}
			[[File:Sodium-chloride-3D-ionic.png\|thumb\|right\|Sodium chloride crystal lattice]]

			The '''lattice energy''' of a crystalline [[solid]] is usually defined as the [[internal energy\|energy]] of formation of the crystal from infinitely-separated [[ion]]s, [[molecule]]s, or [[atom]]s, and as such is invariably negative. The concept of lattice energy was originally developed for [[rocksalt]]-structured and [[sphalerite]]-structured compounds like NaCl and ZnS, where the ions occupy high-symmetry crystal lattice sites. In the case of NaCl, the lattice energy is the energy released by the reaction

			: Na<sup>+</sup> (g) + Cl<sup>−</sup> (g) → NaCl (s)

			which would amount to -786 kJ/mol.<ref name="Johnson"/>

			Some older textbooks define lattice energy with the opposite sign,<ref>{{cite book\|last=Zumdahl\|first=Steven S.\|title=Chemistry\|year=1997\|publisher=Houghton Mifflin\|location=Boston\|isbn=0-669-41794-7\|pages=357–358\|edition=4th ed.}}</ref> i.e. the energy required to convert the crystal into infinitely separated gaseous ions, atoms, or molecules in [[vacuum]], an [[endothermic]] process. Following this convention, the lattice energy of NaCl would be +787 kJ/mol. The lattice energy for ionic crystals such as sodium chloride, metals such as iron, or covalently linked materials such as diamond is considerably greater in magnitude than for solids such as sugar or iodine, whose neutral molecules interact only by weaker [[Dipole-dipole force\|dipole-dipole]] or [[van der Waals force]]s.

			The precise value of the lattice energy may not be determined experimentally, because of the impossibility of preparing an adequate amount of gaseous ions or atoms and measuring the energy released during their condensation to form the solid. However, the value of the lattice energy may either be derived theoretically from [[electrostatics]] or from a thermodynamic cycling reaction, the so-called [[Born–Haber cycle]].

			The relationship between the molar lattice energy and the molar lattice enthalpy is given by the following equation:
			:<math>\Delta _G U=\Delta _G H -p\Delta V_m</math>, where <math>\Delta _G U</math> is the molar lattice energy, <math>\Delta _G H</math> the molar lattice enthalpy and <math>\Delta V_m</math> the change of the volume per mol. Therefore the lattice enthalpy further takes into account that work has to be performed against an outer pressure <math>p</math>.

			==Theoretical treatments==

			===Born–Landé equation===
			{{main\|Born–Landé equation}}
			In 1918<ref>I.D. Brown, ''The chemical Bond in Inorganic Chemistry'', IUCr monographs in crystallography, Oxford University Press, 2002, ISBN 0-19-850870-0</ref> [[Max Born\|Born]] and [[Alfred Landé\|Landé]] proposed that the lattice energy could be derived from the [[electric potential]] of the ionic lattice and a repulsive [[potential energy]] term.<ref name = "Johnson">David Arthur Johnson, ''Metals and Chemical Change'',Open University, Royal Society of Chemistry, 2002,ISBN 0-85404-665-8</ref>

			:<math>E = -\frac{N_AMz^+z^- e^2 }{4 \pi \varepsilon_0 r_0}\left(1-\frac{1}{n}\right),</math>
			where

			:''N''<sub>A</sub> is the [[Avogadro constant]];
			:''M'' is the [[Madelung constant]], relating to the geometry of the crystal;
			:''z''<sup>+</sup> is the charge number of cation;
			:''z''<sup>−</sup> is the charge number of anion;
			:''e'' is the [[elementary charge]], equal to {{val\|1.6022\|e=-19\|u=C}};
			:''ε''<sub>0</sub> is the [[permittivity of free space]], equal to {{val\|8.854\|e=-12\|u=C<sup>2</sup> J<sup>−1</sup> m<sup>−1</sup>}};
			:''r''<sub>0</sub> is the distance to closest ion; and
			:''n'' is the Born exponent, a number between 5 and 12, determined experimentally by measuring the [[compressibility]] of the solid, or derived theoretically.<ref>Cotton, F. Albert; Wilkinson, Geoffrey; (1966). Advanced Inorganic Chemistry (2d Edn.) New York:Wiley-Interscience.</ref>

			The [[Born–Landé equation]] gives a reasonable fit to the lattice energy.<ref name = "Johnson"/>

			{\| class="wikitable" border="1"
			\|-
			! Compound
			! Calculated Lattice Energy
			! Experimental Lattice Energy
			\|-
			\| NaCl
			\| −756 kJ/mol
			\| −787 kJ/mol
			\|-
			\| LiF
			\| −1007 kJ/mol
			\| −1046 kJ/mol
			\|-
			\| CaCl<sub>2</sub>
			\| −2170 kJ/mol
			\| −2255 kJ/mol
			\|}

			From the [[Born–Landé equation]] it can be seen that the lattice energy of a compound is dependent on a number of factors

			* as the charges on the ions increase the lattice energy increases (becomes more negative),
			* when ions are closer together the lattice energy increases (becomes more negative)
			Barium oxide (BaO), for instance, which has the NaCl structure and therefore the same Madelung constant, has a bond radius of 275 picometers and a lattice energy of -3054 kJ/mol, while sodium chloride (NaCl) has a bond radius of 283 picometers and a lattice energy of -786 kJ/mol.

			===Kapustinskii equation===
			The [[Kapustinskii equation]] can be used as a simpler way of deriving lattice energies where high precision is not required.<ref name = "Johnson"/>

			=== Effect of polarisation ===

			For ionic compounds with ions occupying lattice sites with [[crystallographic point groups]] ''C''<sub>1</sub>, ''C''<sub>1</sub>''<sub>h</sub>'', ''C<sub>n</sub>'' or ''C<sub>nv</sub>'' (''n'' = 2, 3, 4 or 6) the concept of the lattice energy and the Born–Haber cycle has to be extended.<ref name= ZPB1995a>{{cite journal \| author = M. Birkholz \| title = Crystal-field induced dipoles in heteropolar crystals – I. concept \| journal = Z. Phys. B \| volume = 96 \| pages = 325–332 \| year = 1995 \| doi = 10.1007/BF01313054 \|bibcode = 1995ZPhyB..96..325B \| url=http://www.mariobirkholz.de/ZPB1995a.pdf}}</ref> In these cases the [[Ionic polarization\|polarization]] energy ''E<sub>pol</sub>'' associated with ions on polar lattice sites has to be included in the Born–Haber cycle and the solid formation reaction has to start from the already polarized species. As an example, one may consider the case of [[pyrite\|iron-pyrite]] FeS<sub>2</sub>, where sulfur ions occupy lattice site of point symmetry group ''C''<sub>3</sub>. The lattice energy defining reaction then reads

			: Fe<sup>2+</sup> (g) + 2 pol S<sup>−</sup> (g) → FeS<sub>2</sub> (s)

			where pol S<sup>−</sup> stands for the polarized, gaseous sulfur ion. It has been shown that the neglection of the effect led to 15% difference between theoretical and experimental thermodynamic cycle energy of FeS<sub>2</sub> that reduced to only 2%, when the sulfur polarization effects were included.<ref name= BJPC1992>{{cite journal \| author = M. Birkholz \| title = The crystal energy of pyrite \| journal = J. Phys.: Condens. Matt. \| volume = 4 \| pages = 6227 \| year = 1992 \| doi = 10.1088/0953-8984/4/29/007 \|bibcode = 1992JPCM....4.6227B \| url=http://www.mariobirkholz.de/JPhysC1992.pdf}}</ref>

			== See also ==
			* [[Bond energy]]
			* [[Born–Haber cycle]]
			* [[Chemical bond]]
			* [[Madelung constant]]
			* [[Ionic conductivity]]

			==References==
			{{reflist}}

			{{chemical solutions}}

			[[Category:Crystallography]]
			[[Category:Solid-state chemistry]]

en>EmausBot: r2.7.2+) (Robot: Adding uk:Сильне просте число

2012-08-27T08:40:46Z

r2.7.2+) (Robot: Adding uk:Сильне просте число

New page

The title of the author is Jayson. I've usually loved living in Alaska. To play domino is some thing I truly enjoy doing. Since he was 18 he's been working as an information officer but he ideas on changing it.<br><br>Have a look at my blog - [http://fashionlinked.com/index.php?do=/profile-13453/info/ are psychics real]