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		<author><name>194.254.117.28</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Torricelli%27s_law&amp;diff=12523</id>
		<title>Torricelli&#039;s law</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Torricelli%27s_law&amp;diff=12523"/>
		<updated>2013-12-09T15:20:52Z</updated>

		<summary type="html">&lt;p&gt;194.254.61.41: Less wrong picture. The one which goes further from the base is the middle hole.&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;{{chembox&lt;br /&gt;
| Watchedfields = changed&lt;br /&gt;
| verifiedrevid = 452010868&lt;br /&gt;
| Name = Uranium trioxide&lt;br /&gt;
| ImageFile = UO3 gamma lattice.png&lt;br /&gt;
| IUPACName = Uranium trioxide&amp;lt;br/&amp;gt;Uranium(VI) oxide&lt;br /&gt;
| OtherNames = Uranyl oxide&amp;lt;br/&amp;gt;Uranic oxide&lt;br /&gt;
| Section1 = {{Chembox Identifiers&lt;br /&gt;
| CASNo = 1344-58-7&lt;br /&gt;
| CASNo_Ref = {{cascite}}&lt;br /&gt;
}}&lt;br /&gt;
| Section2 = {{Chembox Properties&lt;br /&gt;
| Formula = UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&lt;br /&gt;
| MolarMass = 286.29 g/mol&lt;br /&gt;
| Density = 5.5–8.7 g/cm&amp;lt;sup&amp;gt;3&amp;lt;/sup&amp;gt;&lt;br /&gt;
| Appearance = yellow-orange powder&lt;br /&gt;
| Solubility = Partially soluble&lt;br /&gt;
| MeltingPt = ~200–650&amp;amp;nbsp;°C (decomposes)&lt;br /&gt;
}}&lt;br /&gt;
| Section3 = {{Chembox Structure&lt;br /&gt;
| CrystalStruct = &#039;&#039;see text&#039;&#039;&lt;br /&gt;
| SpaceGroup = &#039;&#039;I&#039;&#039;4&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;/amd (&#039;&#039;γ&#039;&#039;-UO&amp;lt;sub&amp;gt;3)&lt;br /&gt;
| Coordination = &lt;br /&gt;
}}&lt;br /&gt;
| Section4 = {{Chembox Thermochemistry&lt;br /&gt;
|   DeltaHf = −1230&amp;amp;nbsp;kJ·mol&amp;lt;sup&amp;gt;−1&amp;lt;/sup&amp;gt;&amp;lt;ref name=b1&amp;gt;{{cite book| author = Zumdahl, Steven S.|title =Chemical Principles 6th Ed.| publisher = Houghton Mifflin Company| year = 2009| isbn = 0-618-94690-X|page=A23}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
|   Entropy = 99&amp;amp;nbsp;J·mol&amp;lt;sup&amp;gt;−1&amp;lt;/sup&amp;gt;·K&amp;lt;sup&amp;gt;−1&amp;lt;/sup&amp;gt;&amp;lt;ref name=b1/&amp;gt;&lt;br /&gt;
 }}&lt;br /&gt;
| Section7 = {{Chembox Hazards&lt;br /&gt;
| ExternalMSDS = [http://www.ibilabs.com/UO3-MSDS.htm External MSDS]&lt;br /&gt;
| EUIndex = 092-002-00-3&lt;br /&gt;
| EUClass = Very toxic (&#039;&#039;&#039;T+&#039;&#039;&#039;)&amp;lt;br/&amp;gt;Dangerous for the environment (&#039;&#039;&#039;N&#039;&#039;&#039;)&lt;br /&gt;
| RPhrases = {{R26/28}}, {{R33}}, {{R51/53}}&lt;br /&gt;
| SPhrases = {{S1/2}}, {{S20/21}}, {{S45}}, {{S61}}&lt;br /&gt;
| NFPA-H = &lt;br /&gt;
| NFPA-F = &lt;br /&gt;
| NFPA-R = &lt;br /&gt;
| NFPA-O = &lt;br /&gt;
| FlashPt = Non-flammable&lt;br /&gt;
| LD50 = &lt;br /&gt;
| PEL = &lt;br /&gt;
}}&lt;br /&gt;
| Section8 = {{Chembox Related&lt;br /&gt;
| OtherAnions = &lt;br /&gt;
| OtherFunctn = [[Uranium dioxide]]&amp;lt;br/&amp;gt;[[Triuranium octoxide]]&lt;br /&gt;
| Function = [[uranium]] [[oxide]]s&lt;br /&gt;
| OtherCpds =&lt;br /&gt;
}}&lt;br /&gt;
}}&lt;br /&gt;
&lt;br /&gt;
&#039;&#039;&#039;Uranium trioxide (UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;)&#039;&#039;&#039;, also called &#039;&#039;&#039;[[uranyl]] oxide&#039;&#039;&#039;, &#039;&#039;&#039;uranium(VI) oxide&#039;&#039;&#039;, and &#039;&#039;&#039;uranic oxide&#039;&#039;&#039;, is the hexavalent [[oxide]] of [[uranium]]. The solid may be obtained by heating [[uranyl nitrate]] to 400&amp;amp;nbsp;°C. Its most commonly encountered [[Polymorphism (materials science)|polymorph]], γ-UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;, is a yellow-orange powder.&lt;br /&gt;
&lt;br /&gt;
==Production and use==&lt;br /&gt;
There are three methods to generate uranium trioxide. As noted below, two are used industrially in the reprocessing of nuclear fuel and uranium enrichment.&lt;br /&gt;
&lt;br /&gt;
[[File:Uranium-trioxide-formation.png|500px|Methods of forming uranium trioxide]]&lt;br /&gt;
&lt;br /&gt;
# U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt; can be oxidized at 500&amp;amp;nbsp;°C with oxygen.&amp;lt;ref&amp;gt;{{cite journal|author= Sheft I, Fried S, Davidson N|title= Preparation of Uranium Trioxide|journal=Journal of the American Chemical Society|year= 1950|volume= 72|issue= 5|pages= 2172–2173|doi= 10.1021/ja01161a082}}&amp;lt;/ref&amp;gt; Note that above 750&amp;amp;nbsp;°C even in 5 atm O&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; decomposes into U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt;.&amp;lt;ref name=&amp;quot;wheeler&amp;quot;&amp;gt;{{cite journal|author= Wheeler VJ, Dell RM, Wait E| title= Uranium trioxide and the UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; hydrates|journal= J. Inorganic Nuclear Chemistry| year= 1964|volume= 26|issue= 11|pages= 1829–1845|doi= 10.1016/0022-1902(64)80007-5}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
# [[Uranyl nitrate]], UO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;(NO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;)&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;·6H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O can be heated to yield UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;. This occurs during the [[Nuclear reprocessing|reprocessing of nuclear fuel]]. Fuel rods are dissolved in [[nitric acid|HNO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;]] to separate [[uranyl nitrate]] from [[plutonium]] and the fission products (the [[PUREX]] method). The pure uranyl nitrate is converted to solid UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; by heating at 400&amp;amp;nbsp;°C. After reduction with hydrogen (with other inert gas present) to [[uranium dioxide]], the uranium can be used in new [[MOX fuel]] rods.&lt;br /&gt;
# [[Ammonium diuranate]] or [[sodium diuranate]] (Na&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;U&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;7&amp;lt;/sub&amp;gt;·6H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O) may be decomposed. [[Sodium diuranate]], also known as [[yellowcake]], is converted to uranium trioxide in the [[enriched uranium|enrichment of uranium]]. [[Uranium dioxide]] and [[uranium tetrafluoride]] are intermediates in the process which ends in [[uranium hexafluoride]].&amp;lt;ref&amp;gt;{{cite journal|author= Dell RM, Wheeler VJ|title= Chemical Reactivity of Uranium Trioxide Part 1. — Conversion to U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt;, UO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; and UF&amp;lt;sub&amp;gt;4&amp;lt;/sub&amp;gt;| journal= Transactions of the Faraday Society|year= 1962|volume= 58|pages= 1590–1607|doi= 10.1039/TF9625801590}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Uranium trioxide is shipped between processing facilities in the form of a gel.&lt;br /&gt;
&lt;br /&gt;
[[Cameco Corporation]], which operates at the world&#039;s largest uranium refinery at [[Blind River, Ontario]], produces high-purity uranium trioxide.&lt;br /&gt;
&lt;br /&gt;
It has been reported that the corrosion of uranium in a silica rich aqueous solution forms both [[uranium dioxide]] and uranium trioxide.&amp;lt;ref&amp;gt;Trueman ER, Black S, Read D, Hodson ME (2003) &amp;quot;Alteration of Depleted Uranium Metal&amp;quot; &#039;&#039;Goldschmidt Conference Abstracts,&#039;&#039; p. A493 [http://www.the-conference.com/2003/Gold2003/abstracts/A493.pdf abstract]&amp;lt;/ref&amp;gt; In pure water, [[schoepite]] (UO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;)&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;(OH)&amp;lt;sub&amp;gt;12&amp;lt;/sub&amp;gt;·12(H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O) is formed&amp;lt;ref&amp;gt;[http://webmineral.com/data/Schoepite.shtml Schoepite]. Webmineral.com. Retrieved on 2011-07-19.&amp;lt;/ref&amp;gt; in the first week and then after four months [[studtite]] (UO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;)O&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;·4(H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O) was produced. Reports on the corrosion of uranium metal have been published by the [[Royal Society]].&amp;lt;ref&amp;gt;Ander L, Smith B (2002) &amp;quot;[http://www.royalsoc.ac.uk/downloaddoc.asp?id=1182 Annexe F: Groundwater transport modelling]&amp;quot; &#039;&#039;The health hazards of depleted uranium munitions, part II&#039;&#039; (London: The Royal Society)&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;Smith B (2002) &amp;quot;[http://www.royalsoc.ac.uk/downloaddoc.asp?id=1183 Annexe G: Corrosion of DU and DU alloys: a brief discussion and review]&amp;quot; &#039;&#039;The health hazards of depleted uranium munitions, part II&#039;&#039; (London: The Royal Society)&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
==Health and safety hazards==&lt;br /&gt;
Like all hexavalent uranium compounds, UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; is hazardous by inhalation, ingestion, and through skin contact. It is a poisonous, slightly radioactive substance, which may cause shortness of breath, coughing, acute arterial lesions, and changes in the chromosomes of [[white blood cell]]s and [[gonads]] leading to [[Congenital disorder|congenital malformations]] if inhaled.&amp;lt;ref name=&amp;quot;morrow&amp;quot;&amp;gt;{{cite journal|author= Morrow, PE, Gibb FR, Beiter HD|title= Inhalation studies of uranium trioxide|journal= Health Physics|year= 1972|volume= 23|pages= 273–280|doi= 10.1097/00004032-197209000-00001|pmid= 4642950|issue= 3}} [http://www.health-physics.com/pt/re/healthphys/abstract.00004032-197209000-00001.htm abstract]&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal|author= Sutton M, Burastero SR|title= Uranium(VI) solubility and speciation in simulated elemental human biological fluids|journal= [[Chemical Research in Toxicology]]|year= 2004| volume= 17| pages= 1468–1480| doi= 10.1021/tx049878k|pmid= 15540945|issue= 11}}&amp;lt;/ref&amp;gt; However, once ingested, uranium is mainly toxic for the [[kidney]]s and may severely affect their function.&lt;br /&gt;
&lt;br /&gt;
==Structure==&lt;br /&gt;
===Solid state structure===&lt;br /&gt;
The only well characterized binary trioxide of any [[actinide]] is UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;, of which several [[polymorphism (materials science)|polymorphs]] are known. Solid UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; loses O&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; on heating to give green-colored [[Triuranium octaoxide|U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt;]]: reports of the decomposition temperature in air vary from 200–650&amp;amp;nbsp;°C. Heating at 700&amp;amp;nbsp;°C under H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; gives dark brown [[uranium dioxide]] (UO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;), which is used in [[MOX fuel|MOX]] [[nuclear fuel]] rods.&lt;br /&gt;
&lt;br /&gt;
====Alpha====&lt;br /&gt;
&amp;lt;table  border=&amp;quot;1&amp;quot; cellpadding=&amp;quot;3&amp;quot; cellspacing=&amp;quot;0&amp;quot;&amp;gt;&lt;br /&gt;
&amp;lt;tr&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
[[File:UO3alphalattice.jpg|center|100px]]&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
&#039;&#039;The α (alpha) form: a layered solid where the 2D layers are linked by oxygen atoms (shown in red)&#039;&#039;&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
Hydrated uranyl peroxide formed by the addition of [[hydrogen peroxide]] to an aqueous solution of [[uranyl nitrate]] when heated to 200–225&amp;amp;nbsp;°C forms an amorphous uranium trioxide which on heating to 400–450&amp;amp;nbsp;°C will form alpha-uranium trioxide.&amp;lt;ref name=&amp;quot;wheeler&amp;quot; /&amp;gt; It has been stated that the presence of nitrate will lower the temperature at which the [[exothermic]] change from the [[amorphous]] form to the alpha form occurs.&amp;lt;ref&amp;gt;{{cite journal|doi= 10.1002/jctb.5010130807|author= Sato T|title=Preparation of uranium peroxide hydrates|journal= Journal of Applied Chemistry|year= 1963|volume= 13|issue= 8|page= 361}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;/tr&amp;gt;&lt;br /&gt;
&amp;lt;/table&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Beta====&lt;br /&gt;
&amp;lt;table  border=&amp;quot;1&amp;quot; cellpadding=&amp;quot;3&amp;quot; cellspacing=&amp;quot;0&amp;quot;&amp;gt;&lt;br /&gt;
&amp;lt;tr&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
[[File:UO3betalattice.jpg|center|150px]]&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
&#039;&#039;β (beta) UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;. This solid has a structure which defeats most attempts to describe it.&#039;&#039;&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
This form can be formed by heating ammonium diuranate, while P.C. Debets and B.O. Loopstra, found four solid phases in the UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;-H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O-NH&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; system that they could all be considered as being UO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;(OH)&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;.H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O where some of the water has been replaced with ammonia.&amp;lt;ref&amp;gt;{{cite journal|author=Debets PC, Loopstra BO|title=On the Uranates of Ammonium II: X-Ray Investigation of the Compounds in the system NH&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;-UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;-H2O|journal= Journal of Inorganic Nuclear Chemistry|year= 1963|volume= 25|issue=8|page= 945|doi=10.1016/0022-1902(63)80027-5}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal|author=Debets PC|title=The Structure of β-UO3|journal= Acta Crystallographica|year= 1966|volume= 21|issue=4|page= 589 |doi =10.1107/S0365110X66003505}}&amp;lt;/ref&amp;gt; No matter what the exact stoichiometry or structure, it was found that [[calcination]] at 500&amp;amp;nbsp;°C in air forms the beta form of uranium trioxide.&amp;lt;ref name=&amp;quot;wheeler&amp;quot; /&amp;gt;&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;/tr&amp;gt;&lt;br /&gt;
&amp;lt;/table&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Gamma====&lt;br /&gt;
&amp;lt;table  border=&amp;quot;1&amp;quot; cellpadding=&amp;quot;3&amp;quot; cellspacing=&amp;quot;0&amp;quot;&amp;gt;&lt;br /&gt;
&amp;lt;tr&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
[[File:UO3 gamma lattice.jpg|center|150px]]&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
&#039;&#039;The γ (gamma) form, with the different uranium environments in green and yellow&#039;&#039;&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
The most frequently encountered polymorph is γ-UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;, whose [[x-ray crystallography|x-ray structure]] has been solved from powder diffraction data. The compound crystallizes in the space group &#039;&#039;I4&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;/amd&#039;&#039; with two uranium atoms in the asymmetric unit. Both are surrounded by somewhat distorted octahedra of oxygen atoms. One uranium atom has two closer and four more distant oxygen atoms whereas the other has four close and two more distant oxygen atoms as neighbors. Thus it is not incorrect to describe the structure as [UO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;]&amp;lt;sup&amp;gt;2+&amp;lt;/sup&amp;gt;[UO&amp;lt;sub&amp;gt;4&amp;lt;/sub&amp;gt;]&amp;lt;sup&amp;gt;2-&amp;amp;nbsp;&amp;lt;/sup&amp;gt;, that is uranyl uranate.&amp;lt;ref&amp;gt;{{cite journal|author= Engmann R, de Wolff PM| title= The Crystal Structure of γ-UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;|journal=Acta Crystallographica|year= 1963|volume=16|issue= 10|page=993| doi= 10.1107/S0365110X63002656}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;/tr&amp;gt;&lt;br /&gt;
&amp;lt;/table&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;table  border=&amp;quot;1&amp;quot; cellpadding=&amp;quot;3&amp;quot; cellspacing=&amp;quot;0&amp;quot;&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
[[File:UO3 gamma env1.jpg|center|150px]]&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
&#039;&#039;The environment of the uranium atoms shown as yellow in the gamma form&#039;&#039;&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
[[File:UO3 gamma rings.jpg|center|150px]]&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
&#039;&#039;The chains of U&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; rings in the gamma form in layers, alternate layers running at 90 degrees to each other. These chains are shown as containing the yellow uranium atoms, in an octahedral environment which are distorted towards square planar by an elongation of the [[Axis of rotation|axial]] [[oxygen]]-[[uranium]] bonds.&#039;&#039;&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;/table&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Delta====&lt;br /&gt;
&amp;lt;table  border=&amp;quot;1&amp;quot; cellpadding=&amp;quot;3&amp;quot; cellspacing=&amp;quot;0&amp;quot;&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
[[File:UO3lattice.jpg|center|150px]]&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;td&amp;gt;&lt;br /&gt;
&#039;&#039;The delta (δ) form is a [[Cubic (crystal system)|cubic]] solid where the oxygen atoms are arranged between the uranium atoms.&#039;&#039;&amp;lt;ref&amp;gt;{{cite journal|author=M. T. Weller, P. G. Dickens, D. J. Penny |year=1988 |title=The structure of δ-UO&amp;lt;sub&amp;gt;3&amp;gt;&amp;lt;/sub&amp;gt; |journal=Polyhedron |volume=7|issue=3 |pages=243–244|doi=10.1016/S0277-5387(00)80559-8}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;/td&amp;gt;&lt;br /&gt;
&amp;lt;/tr&amp;gt;&lt;br /&gt;
&amp;lt;/table&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====High pressure form====&lt;br /&gt;
There is a high-pressure solid form with U&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; and U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; rings in it.&amp;lt;ref&amp;gt;&lt;br /&gt;
{{cite journal| author=Siegel S, Hoekstra HR, Sherry E |year=1966 |title=The crystal structure of high-pressure UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;|journal=Acta Crystallographica |volume=20| issue=2 |pages=292–295 |doi=10.1107/S0365110X66000562}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&amp;lt;ref&amp;gt;&#039;&#039;Gmelin Handbuch&#039;&#039; (1982) &#039;&#039;&#039;U-C1,&#039;&#039;&#039; 129–135.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Hydrates====&lt;br /&gt;
[[File:Uranium Trioxides.jpg|thumb|right|Hydrous and anhydrous forms of UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;]]&lt;br /&gt;
Several [[hydrates]] of uranium trioxide are known, e.g., UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;·6H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O.&amp;lt;ref name=&amp;quot;wheeler&amp;quot; /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Bond valence parameters===&lt;br /&gt;
It is possible by [[bond valence]] calculations&amp;lt;ref&amp;gt;[http://kristall.uni-mki.gwdg.de/softbv/references.html references]. Kristall.uni-mki.gwdg.de. Retrieved on 2011-07-19.&amp;lt;/ref&amp;gt; to estimate how great a contribution a given oxygen atom is making to the assumed valence of uranium.&amp;lt;ref&amp;gt;{{cite journal|author=Zachariasen|journal=J. Less Common Metals|year=1978|volume=62|pages= 1–7|doi=10.1016/0022-5088(78)90010-3|title=Bond lengths in oxygen and halogen compounds of d and f elements}}&amp;lt;/ref&amp;gt; Bond valence calculations use parameters which are estimated after examining a large number of crystal structures of uranium oxides (and related uranium compounds), note that the oxidation states which this method provides are only a guide which assists in the understanding of a crystal structure.&lt;br /&gt;
&lt;br /&gt;
The formula to use is&lt;br /&gt;
&lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
s = e^{-\frac{R-R_O}{B}}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The sum of the &#039;&#039;s&#039;&#039; values is equal to the oxidation state of the metal centre.&lt;br /&gt;
&lt;br /&gt;
For uranium binding to oxygen the constants R&amp;lt;sub&amp;gt;O&amp;lt;/sub&amp;gt; and B are tabulated in the table below. For each oxidation state use the parameters from the table shown below.&lt;br /&gt;
&lt;br /&gt;
{| style=&amp;quot;margin:auto;&amp;quot;&lt;br /&gt;
|-&lt;br /&gt;
!Oxidation state!!R&amp;lt;sub&amp;gt;O&amp;lt;/sub&amp;gt;!!B&lt;br /&gt;
|-&lt;br /&gt;
|U(VI)||2.08 Å||0.35&lt;br /&gt;
|-&lt;br /&gt;
|U(V)||2.10 Å||0.35&lt;br /&gt;
|-&lt;br /&gt;
|U(IV)||2.13 Å||0.35&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
It is possible to do these calculations on paper or software.&amp;lt;ref&amp;gt;[http://www.ccp14.ac.uk/ccp/web-mirrors/i_d_brown/ www.ccp14.ac.uk/ccp/web-mirrors/i_d_brown Free-download software]. Ccp14.ac.uk. Retrieved on 2011-07-19.&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[http://www.ccp14.ac.uk/solution/bond_valence/index.html www.ccp14.ac.uk/solution/bond_valence/ Free-download software mirror]. Ccp14.ac.uk (2001-08-13). Retrieved on 2011-07-19.&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Molecular forms===&lt;br /&gt;
While uranium trioxide is encountered as a polymeric solid under ambient conditions, some work has been done on the molecular form in the gas phase, in matrix isolations studies, and computationally.&lt;br /&gt;
&lt;br /&gt;
====Gas phase====&lt;br /&gt;
At elevated temperatures gaseous UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; is in [[chemical equilibrium|equilibrium]] with solid [[triuranium octaoxide|U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt;]] and molecular [[oxygen]].&lt;br /&gt;
&lt;br /&gt;
::2 U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt;(s) + O&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;(g) {{eqm}} 6 UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;(g)&lt;br /&gt;
&lt;br /&gt;
With increasing temperature the equilibrium is shifted to the right. This system has been studied at temperatures between 900&amp;amp;nbsp;°C and 2500&amp;amp;nbsp;°C. The vapor pressure of monomeric UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; in equilibrium with air and solid U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt; at ambient pressure, about 10&amp;lt;sup&amp;gt;−5&amp;lt;/sup&amp;gt;&amp;amp;nbsp;mbar (1 mPa) at 980&amp;amp;nbsp;°C, rising to 0.1&amp;amp;nbsp;mbar (10&amp;amp;nbsp;Pa) at 1400&amp;amp;nbsp;°C, 0.34&amp;amp;nbsp;mbar (34&amp;amp;nbsp;Pa) at 2100&amp;amp;nbsp;°C, 1.9&amp;amp;nbsp;mbar (193&amp;amp;nbsp;Pa) at 2300&amp;amp;nbsp;°C, and 8.1&amp;amp;nbsp;mbar (809&amp;amp;nbsp;Pa) at 2500&amp;amp;nbsp;°C.&amp;lt;ref&amp;gt;{{cite journal|author= Ackermann RJ, Gilles PW, Thorn RJ|title= High-Temperature Thermodynamic Properties of Uranium Dioxide|journal= Journal of Chemical Physics|year= 1956|volume= 25|issue= 6| page= 1089|doi= 10.1063/1.1743156}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal|author= Alexander CA|title= Volatilization of urania under strongly oxidizing conditions|journal= Journal of Nuclear Materials|year= 2005|volume= 346|issue= 2–3|pages= 312–318| doi= 10.1016/j.jnucmat.2005.07.013}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Matrix isolation====&lt;br /&gt;
Infrared spectroscopy of molecular UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; isolated in an argon matrix indicates a T-shaped structure ([[point group]] &#039;&#039;C&amp;lt;sub&amp;gt;2v&amp;lt;/sub&amp;gt;&#039;&#039;) for the molecule. This is in contrast to the commonly encountered &#039;&#039;D&amp;lt;sub&amp;gt;3h&amp;lt;/sub&amp;gt;&#039;&#039; [[molecular symmetry]] exhibited by most trioxides. From the force constants the authors deduct the U-O bond lengths to be between 1.76 and 1.79 [[ångström|Å]] (176 to 179 [[1 E-12 m|pm]]).&amp;lt;ref&amp;gt;{{cite journal|author= Gabelnick SD, Reedy GT, Chasanov MG| title= Infrared spectra of matrix-isolated uranium oxide species. II: Spectral interpretation and structure of UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;|journal= Journal of Chemical Physics|year= 1973|volume= 59|issue= 12|pages= 6397–6404|doi= 10.1063/1.1680018}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Computational study====&lt;br /&gt;
[[File:Uranium-trioxide-Pyykko-3D-balls-B.png|thumb|right|200px|The calculated geometry of molecular uranium trioxide has C&amp;lt;sub&amp;gt;2v&amp;lt;/sub&amp;gt; symmetry.]]&lt;br /&gt;
Calculations predict that the point group of molecular UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; is &#039;&#039;C&amp;lt;sub&amp;gt;2v&amp;lt;/sub&amp;gt;&#039;&#039;, with an axial bond length of 1.75 Å, an equatorial bond length of 1.83 Å and an angle of 161° between the axial oxygens. The more symmetrical &#039;&#039;D&amp;lt;sub&amp;gt;3h&amp;lt;/sub&amp;gt;&#039;&#039; species is a saddle point, 49 kJ/mol above the &#039;&#039;C&amp;lt;sub&amp;gt;2v&amp;lt;/sub&amp;gt;&#039;&#039; minimum. The authors invoke a second-order [[Jahn–Teller effect]] as explanation.&amp;lt;ref&amp;gt;{{cite journal|author= Pyykkö P, Li J|title= Quasirelativistic pseudopotential study of species isoelectronic to uranyl and the equatorial coordination of uranyl|journal= Journal of Physical Chemistry|year= 1994|volume= 98|issue= 18|pages= 4809–4813|doi = 10.1021/j100069a007}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
==Reactivity==&lt;br /&gt;
Uranium trioxide reacts at 400&amp;amp;nbsp;°C with [[freon-12]] to form [[chlorine]], [[phosgene]], [[carbon dioxide]] and [[uranium tetrafluoride]]. The freon-12 can be replaced with [[freon-11]] which forms [[carbon tetrachloride]] instead of carbon dioxide. This is a case of a hard perhalogenated [[freon]] which is normally considered to be inert being converted chemically at a moderate temperature.&amp;lt;ref&amp;gt;{{cite journal|author= Booth HS, Krasny-Ergen W, Heath RE| title= Uranium Tetrafluoride|journal= [[Journal of the American Chemical Society]]|year= 1946|volume= 68|issue= 10|pages= 1969–1970| doi= 10.1021/ja01214a028}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
:2 CF&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;Cl&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; + UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; → UF&amp;lt;sub&amp;gt;4&amp;lt;/sub&amp;gt; + CO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; + COCl&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; + Cl&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;&lt;br /&gt;
&lt;br /&gt;
:4 CFCl&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; + UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; → UF&amp;lt;sub&amp;gt;4&amp;lt;/sub&amp;gt; + 3 COCl&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; + CCl&amp;lt;sub&amp;gt;4&amp;lt;/sub&amp;gt; + Cl&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Uranium trioxide can be dissolved in a mixture of [[tributyl phosphate]] and [[thenoyltrifluoroacetone]] in [[supercritical carbon dioxide]], ultrasound was employed during the dissolution.&amp;lt;ref&amp;gt;{{cite journal|author= Trofimov TI, Samsonov MD, Lee SC, Myasoedov BF, Wai CM|title= Dissolution of uranium oxides in supercritical carbon dioxide containing tri-&#039;&#039;n&#039;&#039;-butyl phosphate and thenoyltrifluoroacetone|journal= Mendeleev Communications|year= 2001|volume= 11|issue= 4|pages= 129–130|doi= 10.1070/MC2001v011n04ABEH001468}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Electrochemical modification===&lt;br /&gt;
The reversible insertion of [[magnesium]] cations into the [[crystal structure|lattice]] of uranium trioxide by [[cyclic voltammetry]] using a [[graphite]] electrode modified with microscopic particles of the uranium oxide has been investigated. This experiment has also been done for U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt;. This is an example of [[electrochemistry]] of a solid modified [[electrode]], the experiment which used for uranium trioxide is related to a [[carbon paste electrode]] experiment. It is also possible to reduce uranium trioxide with [[sodium]] metal to form sodium uranium oxides.&amp;lt;ref&amp;gt;{{cite journal|doi=10.1149/1.2221232|title=Investigation of the Mechanism of Formation of Insertion Compounds of Uranium Oxides by Voltammetric Reduction of the Solid Phase after Mechanical Transfer to a Carbon Electrode|year=1992|last1=Dueber|first1=R. E.|journal=Journal of the Electrochemical Society|volume=139|issue=9|pages=2363}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
It has been the case that it is possible to insert [[lithium]]&amp;lt;ref&amp;gt;{{cite journal|title =Insertion compounds of uranium oxides| author = Dickens PG, Lawrence SD, Penny DJ, Powell AV| year= 1989| volume= 32–33| doi=10.1016/0167-2738(89)90205-1|pages =77–83|journal =Solid State Ionics}}&lt;br /&gt;
&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal|title =Lithium insertion into αUO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; and U&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;8&amp;lt;/sub&amp;gt;| author = Dickens, P.G. Hawke, S.V. Weller, M.T.| year= 1985| volume= 20| doi=10.1016/0025-5408(85)90141-2|issue =6|pages =635–641|journal =Materials Research Bulletin}}&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;{{cite journal|title =Hydrogen insertion compounds of UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;|author = Dickens, P.G. Hawke, S.V. Weller, M.T.| year= 1984| volume= 19| doi=10.1016/0025-5408(84)90120-X|issue =5|pages =543–547|journal =Materials Research Bulletin}}&amp;lt;/ref&amp;gt; into the uranium trioxide lattice by electrochemical means, this is similar to the way that some [[rechargeable]] [[lithium ion batteries]] work. In these rechargeable cells one of the electrodes is a metal oxide which contains a metal such as [[cobalt]] which can be reduced, to maintain the electroneutrality for each electron which is added to the electrode material a lithium ion enters the lattice of this oxide electrode.&lt;br /&gt;
&lt;br /&gt;
===Amphoterism and reactivity to form related uranium(VI) anions and cations===&lt;br /&gt;
Uranium oxide is [[amphoteric (chemistry)|amphoteric]] and reacts as [[acid]] and as a [[base (chemistry)|base]], depending on the conditions. &lt;br /&gt;
&lt;br /&gt;
;As an acid:&lt;br /&gt;
&lt;br /&gt;
:UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; + H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O → {{chem|UO|4|2−}} + 2 H&amp;lt;sup&amp;gt;+&amp;lt;/sup&amp;gt; &lt;br /&gt;
&lt;br /&gt;
Dissolving uranium oxide in a strong [[base (chemistry)|base]] like [[sodium hydroxide]] forms the doubly negatively charged [[uranate]] [[anion]] ({{chem|UO|4|2−}}). Uranates tend to concatenate, forming [[diuranate]], {{chem|U|2|O|7|2−}}, or other poly-uranates.&lt;br /&gt;
Important diuranates include [[ammonium diuranate]] ((NH&amp;lt;sub&amp;gt;4&amp;lt;/sub&amp;gt;)&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;U&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;7&amp;lt;/sub&amp;gt;), [[sodium diuranate]] (Na&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;U&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;7&amp;lt;/sub&amp;gt;) and&lt;br /&gt;
[[magnesium diuranate]] (MgU&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;7&amp;lt;/sub&amp;gt;), which forms part of some [[yellowcake]]s. It is worth noting that uranates of the form M&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;UO&amp;lt;sub&amp;gt;4&amp;lt;/sub&amp;gt; do &#039;&#039;not&#039;&#039; contain {{chem|UO|4|2−}} ions, but rather flattened UO&amp;lt;sub&amp;gt;6&amp;lt;/sub&amp;gt; octahedra, containing a uranyl group and bridging oxygens.&amp;lt;ref&amp;gt;{{cite book|last= Cotton|first= Simon|year = 1991|title = Lanthanides and Actinides|publisher= Oxford University Press|location = New York|page= 128|isbn= 0-19-507366-5}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
;As a base:&lt;br /&gt;
&lt;br /&gt;
:UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; + H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O → {{chem|UO|2|2+}} + 2 OH&amp;lt;sup&amp;gt;−&amp;lt;/sup&amp;gt;&lt;br /&gt;
Dissolving uranium oxide in a strong acid like [[sulfuric acid|sulfuric]] or [[nitric acid]] forms the double positive charged [[uranyl]] [[cation]]. The [[uranyl nitrate]] formed (UO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;(NO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;)&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;·6H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O) is soluble in [[ether]]s, [[alcohol]]s, [[ketone]]s and [[ester]]s; for example, [[tributylphosphate]]. This solubility is used to separate uranium from other elements in [[nuclear reprocessing]], which begins with the dissolution of [[nuclear fuel]] rods in [[nitric acid]]. The [[uranyl nitrate]] is then converted to uranium trioxide by heating.&lt;br /&gt;
&lt;br /&gt;
From [[nitric acid]] one obtains [[uranyl nitrate]], &#039;&#039;trans&#039;&#039;-UO&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;(NO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;)&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;·2H&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O, consisting of eight-coordinated uranium with two [[bidentate]] nitrato ligands and two water ligands as well as the familiar O=U=O core.&lt;br /&gt;
&lt;br /&gt;
==Uranium oxides in ceramics==&lt;br /&gt;
UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;-based ceramics become green or black when fired in a reducing atmosphere and yellow to orange when fired with oxygen. Orange-coloured [[Fiestaware]] is a well-known example of a product with a uranium-based glaze. UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;-has also been used in formulations of [[Vitreous enamel|enamel]], [[uranium glass]], and [[porcelain]]. &lt;br /&gt;
&lt;br /&gt;
Prior to 1960, UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; was used as an agent of crystallization in crystalline coloured glazes. It is possible to determine with a [[Geiger counter]] if a glaze or glass was made from UO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
{{reflist|2}}&lt;br /&gt;
&lt;br /&gt;
{{Uranium compounds}}&lt;br /&gt;
&lt;br /&gt;
[[Category:Uranyl compounds]]&lt;br /&gt;
[[Category:Oxides]]&lt;/div&gt;</summary>
		<author><name>194.254.61.41</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Separation_logic&amp;diff=13070</id>
		<title>Separation logic</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Separation_logic&amp;diff=13070"/>
		<updated>2013-11-20T12:42:25Z</updated>

		<summary type="html">&lt;p&gt;194.254.61.40: Disambiguated: adjunction → adjunction (category theory)&lt;/p&gt;
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&lt;div&gt;{{nobots}}&lt;br /&gt;
{{Copy to Wikimedia Commons}}&lt;br /&gt;
== Summary ==&lt;br /&gt;
{{Information&lt;br /&gt;
| Description    = {{en|1=Pole-zero diagram with poles at origin and zeros at &amp;lt;math&amp;gt;z_1=-\frac{1}{2}+j\frac{\sqrt{3} }{2}, z_2=-\frac{1}{2}-j\frac{\sqrt{3} }{2}&amp;lt;/math&amp;gt;.}}&lt;br /&gt;
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== Licensing ==&lt;br /&gt;
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		<author><name>194.254.61.40</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Baxter_permutation&amp;diff=28297</id>
		<title>Baxter permutation</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Baxter_permutation&amp;diff=28297"/>
		<updated>2013-07-01T12:58:35Z</updated>

		<summary type="html">&lt;p&gt;194.254.61.45: /* Other properties */ fixed typo&lt;/p&gt;
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&lt;div&gt;{{Underlinked|date=December 2012}}&lt;br /&gt;
&lt;br /&gt;
The &#039;&#039;&#039;Eddy Break-Up Model&#039;&#039;&#039; is used in [[combustion]] engineering.&amp;lt;ref&amp;gt;{{cite web |title = Combustion | work = CFD Online website | publisher = CFD Online | url = http://www.cfd-online.com/Wiki/Combustion#Eddy_Break-Up_model | accessdate = 23 February 2013}}&amp;lt;/ref&amp;gt; &lt;br /&gt;
Combustion modeling is one of the most important topics in demand. Its wide range of applications has attracted the attentions of most of the researchers around the globe. In most of the combustion systems fuel and oxygen (or air) are separately supplied in the combustion chamber. Due to this chemical reaction and combustion occurs simultaneously in the [[combustion chamber]]. However, it is evident that the rate of the chemical reaction is faster than the rate of mixing fuel and oxygen. Which interprets that rate of combustion is controlled by rate of mixing. Such cases where formation of pre-mixture is difficult are called diffusion combustion or diffusion flames.&lt;br /&gt;
&lt;br /&gt;
==Diffusion flames==&lt;br /&gt;
[[Diffusion]] flames can be [[laminar flow|laminar]] diffusion flames (e.g. Candles and matches burning in the air etc.) and [[turbulent diffusion]] flames (e.g. furnaces, turbo-machinery, some liquid-fuel rocket engines and internal combustion engines) depending on the nature of mixed gas flow. Further, the flame shapes of this kind of burning are divided into two categories oxygen-rich diffusion flames and oxygen-deficient. Flame shapes also depend on the discharging velocity of fuel and air. Eddies inside the air flow plays an important role in deciding the flame shapes. With increase in discharging velocity of fuel into the air laminar diffusion flames tends to become turbulent diffusion flames. This leads to an increase in the flame height and finally it gains a maximum height (critical limit). After this limit further increase in the jet velocity of the fuel adds to instability of turbulent flame. The position where laminar flame is changing into turbulent flame is termed as “break-up point”. Increase in jet velocity after critical limit results in decrease of the whole flame height. Beyond this point the height of the break-up point reaches a certain value where even after increasing the jet velocity of fuel the flame height will not change. Studies by Hawthorne et al. prove that the chemical reaction rates in turbulent diffusion flames and in laminar diffusion flames are almost similar.&lt;br /&gt;
&lt;br /&gt;
==Problems==&lt;br /&gt;
Combustion plays a vital role in many applications and a proper understanding of its effect on the system can be very helpful in designing the newer technologies and improving the existing ones. Now a day, numerical modeling is a most effective tool for understanding and studying such problems. But a number of problems are associated with it such as:&lt;br /&gt;
* The control equations associated are very complicated.&lt;br /&gt;
* Large number of control equation need to be studied which include knowledge of both fluid dynamics and chemistry of chemical reactions during combustion.&lt;br /&gt;
* Involvement of huge number of components in the chemical reactions.&lt;br /&gt;
* There are serious time-scale problems because of mismatch in chemical reaction rate and fluid transport rate.&lt;br /&gt;
* Effect of several other processes like heat transfer, radiation, convection and diffusion.&lt;br /&gt;
&lt;br /&gt;
==Combustion models==&lt;br /&gt;
In this context, many combustion models are purposed. There is a big list of such models in literature but due to simplicity “The eddy break-up (EBU) model” proposed by Spalding and later modified by Magnusen and Hiertager became one among the most popular models. It is based on the assumption that the reaction rate is controlled by turbulent mixing because of the dependence of reaction rate on the mixing of the turbulent eddies. It is well documented that there are strong interactions between the turbulent flow and reactions. The heat released during the reaction affects the density and hence turbulence. While the turbulent flow induces changes concentration and temperature pulsations which enhance the mixing and heat transfer and affect the reaction rate. Turbulence also results in deformations in burning flame surface due to which folding occurs and the surface is also broken into pieces of different sizes. This enhances the total surface area of the flame and thus increases the combustible mixed gases burnt per unit time. Due to this, turbulent flames grow at a much faster rate than that of laminar flames.&lt;br /&gt;
&lt;br /&gt;
==Formulae==&lt;br /&gt;
Spalding (1971)&amp;lt;ref&amp;gt;Spalding D.B., Mixing and chemical reaction in steady confined turbulent flames, Thirteenth symposium (international) on combustion, The combustion institute, pp. 649–657, 1971&amp;lt;/ref&amp;gt; defined rate of fuel consumption as a function of local flow properties of fuel and oxidant. This model is based on a single step global infinitely fast stochiometric chemical reaction.&lt;br /&gt;
&lt;br /&gt;
1&amp;amp;nbsp;kg of fuel + s kg of oxidant→(1+s) kg of products&lt;br /&gt;
&lt;br /&gt;
For &#039;&#039;m&amp;lt;sub&amp;gt;j&amp;lt;/sub&amp;gt;&#039;&#039; as mass fraction and &#039;&#039;M&amp;lt;sub&amp;gt;j&amp;lt;/sub&amp;gt;&#039;&#039; as molecular weight of species &#039;&#039;j&#039;&#039;; local density of mixture &#039;&#039;(ρ)&#039;&#039; is dependent on concentration of reactant and products and temperature of mixture. It can be mathematically calculated as:&lt;br /&gt;
&lt;br /&gt;
{{NumBlk|:|&amp;lt;math&amp;gt;{\rho = \frac{P}{RT\sum_{\text{for all j}}{\frac{m_j}{M_j}}}}&amp;lt;/math&amp;gt;|{{EquationRef|1}}}}&lt;br /&gt;
                                                                                                                 &lt;br /&gt;
It is used to express the turbulent dissipation rate of fuel &#039;&#039;(R&amp;lt;sub&amp;gt;fu&amp;lt;/sub&amp;gt;)&#039;&#039;, oxygen &#039;&#039;(R&amp;lt;sub&amp;gt;ox&amp;lt;/sub&amp;gt;)&#039;&#039; and products &#039;&#039;(R&amp;lt;sub&amp;gt;pr&amp;lt;/sub&amp;gt;)&#039;&#039; as:&lt;br /&gt;
{{NumBlk|:|&amp;lt;math&amp;gt;{R_{fu}=-C_R\rho m_{fu}\frac{\varepsilon}{k}}&amp;lt;/math&amp;gt;|{{EquationRef|2}}}}&lt;br /&gt;
{{NumBlk|:|&amp;lt;math&amp;gt;{R_{ox}=-C_R\rho \frac{m_{ox}}{s}\frac{\varepsilon}{k}}&amp;lt;/math&amp;gt;|{{EquationRef|3}}}}&lt;br /&gt;
{{NumBlk|:|&amp;lt;math&amp;gt;{R_{pr}=-C_R^{&#039;}\rho \frac{m_{pr}}{(1+s)}\frac{\varepsilon}{k}}&amp;lt;/math&amp;gt;|{{EquationRef|4}}}}            &lt;br /&gt;
Where &#039;&#039;k&#039;&#039; is the turbulent kinetic energy,   is the rate of dissipation of &#039;&#039;k&#039;&#039;,   and   are model constants (value varies from .35 to 4). The reaction rate of fuel is considered to be smallest among all and is given by:&lt;br /&gt;
{{NumBlk|:|&amp;lt;math&amp;gt;{S_{fu}=-\rho\frac{\varepsilon}{k}min\biggl[C_Rm_{fu}, C_R\frac{m_{ox}}{s}, C_R^{&#039;}\frac{m_{pr}}{1+s}\biggr]}&amp;lt;/math&amp;gt;|{{equationRef|5}}}}&lt;br /&gt;
The model also leads to determination of mass fraction of product and oxygen using transport equation of mixture fraction &#039;&#039;(f)&#039;&#039;&lt;br /&gt;
{{NumBlk|:|&amp;lt;math&amp;gt;{ f=\frac{[{sm}_{fu}-m_{ox}]-{[{sm}_{fu}-m_{ox}]}_0}{{[{sm}_{fu}-m_{ox}]}_1-{[{sm}_{fu}-m_{ox}]}_0}}&amp;lt;/math&amp;gt;|{{EquationRef|6}}}}&lt;br /&gt;
For solving we need to first find the stociometric mixture fraction &#039;&#039;(f&amp;lt;sub&amp;gt;st&amp;lt;/sub&amp;gt;)&#039;&#039; when there is no fuel and oxygen is present in products, which is defined as &lt;br /&gt;
{{NumBlk|:|&amp;lt;math&amp;gt;{f_{st}=\frac{m_{ox, 0}}{{sm}_{fu, 1}+m_{ox, 0}}}&amp;lt;/math&amp;gt;|{{EquationRef|7}}}}&lt;br /&gt;
{{NumBlk|:|&amp;lt;math&amp;gt;{{If} f &amp;lt; f_{st}, f=\frac{-m_{ox}+m_{ox, 0}}{{sm}_{fu, 1}+m_{ox, 0}}}&amp;lt;/math&amp;gt;|{{EquationRef|8}}}}&lt;br /&gt;
{{NumBlk|:|&amp;lt;math&amp;gt;{{If} f &amp;gt; f_{st}, f=\frac{-{sm}_{fu, 1}+m_{ox, 0}}{{sm}_{fu, 1}+m_{ox, 0}}}&amp;lt;/math&amp;gt;|{{EquationRef|9}}}}&lt;br /&gt;
Above equations (7-9) not only shows linear relation of mixture fraction with mass fraction of oxygen and products but also helps in predicting their values. Magnussen and Hjertager (1976)&amp;lt;ref&amp;gt;Magnussen B.F. and Hjertager B.H., On the mathematical modeling of turbulent combustion with special emphasis on shoot formation and combustion, Sixth symposium (International) on combustion, The combustion institute, pp. 719–729, 1976&amp;lt;/ref&amp;gt; utilize this model and conformity of experimental results with predictions supports this model. Several other researchers also justified the beauty of this model for fairly good predictions which are quite close to experimental results. Hence, this model is a topic of top priority due to its simplicity, steady convergence, and implementation in [[computational fluid dynamics]] (CFD) procedures.&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&amp;lt;references/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
==Further reading==&lt;br /&gt;
1. Versteeg H.K. and Malalasekera W., An introduction to computational fluid dynamics, ISBN 978-81-317-2048-6.&lt;br /&gt;
&lt;br /&gt;
2. Gao Y. and Chow W.K., A Brief Review on combustion modeling, International Journal on Architectural Science, Volume 6, Number 2, p.&amp;amp;nbsp;38-69, 2005.&lt;br /&gt;
&lt;br /&gt;
[[Category:Combustion]]&lt;br /&gt;
[[Category:Combustion engineering]]&lt;/div&gt;</summary>
		<author><name>194.254.61.45</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Shifted_Gompertz_distribution&amp;diff=253426</id>
		<title>Shifted Gompertz distribution</title>
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		<updated>2012-07-31T16:13:47Z</updated>

		<summary type="html">&lt;p&gt;194.254.137.114: &lt;/p&gt;
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