Crown graph - Revision history

en>David Eppstein: Undid revision 614468491 by 122.166.186.254 (talk) that's not an application, Ferrers is for digraphs, and boxicity needs a source

2014-06-26T06:55:27Z

Undid revision 614468491 by 122.166.186.254 (talk) that's not an application, Ferrers is for digraphs, and boxicity needs a source

← Older revision		Revision as of 08:55, 26 June 2014
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	~~The '''Girolami method'''~~,~~<ref>Gregory S. Girolami~~, ~~A Simple "Back of the Envelope" Method for Estimating the Densities~~ and ~~Molecular Volume of Liquids and Volumes, J~~. ~~of Chemical Education, 71(11), 962-964 (1994)</ref> named after Gregory Girolami, is~~ a ~~predictive method for estimating [[Density\|densities]] of pure [[liquid]] components at [[room temperature]]. The objective of this method is~~ the ~~simple prediction of~~ the ~~density and not high precision.~~		Nurse Researcher Morgado from La Prairie, has pastimes which includes sun tanning, vigrx plus and collecting artwork. Has completed a great around the world voyage that included visting the Dong Phayayen-Khao Yai Forest Complex.<br><br>My web-site :: top male enlargement pills; [http://vigrxplusreview.info http://vigrxplusreview.info],

	~~== Procedure ==~~
	The method uses purely additive volume contributions for single atoms and additional correction factors for components with special [[functional group]]s which cause a volume contraction and therefore a higher density. The Girolami method can be described as a mixture of an atom and [[group contribution method]].

	~~=== Atom contributions ===~~
	~~The method uses the following contributions for the different atoms:~~

	~~{\| class="wikitable"~~
	~~! Element !! Relative volume~~ <br />V<~~sub>i</sub~~>
	\|-
	~~\| [[Hydrogen]] \|\| 1~~
	\|-
	~~\| [[Lithium]] to [[Fluorine]] \|\| 2~~
	\|-
	~~\| [[Sodium]] to [[Chlorine]] \|\| 4~~
	\|-
	~~\| [[Potassium]] to [[Bromine]] \|\| 5~~
	\|-
	~~\| [[Rubidium]] to [[Iodine]] \|\| 7.5~~
	\|-
	~~\| [[Cesium]] to [[Bismuth]] \|\| 9~~
	\|}

	~~A scaled molecular volume is calculated by~~

	:~~<math>V_S \,=\, \sum_i V_i</math>~~

	~~and the density is derived by~~

	:~~<math>d \,=\, \frac{M}{5 \cdot V_S}</math>~~

	~~with the [[molecular weight]] M. The scaling factor '''5''' is used to obtain the density in g·cm<sup>−3</sup>.~~

	~~=== Group contribution ===~~
	~~For some components Girolami found smaller volumes and higher densities than calculated solely by the atom contributions. For components with~~
	* a hydroxylic function ([[Alcohol]]s)
	* a carboxylic function ([[Carboxylic acid]]s)
	* a primary or secondary [[amine]] function
	* an [[amide]] group (incl. amides substituted at the [[nitrogen]])
	* a [[sulfoxide]] group
	* a [[sulfone]] group
	* a ring (non-condensed),
	~~it is sufficient to add 10% to the density obtained by the main equation. For sulfone groups it is necessary to use this factor twice (20%).~~

	~~Another specific case are condensed ring systems like [[Naphthalene]]. The density has to increased by 7.5% for every ring~~; ~~for Naphthalene the resulting factor would be 15%.~~

	~~If multiple corrections are needed their factors have to be added but not over 130% in total.~~

	~~== Example calculation ==~~

	~~{\| class="wikitable"~~
	~~! Component~~
	~~! M<br />~~[~~<sup>g</sup>/<sub>mol</sub>]~~
	~~! Volume V<sub>S</sub>~~
	~~! Corrections~~
	~~! Calculated density<br />[g·cm<sup>−3</sup>]~~
	~~! Exp. density<br />[g·cm<sup>−3</sup>]~~
	\|-
	~~\| [[Cyclohexanol]]~~
	~~\| 100~~
	~~\| (6×2)+(13×1)+(1×2)=26~~
	~~\| One ring and a hydroxylic group = 120%~~
	~~\| d=<sup>1.2*100</sup>/<sub>5×26</sub>=0.92~~
	~~\| 0.962~~
	\|-
	~~\| Dimethylethylphosphine~~
	~~\| 90~~
	~~\| (4×2)+(11×1)+(1×4)=23~~
	~~\| No corrections~~
	~~\| d=<sup>90</sup>/<sub>5×23</sub>=0.78~~
	~~\| 0.76~~
	\|-
	~~\| [[Ethylenediamine]]~~
	~~\| 60~~
	~~\| (2×2)+(8×1)+(2×2)=16~~
	~~\| Two primary amine groups = 120%~~
	~~\| d=<sup>1.2×60</sup>/<sub>5×16</sub>=0.90~~
	~~\| 0.899~~
	\|-
	~~\| [[Sulfolane]]~~
	~~\| 120~~
	~~\| (4×2)+(8×1)+(2×2)+(1×4)=24~~
	~~\| One ring and two S=O bonds = 130%~~
	~~\| d=<sup>1.3×120</sup>/<sub>5×24</sub>=1.30~~
	~~\| 1.262~~
	\|-
	~~\| [[1-Bromonaphthalene]]~~
	~~\| 207~~
	~~\| (10×2)+(7×1)+(1×5)=32~~
	~~\| Two condensed rings = 115%~~
	~~\| d=<sup>1,15×207</sup>/<sub>5×32<~~/~~sub>=1.49~~
	~~\| 1.483~~
	\|}

	~~== Quality ==~~
	~~The author has given a mean quadratic error (RMS) of 0.049 g·cm<sup>−3<~~/~~sup> for 166 checked components. Only for two components ([[acetonitrile]] and [[dibromochloromethane]]) has an error greater than 0~~.~~1 g·cm <sup>−3</sup> been found.~~

	~~==References==~~
	~~<references />~~

	~~== External links ==~~
	* [http://~~www.aim.env.uea~~.~~ac.uk/aim/density/density.php Online calculator for the Girolami model]~~

	~~{{DEFAULTSORT:Girolami Method}}~~
	~~[[Category:Thermodynamic models]~~]

en>David Eppstein: /* Examples */ Schläfli double six

2013-07-28T18:42:42Z

Examples: Schläfli double six

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en>MathsPoetry: interwiki

2012-04-20T08:48:32Z

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