134.96.223.199: /* L2,1 norm */

2014-11-27T14:51:21Z

L2,1 norm

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147.27.11.39: /* Induced norm */

2014-02-21T11:50:52Z

Induced norm

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76.111.160.24: /* Equivalence of norms */ Changed HTML to LaTeX

2014-01-12T20:31:46Z

Equivalence of norms: Changed HTML to LaTeX

← Older revision		Revision as of 22:31, 12 January 2014
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	Hi, I'~~m Tomiko household~~ is ~~not~~ the ~~name on my birth certificate~~. ~~What his friends~~ and ~~him love~~ is to ~~arrange flowers~~ an ~~excellent he~~ is ~~intending~~ to ~~cash~~ with ~~things~~. ~~After being~~ the ~~particular my project~~ for ~~years I came~~ to be ~~a hour or so assistant~~. ~~For a while I~~'~~ve held it~~'~~s place in Maine~~. ~~I am running and maintaining a blog here~~: ~~http~~://~~beautifulblankpages~~.~~blogspot~~.~~com~~
			In a [[chemical reaction]], the '''limiting reagent''', also known as the "'''limiting reactant'''", is the [[chemical substance\|substance]] which is totally consumed when the chemical reaction is complete. The amount of product formed is '''limited''' by this reagent since the reaction cannot proceed further without it. If one or more other reagents are present in excess of the quantities required to react with the limiting reagent, they are described as ''excess reagents'' or excess reactants.

			The limiting reagent must be identified in order to calculate the [[yield (chemistry)\|percentage yield]] of a reaction, since the theoretical yield is defined as the amount of product obtained when the limiting reagent reacts completely.

			Given the balanced [[chemical equation]] which describes the reaction, there are several equivalent ways to identify the limiting reagent and evaluate the excess quantities of other reagents.

			==Method 1: Comparison of reactant amounts==
			This method is most useful when there are only two reactants. One reactant (A) is chosen, and the balanced chemical equation is used to determine the amount of the other reactant (B) necessary to react with A. If the amount of B actually present exceeds the amount required, then B is in excess and A is the limiting reagent. If the amount of B present is less than required, then B is the limiting reagent.

			=== Example for two reactants ===
			Consider the [[combustion]] of [[benzene]], represented by the following [[chemical equation]]:

			:<math>2 C_6H_6 + 15 O_2 \rightarrow 12 CO_2 + 6 H_2O</math>
			This means that 15 [[Mole (unit)\|mol]] molecular [[oxygen]] (<math>O_2</math>) is required to react with 2 mol benzene (<math>C_6H_6</math>)

			The amount of oxygen required for other quantities of benzene can be calculated using [[cross-multiplication]] (the rule of three). For example,
			if 1.5 mol <math>C_6H_6</math> is present, 11.25 mol <math>O_2</math> is required:

			:<math> 1.5 \ \mbox{mol}\,C_6H_6 \times \frac{15 \ \mbox{mol}\,O_2}{2 \ \mbox{mol}\,C_6H_6} = 11.25 \ \mbox{mol}\,O_2\ </math>

			If in fact 18 mol <math>O_2</math> are present, there will be an excess of (18 - 11.25) = 6.75 mol of unreacted oxygen when all the benzene is consumed. Benzene is then the limiting reagent.

			This conclusion can be verified by comparing the mole ratio of <math>O_2</math> and <math>C_6H_6</math> required by the balanced equation with the mole ratio actually present:

			:required: <math>\frac{mol O_2}{mol C_6H_6}</math> = <math>\frac{15 mol O_2}{2 mol C_6H_6}=7.5 mol O_2</math>

			:actual: <math>\frac{mol O_2}{mol C_6H_6}</math> = <math>\frac{18 mol O_2}{1.5 mol C_6H_6}=12 mol O_2</math>

			Since the actual ratio is larger than required, <math>O_2</math> is the reagent in excess, which confirms that benzene is the limiting reagent.

			==Method 2: Comparison of product amounts which can be formed from each reactant==
			In this method the chemical equation is used to calculate the amount of one product which can be formed from each reactant in the amount present. The limiting reactant is the one which can form the smallest amount of the product considered. This method can be extended to any number of reactants more easily than the first method.

			===Example===

			20.0 g of [[iron (III) oxide]] (Fe<sub>2</sub>O<sub>3</sub>) are reacted with 8.00 g [[aluminium]] (Al) in the following [[thermite reaction]]?

			:<math>Fe_2O_3(s) + 2 Al(s) \rightarrow 2 Fe(l) + Al_2O_3(s)\,</math>

			Since the reactant amounts are given in grams, they must be first converted into moles for comparison with the chemical equation, in order to determine how many moles of Fe can be produced from either reactant.

			''Moles produced of <math>Fe</math> from reactant <math>Fe_2O_3</math>''

			:<math>mol Fe_2O_3 = \frac{grams Fe_2O_3}{g/mol Fe_2O_3}\,</math>

			:<math>mol Fe_2O_3 = \frac{20.0 g}{159.7 g/mol} = 0.125 mol\,</math>

			:<math> mol Fe = 0.125 \ \mbox{mol}\,Fe_2O_3 \times \frac{2 \ \mbox{mol}\,Fe}{1 \ \mbox{mol}\,Fe_2O_3} = 0.250\ \mbox{mol}\,Fe\ </math>

			''Moles produced of <math>Fe</math> from reactant <math>Al</math>''

			:<math>mol Al = \frac{grams Al}{g/mol Al}\,</math>

			:<math>mol Al = \frac{8.00 g}{26.98 g/mol} = 0.297 mol\,</math>

			:<math> mol Fe = 0.297 \ \mbox{mol}\,Al \times \frac{2 \ \mbox{mol}\,Fe}{2 \ \mbox{mol}\,Al} = 0.297\ \mbox{mol}\,Fe\ </math>

			There is enough Al to produce 0.297 mol Fe, but only enough Fe<sub>2</sub>O<sub>3</sub> to produce 0.250 mol Fe. This means that the amount of Fe actually produced is limited by the Fe<sub>2</sub>O<sub>3</sub> present, which is therefore the limiting reagent.

			===Shortcut===
			It can be seen from the example above that the amount of product (Fe) formed from each reagent X (Fe<sub>2</sub>O<sub>3</sub> or Al) is proportional to the quantity

			<math>\frac{\mbox{Moles of Reagent X }}{\mbox{Coefficient of Reagent X}}</math>

			This suggests a shortcut which works for any number of reagents. Just calculate this formula for each reagent, and the reagent that has the lowest value of this formula is the limiting reagent.

			==References==
			*Zumdahl, Steven S. ''Chemical Principles''. 4th ed. New York: Houghton Mifflin Company, 2005. ISBN 0-618-37206-7.

			==See also==
			* [[Limiting factor]]

			[[Category:Chemical reactions]]
			[[Category:Stoichiometry]]

87.0.228.42: /* Definition */

2012-08-17T01:40:26Z

Definition

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Hi, I'm Tomiko household is not the name on my birth certificate. What his friends and him love is to arrange flowers an excellent he is intending to cash with things. After being the particular my project for years I came to be a hour or so assistant. For a while I've held it's place in Maine. I am running and maintaining a blog here: http://beautifulblankpages.blogspot.com

Matrix norm - Revision history

134.96.223.199: /* L2,1 norm */

147.27.11.39: /* Induced norm */

76.111.160.24: /* Equivalence of norms */ Changed HTML to LaTeX

87.0.228.42: /* Definition */