en>Tobias Bergemann: Disambiguate John Mather to John Mather (mathematician)

2013-05-29T11:12:48Z

Disambiguate John Mather to John Mather (mathematician)

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	~~Alyson~~ is ~~what my spouse enjoys~~ to ~~contact me~~ but ~~I don't like when people use my full title~~. He is an ~~information officer~~. ~~clairvoyants -~~ [~~http~~:/~~/www~~.~~article~~-~~galaxy~~.~~com~~/~~profile~~.~~php?a~~=~~143251 www~~.~~article~~-~~galaxy~~.~~com~~] ~~- My wife~~ and ~~I reside~~ in ~~Mississippi~~ and ~~I love every working day residing here~~. ~~The preferred pastime~~ for ~~him and his children~~ is to ~~play lacross~~ and ~~he'll be beginning some thing else alongside~~ with it.<br><br>~~Take a~~ ~~psychics online;~~ [~~http~~://si.~~dgmensa~~.~~org~~/xe/~~index~~.~~php?document_srl=48014&mid=c0102 si~~.~~dgmensa~~.~~org~~]~~, look at my web page~~ ... ~~best psychics (~~[http://~~brazil~~.~~amor-amore~~.~~com~~/~~irboothe click the next web page~~])		The '''Mayo–Lewis equation''' or '''copolymer equation''' in [[polymer chemistry]] describes the distribution of [[monomer]]s in a [[copolymer]]:<ref>''Copolymerization. I. A Basis for Comparing the Behavior of Monomers in Copolymerization; The Copolymerization of Styrene and Methyl Methacrylate'' Frank R. Mayo and Frederick M. Lewis [[J. Am. Chem. Soc.]]; '''1944'''; 66(9) pp 1594 - 1601; {{DOI\|10.1021/ja01237a052}}</ref> It is named for [[Frank R. Mayo]] and [[Frederick M. Lewis]].

			Taking into consideration a monomer mix of two components <math>M_1\,</math> and <math>M_2\,</math> and the four different reactions that can take place at the reactive chain end terminating in either monomer (<math>M^*\,</math>) with their [[reaction rate constant]]s <math>k\,</math>:

			:<math>M_1^* + M_1 \xrightarrow{k_{11}} M_1M_1^* \,</math>
			:<math>M_1^* + M_2 \xrightarrow{k_{12}} M_1M_2^* \,</math>
			:<math>M_2^* + M_2 \xrightarrow{k_{22}} M_2M_2^* \,</math>
			:<math>M_2^* + M_1 \xrightarrow{k_{21}} M_2M_1^* \,</math>
			and with '''reactivity ratios''' defined as:

			:<math>r_1 = \frac{k_{11}}{k_{12}} \,</math>
			:<math>r_2 = \frac{k_{22}}{k_{21}} \,</math>

			the copolymer equation is given as:

			:<math>\frac {d\left [M_1 \right]}{d\left [M_2\right]}=\frac{\left [M_1\right]\left (r_1\left[M_1\right]+\left [M_2\right]\right)}{\left [M_2\right]\left (\left [M_1\right]+r_2\left [M_2\right]\right)}</math>

			with the [[concentration]] of the components given in [[square bracket]]s. The equation gives the copolymer composition at any instant during the polymerization.

			==Limiting cases==
			From this equation several [[limiting case]]s can be derived:
			* <math>r_1 = r_2 >> 1 \,</math> with both reactivity ratios very high the two monomers have no inclination to react to each other except with themselves leading to a mixture of two [[homopolymer]]s.
			* <math>r_1 = r_2 > 1 \,</math> with both ratios larger than 1, homopolymerization of component M_1 is favored but in the event of a crosspolymerization by M_2 the chain-end will continue as such giving rise to [[block copolymer]]
			* <math>r_1 = r_2 \approx 1 \,</math> with both ratios around 1, monomer 1 will react as fast with another monomer 1 or monomer 2 and a [[random copolymer]] results.
			* <math>r_1 = r_2 \approx 0 \,</math> with both values approaching 0 the monomers are unable to react in homopolymerization and the result is an [[alternating polymer]]
			* <math>r_1 >> 1 >> r_2 \,</math> In the initial stage of the copolymerization monomer 1 is incorporated faster and the copolymer is rich in monomer 1. When this monomer gets depleted, more monomer 2 segments are added. This is called '''composition drift'''.

			An example is [[maleic anhydride]] and [[stilbene]], with reactivity ratio:
			* Maleic anhydride (<math>r_1\,</math> = 0.08) & cis-stilbene (<math> r_2 ,</math> = 0.07)
			* Maleic anhydride (<math>r_1\,</math> = 0.03) & trans-stilbene (<math> r_2 ,</math> = 0.03)

			Both of these compounds do not homopolymerize and instead, they react together to give exclusively alternating copolymer.

			Another form of the equation is:

			<math>F_1=1-F_2=\frac{r_1 f_1^2+f_1 f_2}{r_1 f_1^2+2f_1 f_2+r_2f_2^2}\,</math>

			where <math>F\,</math> stands the [[mole fraction]] of each monomer in the copolymer:

			<math>F_1 = 1 - F_2 = \frac{d M_1}{d (M_1 + M_2)} \,</math>

			and <math>f\,</math> the mole fraction of each monomer in the feed:

			<math>f_1 = 1 - f_2 = \frac{M_1}{(M_1 + M_2)} \,</math>

			When the copolymer composition has the same composition as the feed, this composition is called the ''azeotrope''.

			==Calculation of reactivity ratios==
			The reactivity ratios can be obtained by rewriting the copolymer equation to:

			<math> \frac{f(1-F)}{F} = r_2 - r_1\left(\frac{f^2}{F}\right) \,</math>

			with

			<math> f = \frac{[M_1]}{[M_2]} \,</math> in the feed

			and

			<math> F = \frac{d[M_1]}{d[M_2]} \,</math> in the copolymer

			A number of copolymerization experiments are conducted with varying monomer ratios and the copolymer composition is analysed at low conversion. A plot of <math>\frac{f(1-F)}{F}\,</math> versus <math>\frac{f^2}{F}\,</math> gives a straight line with slope <math>r_1\,</math> and intercept <math>r_2\,</math>.

			A semi-empirical method for the determination of reactivity ratios is called the [[Q-e scheme]].

			==Equation derivation==
			Monomer 1 is consumed with [[reaction rate]]:<ref>{{cite book\|last=Young\|first=Robert J.\|title=Introduction to polymers\|year=1983\|publisher=Chapman and Hall\|location=London\|isbn=0-412-22170-5\|edition=[Reprinted with additional material]}}</ref>

			<math>\frac{-d[M_1]}{dt} = k_{11}[M_1]\sum[M_1^] + k_{21}[M_1]\sum[M_2^] \,</math>

			with <math>\sum[M_x^*]</math> the concentration of all the active centers terminating in monomer 1 or 2.

			Likewise the rate of disappearance for monomer 2 is:

			<math>\frac{-d[M_2]}{dt} = k_{12}[M_2]\sum[M_1^] + k_{22}[M_2]\sum[M_2^] \,</math>

			Division of both equations yields:

			<math>\frac{d[M_1]}{d[M_2]} = \frac{[M_1]}{[M_2]} \left( \frac{k_{11}\frac{\sum[M_1^]}{\sum[M_2^]} + k_{21}} {k_{12}\frac{\sum[M_1^]}{\sum[M_2^]} + k_{22}} \right) \,</math>

			The ratio of active center concentrations can be found assuming [[steady state (chemistry)\|steady state]] with:

			<math>\frac{d\sum[M_1^]}{dt} = \frac{d\sum[M_2^]}{dt} \approx 0\,</math>

			meaning that the concentration of active centres remains constant, the rate of formation for active center of monomer 1 is equal to the rate of their destruction or:

			<math>k_{21}[M_1]\sum[M_2^] = k_{12}[M_2]\sum[M_1^] \,</math>

			or

			<math> \frac{\sum[M_1^]}{\sum[M_2^]} = \frac{k_{21}[M_1]}{k_{12}[M_2]}\,</math>

			==External links==
			* copolymer equation applet @eng.utah.edu [http://www.eng.utah.edu/~nairn/applets/Copoly.html Link]
			* copolymers @zeus.plmsc.psu.edu [http://zeus.plmsc.psu.edu/~manias/MatSE443/chapter5.pdf Link]

			==References==
			{{Reflist}}

			{{DEFAULTSORT:Mayo-Lewis equation}}
			[[Category:Polymer chemistry]]
			[[Category:Equations]]

en>Nbarth: boundary, corners

2012-08-03T06:00:19Z

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Alyson is what my spouse enjoys to contact me but I don't like when people use my full title. He is an information officer. clairvoyants - [http://www.article-galaxy.com/profile.php?a=143251 www.article-galaxy.com] - My wife and I reside in Mississippi and I love every working day residing here. The preferred pastime for him and his children is to play lacross and he'll be beginning some thing else alongside with it.<br><br>Take a psychics online; [http://si.dgmensa.org/xe/index.php?document_srl=48014&mid=c0102 si.dgmensa.org], look at my web page ... best psychics ([http://brazil.amor-amore.com/irboothe click the next web page])

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en>Tobias Bergemann: Disambiguate John Mather to John Mather (mathematician)

en>Nbarth: boundary, corners