Kepler problem

2013-09-22T19:53:07Z

83.30.195.118: /* Solution of the Kepler problem */ There was a mistake in formula for e and also in E for circular orbit. You may check units. Also mass m should not affect e at all.

The '''degree of polymerization''', or DP, is usually defined as the number of [[structural unit|monomeric unit]]s in a [[macromolecule]] or [[polymer]] or [[oligomer]] molecule.<ref>[http://goldbook.iupac.org/D01569.html IUPAC Definition] in [[Compendium of Chemical Terminology]] (IUPAC Gold Book)</ref><ref>Cowie J.M.G. "Polymers: Chemistry and Physics of Modern Materials" (2nd edn Blackie 1991), p.10</ref><ref>Allcock H.R., Lampe F.W. and Mark J.P. "Contemporary Polymer Chemistry" (3d edn Pearson Prentice-Hall 2003), p.316</ref>

For a homopolymer, there is only one type of monomeric unit and the ''number-average'' degree of polymerization is given by <math>DP_n\equiv X_n=\frac{M_n}{M_0}</math>,
where Mn is the [[Molar mass distribution#Number average molecular weight|number-average molecular weight]] and M0 is the molecular weight of the monomer unit. For most industrial purposes, degrees of polymerization in the thousands or tens of thousands are desired.

Some authors, however, define DP as the number of [[repeat unit]]s, where for [[copolymer]]s the repeat unit may not be identical to the monomeric unit.<ref>Fried J.R. "Polymer Science and Technology" (Pearson Prentice-Hall, 2nd edn 2003), p.27</ref><ref>Rudin A. "Elements of Polymer Science and Engineering" (Academic Press 1982), p.7</ref> For example, in [[nylon-6,6]], the repeat unit contains the two monomeric units —NH(CH2)6NH— and —OC(CH2)4CO—, so that a chain of 1000 monomeric units corresponds to 500 repeat units. The degree of polymerization or chain length is then 1000 by the first (IUPAC) definition, but 500 by the second.

In [[step-growth polymerization]], in order to achieve a high degree of polymerization (and hence molecular weight), ''X''n, a high fractional monomer conversion, ''p'', is required, as per [[Carothers' equation]]: ''X''n = 1/(1−''p''). A monomer conversion of ''p'' = 99% would be required to achieve ''X''n = 100. For [[chain-growth polymerization]], however, this is not generally true and long chains are formed for much lower monomer conversions.

{{Quote box
|title = IUPAC definition
|quote = The number of monomeric units in a macromolecule an oligomer molecule, a block, or a chain.<ref>{{cite journal|title=Glossary of basic terms in polymer science (IUPAC Recommendations 1996)|journal=[[Pure and Applied Chemistry]]|year=1996|volume=68|issue=12|pages=2287–2311|doi=10.1351/pac199668122287|url=http://pac.iupac.org/publications/pac/pdf/1996/pdf/6812x2287.pdf}}</ref>
}}

==Correlation with physical properties==
[[File:Degree of polymerization.png|thumb|right|200px|Relationship between degree of polymerization and melting temperature for polyethylene. Data from Flory (1963).]]

Polymers with identical composition but different total molecular weights may exhibit different physical properties. In general, increasing degree of polymerization correlates with higher melting temperature <ref>Flory, P.J. and Vrij, A. J. Am. Chem. Soc.; 1963; 85(22) pp3548-3553</ref> and higher mechanical strength.

== Kinds of degree of polymerization ==

Mainly, there are two types used to measure the degree of polymerization, number average degree of polymerization and weight average degree of polymerization.
Number Average degree of polymerization is found by finding the [[Weighted mean]] of mole fraction o. While the weight average degree of polymerization is found by finding the weighted mean of weight fraction .<ref>{{cite book|first=Paul Painter & Michael Coleman|title=Fundamentals of Polymer Science}}</ref>

==See also==
* [[Carothers equation]]
* [[Kinetic chain length]]

==References==
{{reflist}}

[[Category:Polymer chemistry]]

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Kepler problem