Research: Science and Education
Molecular Weight Averages and Polydispersity of Polymers Wen-Shyan Sheu Department of Chemistry, Fu-Jen Catholic University, Taipei, Taiwan 242, ROC;
[email protected] Polymer science is a very important field in chemistry owing to its many practical applications and its fundamental contributions to understanding the biological functions of macromolecules such as proteins. Therefore, it is part of the undergraduate curriculum in many universities either in the physical chemistry course (1) or in a more specialized course on introductory polymer science (2). Because of the kinetics of polymerization, all synthetic polymers and most natural polymers (except proteins) have a distribution of molecular weights. Since the molecular weights of polymers have a great effect on their physical and mechanical properties, it is very important to characterize the molecular weight distributions by various average quantities. Among the averages used, the number-average molecular weight (NAMW, denoted by Mn) and weight-average molecular weight (WAMW, denoted by Mw) are the two most important quantities. This is partly because they can be directly measured from experiments. For example, NAMW can be determined from colligative properties such as osmotic pressure and boiling point elevation (1, 2), and WAMW can be measured from sedimentation equilibria and light-scattering experiments (1, 2). It is also well known that NAMW is never greater than WAMW and the width of a molecular-weight distribution can be characterized by the polydispersity index, Mw/Mn (2). The larger the index is, the wider the distribution is. However, these facts are stated without proof in most textbooks, which makes it difficult for students to fully understand these concepts. Here, we give a simple statistical explanation, which should be comprehensible to most undergraduate students. First, the probability distribution fi is defined as the probability of a polymer having molecular weight Mi in a molecular weight distribution. Therefore,
fi ≡
Ni
Σi N i
(2)
and
554
This is because the mass difference, Mi – Mn, determines how molecular weight Mi deviates from the NAMW. Since only the magnitude of the deviation is needed to measure the width of a distribution, the square is taken in eq 4. Finally, the proper distribution weight fi of Mi is taken and summed over all possible molecular weights to give the standard deviation of a distribution. Note that, from the definition, δ2 is always positive. Equation 4 can be expanded and simplified: 2 δ 2 ≡ Σ M i2 – 2M i M n + M n2 f i = i
where
Σi M i2 f i
– M n2
(5)
Σi f i = 1
and eq 2 have been used. Substituting eq 3 into the equation, the following result is obtained: δ 2 = Mn M w – M n2
(6)
Therefore,
M n2
≡
Mw Mn
–1
(7)
+1
(8)
or
as is required by the condition for a probability distribution. With these definitions, NAMW and WAMW can be expressed in terms of Ni or fi:
Σi Ni M i2 Σi Mi 2 f i Σi Mi 2 f i Mw ≡ = = Σi Ni M i Σi Mi f i M n
(4)
i
(1)
Σi f i = 1
Σi Ni M i = ΣM f i Σi Ni i i
δ2 ≡ Σ Mi – Mn 2f i
δ2
where Ni is the number of molecules with molecular weight Mi. Note that
Mn ≡
Hence NAMW and WAMW are related to the first and second moments of the distribution, respectively. Since the polydispersity index is used in textbooks as a measure of the width of the molecular weight distribution, which is usually measured by the standard deviation δ of a distribution, it is interesting to find the relationship between the polydispersity index and the standard deviation. The standard deviation is defined in this case as
(3)
Mw Mn
=
δ2 M n2
Hence, the relationship between the polydispersity and the standard deviation is obtained. Since Mw /Mn is directly related to the standard deviation, it can be used as a measure of the distribution width. The distribution width increases with the polydispersity ratio as previously described. Moreover, because δ2 is always positive, Mw/Mn ≥ 1 or Mw ≥ Mn is obtained. That is to say, NAMW is never greater than WAMW. When Mw = Mn, the standard deviation is zero, which means that the distribution has zero width, or, the polymer is monodisperse, as stated in textbooks. In summary, a simple statistical explanation for the relationship between the polydispersity index and the width of the polymer molecular weight distribution is given. This method should help students to understand the physical meaning of the polydispersity index.
Journal of Chemical Education • Vol. 78 No. 4 April 2001 • JChemEd.chem.wisc.edu
Research: Science and Education
Acknowledgment
Literature Cited
The work reported in this article was funded by the National Science Council, ROC, under Contract No. NSC 89-2113-M-030-013, and this support is gratefully acknowledged.
1. Atkins, P. W. Physical Chemistry, 6th ed.; Oxford University Press: Oxford, 1998 (or earlier editions). 2. Sperling L. H. Introduction to Physical Polymer Science, 2nd ed.; Wiley: New York, 1993.
JChemEd.chem.wisc.edu • Vol. 78 No. 4 April 2001 • Journal of Chemical Education
555
