An Experimental Introduction to Molecular Weight Averages of Polymers A Simple Experiment That Uses Paper Clips Maria Pilar Tarazona and Enrique Saiz Departamento de Quimica Fisica, Universidad de Alcala de Henares, 28871 - Alcala de Henares, Madrid, Spain
Molecular weight and molecular weight distribution (MWD) are two variables that play a dominant role in determining the physical, rheological, and mechanical properties of a polymer sample. Polymer textbooks and several ' ~ detailed discussions of the articles in this J o ~ r n a l have different average molecular weights, MWD, and experimental methods that can be applied to the characterization of polymers. However, students in introductory polymer courses usually have a hard time understanding why a polymer, unlike other molecules, can have more than one molecular weight. We have devised an experimental introduction to the topic that gets the student actively involved.
2. Weight Distribution and Weight-Average Molecular Weight The wei&t distribution w iand weight-average molecular weight M , are obtained by the following expressions.
and
The Experiment Polymer molecular weights can be easily introduced to the student with the aid of a few simple items.
several paper clips of different sizes a ruler a balance with an accuracy of at least 0.01 g The student is requested to prepare a sample by picking up a few clips of each size. We recommend that they pick more paper clips of the intermediate sizes so that the sample resembles a polymer distribution. Thus, the students obtain the following five sets of data.
1. Number Distribution and Nurnber-Average Molecular Weight The n u m b distribution ni and number-average molecular weight M. are obtained according to the following expressions.
Weight of one clip (Mi) Figure 1. Number distribution of a sample of paper clips.
and
where Ni is the number of clips of size i contained in the sample. Each of these clips has a weight of Mi. The difference between the values of Mi for clips of the same sue is always less than 3 rng, so the values can be taken as equal, with an accuracy of 0.01g. (See Fig. 1.) 'Ward. T. C. J. Chem. Educ. 1981,58,867. 2~lumenstein, R.; Carraher, C. E.; Coker, H.: Fowkes, F.; Hellmuth, E.: Karl, D.; Mandelkern, L.; Mark. J. E.; Mattice, W.; Rodriguez, F.; Rooers. C.: Soerlina. L.: Stein. R. J. Chem. Educ. 1985.62,1030.
Weight of one clip
(Mi)
Figure 2. Weight distribution of the same sample of paper clips as in Figure 1. Volume 69 Number 9 September 1992
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Typical Student Results
Size Ni
Mi
ni
mi (9)
Wlip) 1 2 3
4 5 2
wi
h
Mu
N~M
NiMi
r = r
5 6 11
0.26 0.39 0.59
0.161 0.194 0.355
0.068 0.15 0.35
1.31 2.35 6.48
0.066 0.118 0.326
9.0 13.2 31.8
1.30 2.34 6.49
7 2 3
0.97 1.49 3.70
0.226 0.065 1.00
0.94 2.22 3.73
6.78 2.98 19.90
0.341 0.150 1.00
24.0 9.6
6.79 2.98
87.6
19.90
where the mass of a11 the clips of size i can be determined either by weighting together all the Nj clips of the same size to obtain mi or by multiplying Niby the average weight of one clip of class i: Mi. (See Fig. 2.) It can be seen in the table that the results of both procedures agree within the limits of experimental error.
0.34
Mn
IiMi
(em) 2.34
5. Standard Deviation
The standard deviation o of the number distribution, which is shown below, 18.76 is defined as the souare root of the dif6.58 23.28 ference between the second moment and the square of the first moment. 4.44 14.30 0'92 3.82
16.10
"I5
6.8 o
(
n
n
i
M
2
112
)
=Mnp-l] 112
The results obtained for one sample are given in the table from which the student can obtain the number and weight distributions shown in the figures. The following values of the averages can also be obtained. a,, = 0.64
3. Length-Average Molecular Weight
The length-average molecular weight is obtained by the following expression.
where li is the length of a chain formed by linking all the clips of size i. The length of this chain is smaller than the sum of the individual lengths of each clip of size i. 4. Polydispersity Ratio
The polydispersity ratio is computed as below.
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Journal of Chemical Education
Different experimental techniques (i.e., counting, weighting, and measuring the length) yield different weight averages due to the polydispersity of the paper clips system. This behavior can be extrapolated to experimental techniques such as osmometry, light scattering, viscosity measuremen&, e&., and to the different molecular weight averages M,, M,, Mu, etc., thus obtained for polymeric chains.
