AVERAGE MOLECULAR WEIGHTS OF HIGH POLYMERS
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SANTI R. PALIT Polytechnic Institute of Brooklyn, Brooklyn, N. Y.
IN
THE study of high polymers a number of diierent types of average molecular weights are introduced which offer great dificulty to beginners. The following approach has been found helpful to the teaching and understanding of these concepts. The term "average" has in ordinary usage only one significance, the familiar concept of the mean value. But, mathematically, i t has an extensive sense. Suppose we have a series of pairs of numbers (h9y3,(%Yr), (%YJ) . . . (zi,Yd) ...(h,Yn) Clearly, we have
where f(x,y) is any function of x and y. The average value of x is defined as 5,where
and this average is to be called average value of x with respect to f(x,y) and symbolized as @cz,ul. .In other words, whenever a fraction simplifies to x, its summation value, the numerator and the denominator having been summed separately, is an average of x with respect to that denominator. If M, is the molecular weight of the i polymer, whose number is Ni, we have M
=
MN/N
and therefore, M ~ N ~ QN= ~zlvi r
(2)
This is the number average molecdar weight, which is evidently the same as the mean or average molecular weight as usually understood. Now, we might multiply both the numerator and the denominator by M when we get another average moleculm weight-that is, we have M e -MXN MN
and therefore -
M M N=
zMIW~
(3)
this since M,N, is the total weight of the i average is called weight average molecular weight and isusually symbolized M,. By a similar process and by using different types of functions, we can have an infinite number of types of average molecular weights, for example
Therefore,
This leads to another average molecular weight which is the ath root of the weight average value of Ma, and this is called viscosity average molecular weight M, for reasons to be explained shortly. NECESSITY OF THE DIFFERENT MOLECULAR WEIGHTS
These different molecular weights bad to be introduced becauseof the diierencein the methodsof molecular weight determination. In any such method we measure a property which is related to the molecular weight and therefrom calculate the molecular weight. I n a homogeneous substance the methods do not matter and we arrive a t the same value of themolecular weights because any kind of averaging will lead to the same value. I n a polydisperse system this is not generally true, however. For example, the osmotic pressure exerted by a small molecule is the same as that by a big molecule and so, since the osmotic pressure is proportional irrespective of only to the number of units in their weights. - . we arrive a t a number average molecular weight by such measurements. On the contrary, in light scattering measurements, the bigger particle will scatter more than the smaller particle, and so the total scattering is not proportional to the total number. However, some function of the scattering by each molecule is pbtainable which is proportional to the weight of each molecule. Hence, the calculated moleular weight is a weight average molecular weight. Similarly, intrinsic viscosity has been shown to be proportional, not to the moiecular weight, but to its ath .power, [?I = Mu,where a is a constant. This, on calculation, will lead to another type of molecular weight, the weight average of ath power of molecular weight (A@),, the ath root of which for simplicity is called viscosity averagemolecular weight, M,. Suppose we could measure a property of a polymer solution which is proportional to the logarithm of the molecular weight. The calculated molecular weight will be the weight a v J e value of logarithm of molecular weight, i.!., (log M),, and the value raisedto the power of e w111 give a new type of average molecular weight. This shows the necessity of introducing the various average molecular weights.
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JOURNAL OF CHEMICAL EDUCATION SIGNIFICANCE AND RELATIONSHIP
The above line of reasoning clearly shows that averaging is a mathematical process involving a division, and therefore i t can be visualized only if the terms involved in the division can be visualized. In equation (2) M,N, is a weight and N, is a number, and therefore we can visualize i t as a weight per unit number, i. e., the weight was divided equally among all the numbers. But we cannot imagine weight X weight, which is the numerator of equation (3), and therefore we cannot visualize the meaning of the weight average molecular weight except to call it molecular weight per unit weight.
That number average is always less than the weight average can be easily shown. Suppose in a sample of polymer we introduce n more polymer molecules of zero weight each. The sample apparently remains unchanged, but its number average molecular weight goes down whereas the weight average remains unchanged. This shows clearly that the introduction of a low molecular weight fraction decreases the number average more than the weight average. Since any polymer may be regarded as t o be formed by introduction of low molecular weight fractions to the highest molecular weight fraction, the number average will always be less than the weight average.
