Arithmetic Mean

CHAPTER

2.3 Arithmetic Mean

The distribution of molecular sizes in a polymer sample is usually expressed as the proportions of the sample with particular molecular weights. The mass of data contained in the distribution can be understood more readily by condensing the

information into parameters descriptive of various aspects of the distribution.

Such parameters evidently must contain less information than the original distri- bution, but they present a concise picture of the distribution and are indispensable for comparing different distributions.

One such summarizing parameter expresses the central tendency of the distri- bution. A number of choices are available for this measure, including the median, mode, and various averages, such as the arithmetic, geometric, and harmonic means. Each may be most appropriate for different distributions. The arithmetic mean is usually used with synthetic polymers. This is because it was very much easier, until recently, to measure the arithmetic mean directly than to characterize the whole distribution and then compute its central tendency. The distribution must be known to derive the mode or any simple average except the arithmetic mean. (Some methods like those based on measurement of sedimentation and diffusion coefficients measure more complicated averages directly. They are not used much with synthetic polymers, however, and will not be discussed in this text.)

Various molecular weight averages are current in polymer science. We show here that these are simply arithmetic means of molecular weight distributions.

It may be mentioned in passing that the concepts of small particle statistics that are discussed here apply also to other systems, such as soils, emulsions, and car- bon black, in which any sample contains a distribution of elements with different sizes.

To define any arithmetic mean A, let us assume unit volume of a sample of Npolymer molecules comprisingn1molecules with molecular weightM1,n2mole- cules with molecular weightM2,. . .,njmolecules with molecular weightMj.

n11n21?1nj5N (2-1)

A5n1M11n2M21?1njMj

n11n21?1n_j 5n1M11n2M21?1njMj

N (2-2)

A5n1

NM11n2

NM21?1nj

NMj (2-3)

The arithmetic mean molecular weight A is given as usual by the total mea- sured quantity (M) divided by the total number of elements. That is, the ration/N is the proportion of the sample with molecular weightMi. If we call this propor- tionfi, the arithmetic mean molecular weight is given by

A5f1M11f2M21?1f_jM_j5 X

i

f_iM_i (2-4)

Equation (2-4) defines the arithmetic mean of the distribution of molecular weights. Almost all molecular weight averages can be defined from this equation.

65 2.3 Arithmetic Mean

2.3.1 Number Distribution, M

n

The distribution we have just assumed to define the arithmetic mean is a number distribution, since the record consists of numbers of molecules of specified sizes.

The sum of these numbers comprises the integral (cumulative) number distribu- tion. Figure 2.1 represents one such distribution. The scale along the abscissa is the molecular weight while that on the ordinate could be the total number of molecules with molecular weights less than or equal to the corresponding value on the abscissa. However, it is easier to compare different distributions if the cumulative figures along the ordinate are expressed as fractions of the total num- ber of molecules in each sample, and Fig. 2.1is drawn in this way. The units of the ordinate are therefore mole fractions and extend from 0 to 1; the integral distribution is now said to be normalized.

In mathematical terms, the cumulative number (or mole) fraction X(M) is defined as

XðMÞ5 X^M

i

x_i (2-5)

where xi is the fraction of molecules with molecular weight Mi. The differential number function is simply the mole fraction xi, and a plot of these values against correspondingMi’s yields a differential number distribution curve, as inFig. 2.2.

If the distribution is normalized, the area under the xiMi curve inFig. 2.2 will be unity. (See Section 2.4.2 for units.)

To compile the number distribution we have expressed the proportion of species with molecular weight Mi as the corresponding mole fraction xi.

0.8 1.0

0.6 0.4 0.2 0 X (M), mole fraction with molecular weight ≤ Mi

Molecular weight, M_i FIGURE 2.1

A normalized integral distribution curve.

Substitution ofxiforfiinEq. (2-4)shows that the arithmetic mean of the number distribution is

A5X

i

xiMi5Mn (2-6)

This is the definition of number average molecular weight Mn. Equivalent definitions follow from simple arithmetic. Since

xi5ni

N 5ni

Xni (2-7)

Mn5X

n_iM_i X

n_i (2-8)

whereniis defined, as above, as the number of polymer molecules per unit volume of sample with molecular weightMi. Also, ifciis the total weight of thenimolecules, each with molecular weightMi, andwiis the corresponding weight fraction, then

c_i5n_iM_i (2-9)

wi5ci

Xci5niMi

XniMi (2-10)

and

Mn5X ci

Xci

M_i 51 Xwi

M_i (2-11)

Since polymer solutions are used for direct determinations of average molecu- lar weights, the symbolsniandci will usually refer respectively to the molar and weight concentrations of macromolecules in such solutions.

Mole fraction xi with molecular weight ≤ Mi

Molecular weight, Mi

FIGURE 2.2

A normalized differential number distribution curve.

67 2.3 Arithmetic Mean

2.3.2 Weight Distribution, M

w

If we had recorded the weight of each species in the sample, rather than the number of molecules of each size, the array of data would be a weight distribution. The situa- tion corresponds to that described for a number distribution.Figure 2.3depicts a sim- ple integral weight distribution, normalized by recording fractions of the total weight rather than actual weights of the different species.

The integral (cumulative) weight fractionW(M) is given by WðMÞ5X

i

wi (2-12)

and is equal to the weight fraction of the sample with molecular weight not greater thanMi. A plot ofwiagainstMiyields a differential weight distribution curve, as in Fig. 2.4. As in the case of the number distribution, ifW(M) is normalized, the scale of the ordinate in this figure goes from 0 to 1 and the area under the curve equals unity.

1.0 0.8

W (M), weight fraction with molecular weight ≤ Mi

0.6 0.4 0.2

0 Molecular weight, M_i

FIGURE 2.3

A normalized integral weight distribution curve.

Wi, weight fraction with molecular weight Mi

Molecular weight, Mi

FIGURE 2.4

A normalized differential weight distribution curve.

The proportion of the sample with size Mi is expressed in the present case as the corresponding weight fraction. Equating wi and fi in Eq. (2-4) produces the following expression for the arithmetic mean of the weight distribution:

A5X

i

wiMi5Mw (2-13)

where Mw is the weight average molecular weight, which from Eqs. (2-9) and (2-10)may also be expressed as

Mw

XM_ic_i=X

c_i5X

M_i²n_i X

M_in_i (2-14)

EXAMPLE 2-1

Given that a polymer sample contains two moles of chains with one mole having a molecular weight of 5000 and the other 10,000, calculate itsMnandMw. A different polymer sample also contains two moles of chains but in this sample, one mole of chains has a molecular weight of 2500 while the other has 12,500; what are itsMnandMw?

For the first sample, Mn51

23500011

231000057500 andMw51

33500012

331000058333 For the second sample,

Mn51

23250011

231250057500 andMw51

63500015

6310000510833 Both samples have the sameMnbut the second sample has a higherMw, indicating that the second sample has a wider distribution of molecular weight.