Molecular Weight Averages as Ratios of Moments

CHAPTER

2.4 Molecular Weight Averages as Ratios of Moments

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.

In this case it is commonly known as the torque. The concept has been extended to more abstract applications such as the moment of an area with respect to a plane and moments of statistical distributions. It is then referred to as the appro- priate first moment (the termtorqueis not used).

The second moment of force about the same axis is the product of the force and the square of the distance between its line of action and the axis. This is the moment of inertia. The most direct example of its use is possibly connected with the motion of a rotating body, for which the rotational acceleration caused by an applied torque is calculated by dividing the torque by the moment of inertia of the body. The concept of a second moment has been extended to other less read- ily pictured applications such as computation of stresses in beams from second moments of cross-sectional areas about particular axes.

By extending the above examples we can say that a moment in mechanics is generally defined as

U_j^a5Fd^j (2-15)

whereUjis thejth moment, about a specified line or planeaof a vector or scalar quantity F (for example, force, weight, mass, area), d is the distance from F to the reference line or plane, andjis a number. The moment is named according to the power jto whichd is raised. If Fis composed of elementsFieach located a distance di from the same reference, the moment is the sum of the individual moments of each element

U_j^a5 X

i

F_id_i^j (2-16)

Mathematically, there is no restriction on the choice of F or j, but use of moments to solve practical mechanics problems usually confines F to the exam- ples listed above and j to values of 1 or 2. The reference line or plane must be specified when the value of the moment is quoted.

In polymer science the mathematical formulation for moments corresponds to that in Eq. (2-16). While the reference line may be located anywhere, the useful- ness of choosing the ordinate (M50) in the graph of the molecular weight distri- bution (Figs. 2.2 and 2.4) is so great that this reference is usually not mentioned explicitly. The distance d from the reference line is measured along the abscissa in terms of the molecular weightM, and the quantityFis replaced byfi, the pro- portion of the polymer with molecular weightMi. As a matter of utility,jassumes a wider range of values in polymer science than in mechanics. With these differ- ences, which are mainly matters of emphasis, the concepts of moments corre- spond closely in both disciplines. A general definition of a statistical moment of a molecular weight distribution taken about zero is then

U⁰_jX

qiM^j_i (2-17)

whereqiis the quantity of polymer in unit volume of the sample with molecular weightMiand respective values ofqi5ni(number of molecules, or moles) for an unnormalized number distribution,5xi (mole fraction) for a normalized number distribution,5ci (number of grams) for an unnormalized weight distribution, or5wi (weight fraction) for a normalized weight distribution. In addition, we shall use the notation_nUto refer to a moment of the number distribution and_wU to denote a moment of the weight distribution.

Weight distributions will usually be encountered during analyses of polymer samples. Considerations of polymerization kinetics are often easier in terms of number distributions.

2.4.2 Dimensions

Molecular weight itself is dimensionless. It is the sum of the atomic weights in the formula of the molecule. Atomic weights, in turn, are expressed in terms of dimensionless atomic mass units (amu) which are ratios (312) of the masses of the particular atoms to that of the most abundant carbon isotope¹²C to which a mass of 12 is assigned. A gram molecular weight, or gram-mole, is the amount of polymer whose weight in grams is numerically equal to the molecular weight (in amu). It is just as correct to use pound-moles or ton-moles if the circumstances so dictate.

The moments of normalized distributions are products of dimensionless fre- quencies and dimensionless molecular weights or of gram-moles with dimensions of mass. The former moments will be unitless, and the units of the latter will depend on the moment number and on the units of the distribution. Most equa- tions in polymer science imply use of gram-moles, but this is not universal and the dimensions of the particular equation should be checked to determine which units, if any, are being used for molecular weight and concentration quantities.

2.4.3 Arithmetic Mean as a Ratio of Moments

As a general case the ratio of the first moment to the zeroth moment of any distri- bution defines the arithmetic mean. For an unnormalized number distribution,ni

is the number of moles per unit volume with molecular weightMiand the zeroth and first moments of the distribution about zero are given, respectively, by

nU⁰₀5X

i

ðM_iÞ⁰n_i5X

n_i (2-18)

nU⁰15X

i

ðMiÞ¹ni5X

Mini (2-19)

In these symbols the subscript n shows that the moment refers to a number distribution, the numerical subscript is the moment order, and the prime super- script indicates that the moment is taken about the M50 axis. These equations

71 2.4 Molecular Weight Averages as Ratios of Moments

follow from the definition in Eq. (2-17). The arithmetic mean of the number distribution is the ratio of these moments:

A5ⁿU⁰1 nU⁰0

5X Mini

Xni5Mn (2-20)

(CompareEq. 2-8.)

The arithmetic mean of a weight distribution (the count is in terms of the weight ci, rather than number of moleculesni of each species) is likewise given by the ratio of the first to the zeroth moment of the particular distribution about zero. (The notation for moments of weight distributions follows that for number distributions except that the subscript n is replaced by a w.)

In these last examples we have chosen unnormalized distributions. If the dif- ferential number or weight distribution is normalized, the area under the curve in Figs. 2.2 and 2.4equals unity. That is,

nU⁰05_wU⁰051ðnormalized distributionsÞ (2-21) The arithmetic mean is then numerically equal to the first moment of the nor- malized distribution, as expressed inEqs. (2-6) and (2-13).

2.4.4 Extension to Other Molecular Weight Averages

We have seen thatMn;the arithmetic mean of the number distribution, is equal to the ratio of the first to the zeroth moment of this distribution (Eq. 2-20). If we take ratios of successively higher moments of the number distribution, other aver- age molecular weights are described:

nU⁰2 nU⁰1

5X M_i²ni

XMini5Mw (2-22)

nU⁰3

nU⁰₂ 5 X

M_i³ni

XM²_ini5Mz (2-23)

nU⁰4 nU⁰3

5X

M⁴_in_i X

M³_in_i5M_z11 (2-24) We may define an average in general as the ratio of successive moments of the distribution. Mn and Mw are special cases of this definition. The process of taking ratios of successive moments to compute higher averages of the distribu- tion can continue without limit. In fact, the averages usually quoted are limited to Mn, Mw, Mz, and the viscosity average molecular weight Mv, which is defined later in Section 3.3. We can measure Mn,Mw, andMv directly, but it is usually necessary to measure the detailed distribution to estimateMzand higher averages.

Table 2.1 lists averages of the number and weight distributions in terms of these moments.

The reader may notice that any moment about zero of a normalized distribution

nU⁰_j5 X

x_iðM_iÞ^j or _wU⁰_j5 X w_iðM_iÞ^j

corresponds to the arithmetic mean of the number or weight distribution of (Mi)^j, respectively. Respectively, Mn and Mw are arithmetic means of the number and weight distributions and the source of their names is obvious. TheMz,Mz11, and so on, are arithmetic means of the z, z11, etc., distributions. Operational models of these distributions would be too complicated to be useful in polymer science.

Table 2.1 Moments about Zero and Molecular Weight Averages (a) Number distribution

Not normalized Normalized Averages

nU⁰₀5 X

i

ni 5X

i

xi51

nU⁰₁5 X

i

Mini 5X

i

Mixi M_n5nU⁰₁=nU⁰₀

nU⁰₂5 X

i

M_i²ni5M_wM_nnU⁰₀ 5X

i

M²_ixi5M_wM_n M_w5nU⁰₂=nU⁰₁

nU⁰₃5 X

i

M_i³ni5MzM_wM_nUnU⁰₀ 5P

M_i³xi5MzM_wM_n Mz5nU⁰₃=nU⁰₂

M_v5nU⁰_a011=nU⁰₁1=a nU⁰

j5 P

M_i^jni 5P M_i^jxi

(b) Weight distribution

Not normalized Normalized Averages

wU⁰₂₁5 X

i

ciM²¹ 5X

i

wiM_i²¹

wU⁰₀5 X

i

ci 5X

i

wi51 M_n5wU⁰₀=wU⁰₂₁

wU⁰₁5 X

i

ciMi5M_wUwU⁰₀ 5P

wiMi5M_w M_w5wU⁰₁=wU⁰₀

wU⁰₂5 X

i

ciM²_i 5MzMwUwU⁰₀ 5X

i

wiM_i²5MzM_w Mz5wU⁰₂=wU⁰₁

M_v^a5ðwU⁰_aÞ^1=a

wU⁰_j5 P

ciM_i^j 5P wiM_i^j

aM_vis derived from solution viscosity measurements through the MarkHouwink equation½n5KM_v^a; where[n]is the limiting viscosity number and K and1are constants which depend on the polymer, solvent, and experimental conditions, but not on M(Section 3.3.1).

73 2.4 Molecular Weight Averages as Ratios of Moments

Table 2.2 lists various average molecular weights in terms of moments of the number and weight distributions, where the quantity of polymer species with par- ticular sizes are counted in terms of numbers of moles or weights, respectively.

Note that in general a given average is given by M_z1k5_nU⁰_k13

nU⁰_k125_wU⁰_k12

wU⁰_k11 (2-25) The moment orders in the weight distribution are one less than the corres- ponding orders in the number distributions. (CompareMnformulas.) This symme- try arises because molar and weight concentrations are generally related by Eqs. (2-9) and (2-10). Thus,

nU⁰_k5 X

niM_i^k5X

ðniMiÞM^k2_i ¹5X

ciM_i^k215_wU⁰_k21 (2-26) The viscosity average molecular weight Mv, which will be discussed later in Section 3.3, is the only average listed in these tables that is not a simple ratio of successive moments of the molecular weight distribution.