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Molecular Weight
Nomenclature
The term molecular weight is commonly used to describe the mass of a polymer molecule. With the advent of advanced detection methods, the meaning of the term has become more obscure as the properties of individual molecules can now be observed. All elements have a defined mass dependent upon their isotopic composition. The atomic mass or less precisely the atomic weight of an element can be defined as the weighted average of all the naturally occurring isotopes using the carbon 12 scale and is given by the formula:
Where Ai = Abundance of a specific isotope Mi = Mass of a specific isotope
This can be contrasted with the monoisotopic mass which is the exact mass of the most abundant isotope of an element. (1)
Molecules (including polymers) are formed by bonding multiple atoms together. The relative molecular mass is equivalent to the atomic mass but it applies to molecules. It is obtained by summing the atomic mass of all of the atoms in the molecule. (2)^ The molecular weight , molar mass and relative molecular mass are often used as synonymous terms to describe the weight or mass of a particular type of molecule. The molecular mass is synonymous to the monoisotopic mass of a compound and is specific for one isotope.
Polymers are large molecules formed using a repeating subunit. In most contexts, the molecular weight of a polymer refers to the relative molecular mass. (3)^ The monoisotopic mass can also be determined in some cases. It is typically observed that monoisotopic composition is reserved for determinations made by LCMS while most other methods result in a relative molecular mass.
Polymer Molecular Weight
The term molecular weight as it is applied to polymeric materials implies something different from what is generally meant for small molecules. This is due to the fact that a polymer sample does not have a single molecular weight but rather a range of values (See Jordi white paper “Definition of a Polymer” for more information). In a given sample there may be polymer chains which contain widely different numbers of repeat units. In fact, most polymers contain some residual monomer (Degree of Polymerization “DP” = 1) and some chains which contain several hundred repeat units (DP > 100). These molecules will differ in relative molecular mass by several orders of magnitude.
For this reason, the molecular weight of a polymer is reported using averages. These averages are intended to describe the distribution of molecular weight values for the polymer chains. Three different molecular weight averages are commonly used to provide information about polymers. These are the number average molecular weight (Mn) , weight average molecular weight (Mw) and viscosity average molecular weight. In addition, several other averages are used to lesser extents including the Z average molecular weight (Mz) and the Z+1 average molecular weight (Mz+1). The averages are defined mathematically using the following formulas:
Number Average Molecular Weight = ܯ^ തതതത^ ൌ
∑ ேெ ∑ ே
Weight Average Molecular Weight = ܯ^ തതത௪തത^ ൌ
∑ ேெమ ∑ ேெ
Z Average or Size Average Molecular Weight = ܯ^ തതതത^ ൌ
∑ ேெయ ∑ ேெమ
Viscosity Average Molecular Weight = ܯ௩ ൌ ቂ
∑ ேெೌశభ
∑ ேெ^ ቃ^
ଵ/
Where the summation is over all the chain lengths from i = 1 to I = ∞ and Ni is the number of molecules whose weight is Mi. (4)^ The viscosity average molecular weight also makes use of an additional term “a.” This parameter is an empirical constant that is dependent upon the hydrodynamic volume or “the effective volume of the solvated polymer molecule in solution, and varies with polymer, solvent, and temperature.” (4) (^) It should be noted that M v and Mw are the same when “a” has a value of 1.
Figure I: Plot of the distribution of molecular weights in a typical polymer sample.
From the definitions of the molecular weight averages it can be observed that for a monodisperse polymer (polymers with chains of only one mass) the four averages are equal Mn = Mw = Mv = Mz. If however the polymer has many different sized chains (this is typically true) then the relationship between the averages is Mn ≤ Mv ≤ Mw ≤ Mz. The presence of the additional Mi term in each succeeding average results in greater emphasis being placed on the highest molecular weight molecules. For step growth polymers which have a most probable distribution of molecular sizes the ratio of Mz:Mw:Mn is 3:2:1. (3)
To provide a reasonable estimate of the molecular weight, it is necessary to provide more than one average (unless the polymer is monodisperse). To fully describe the molecular weight distribution it is necessary to prepare a plot showing the weight fraction of the polymer chains which have a given molecular weight.
Molecular Weight Averages
To explain the meaning of the molecular weight averages, it is helpful to consider an analogy. Let’s consider the average weight of a player on the offensive squads for two competing football teams. Now let’s assume that these teams are identical accept for the weight of the center on each team. One team has a center who is 250 lbs and on the other team the center is 400 lbs. What effect will this have on the three types of averages? Figure I shows the players which make up the offensive squads of both teams. If we know the weights of the various players on each team, then we can calculate the number average (Mn), weight average (Mw) and Z-average (Mz) weights for the players on both teams. To begin, we will tabulate the data for each team in groups by their weights (Table I).
Table I
Number of Players Weight Team^ of^ Players 1 Weight Team^ of^ Players 2 Ni Mi Mi 2 100 100 6 150 150 2 200 200 1 250 400
Figure III: Offensive squad of a football team by player weight.
Table II summarizes the equations used to calculate each average and the resulting values for each team. To calculate the number average molecular weight (Mn), we take the sum of the weight of all the members of each team and divide it by the number of players on that team. This is the typical numerical average. We note that the Mn values for both teams are reasonably similar because most of the players have the same weights and Mn is not very sensitive to the largest members of the team.
ܯ ݂ ൌ ͳ ݉ܽ݁ܶ ݎ
ሺʹ · ͳͲͲሻ ሺ · ͳͷͲሻ ሺʹ · ʹͲͲሻ ሺͳ · ʹͷͲሻ ͳͳ
ൌ ͳͷͻ
Number Average Weights Weight of Team 1 1750 Weight of Team 2 1900 Number of Players on each team 11 Mn Team 1 159 Mn Team 2 173 Percent Difference (Team 1 vs 2) 8%
In the number average, we saw that the weight classes (100, 150, 200, 250, 400 lbs) were weighted by the numerical fraction of the players of that weight. In the weight average, each different weight class is weighted by the weight fraction of players on the team who have that weight. The Mw values for both teams are then 170 and 208. The Mw values for the two teams differ to a greater extent than did the Mn values because Mw is more sensitive to the largest players. We note that the weight fraction contributed by the heaviest players can be large even if they are few in number.
ܯ௪ ݂ ൌ ͳ ݉ܽ݁ܶ ݎ
ሺʹ · ͳͲͲଶሻ ሺ · ͳͷͲଶሻ ሺʹ · ʹͲͲଶሻ ሺͳ · ʹͷͲଶሻ ሺʹ · ͳͲͲሻ ሺ · ͳͷͲሻ ሺʹ · ʹͲͲሻ ሺͳ · ʹͷͲሻ ൌ ͳͲ
Weight Average Weights Mw Team 1 170 MwTeam 2 208 Percent Difference (Team 1 vs 2) 20%
The difference between Mn and Mw can be better understood when one considers that on team one the total weight of the players which have a weight less than 170 is 1100 lbs (9 of 11 players) while those greater than this weight account for 650 lbs (3 players). Thus the three heaviest players on team one make up 37% of the total weight of team but only account for 27% of the people. Similarly on team two, the weight fraction of players above 208 lbs is 24% but the number fraction is only 9% (1 of 11 players). The presence of the one very large player on team two shifts the Mw average significantly toward higher weights. This demonstrates the increased sensitivity of the weight average to higher weight components.
Weight Fraction and Mw Team Number Fraction ofplayers < Mw Weight Fraction ofplayers < Mw Number Fraction ofplayers > Mw Weight Fraction ofplayers > Mw 1 73% 63% 27% 37% 2 91% 79% 9% 21%
Finally, we can calculate the Z-average molecular weight by introducing yet another term for the weight of each player. This is done by multiplying the number of players of each weight by their weight three times. This average strongly emphasizes the largest members of the team to the exclusion of the smaller players. The Mz values for the two teams were 181 and 259. From the values we note that the averages for both teams are in the following order Mn < Mw < Mz. This is always observed. Table II summarizes the results for all of the averages.
ܯ௭ ݂ ൌ ͳ ݉ܽ݁ܶ ݎ
ሺʹ · ͳͲͲଷሻ ሺ · ͳͷͲଷሻ ሺʹ · ʹͲͲଷሻ ሺͳ · ʹͷͲଷሻ ሺʹ · ͳͲͲଶሻ ሺ · ͳͷͲଶሻ ሺʹ · ʹͲͲଶሻ ሺͳ · ʹͷͲଶሻ ൌ ͳͺͳ
Z-Average Weight Mz Team 1 181 MzTeam 2 259 Percent Difference (Team 1 vs 2) 35%
