PolySciTech (www.polyscitech.com) provides a wide
variety of polymers. One of the most common questions I get is about polymer molecular
weight. The problem with polymer molecular weight is that it is not truly a
number but rather a distribution. As polymers are synthesized the individual
chains grow and initiate at different rates and different times respectively.
For this reason, polymers always exhibit a polydispersity often, but not
always, in a roughly bell-curve shaped format. Due to the distribution of
polymers and the historical difficulties in actually ‘measuring’ the molecular
weight of polymers there are many different ‘numbers’ used to describe the
molecular weight of a polymer set. These include number average (Mn), weight
average (Mw), peak average (Mp), viscosity average (Mv), Mz, Mz+1, and many
more. These numbers are used broadly in different fields to describe different
polymer systems. A good way to initiate a fist-fight in a room full of polymer
chemists is to simply ask which number represents the ‘true’ molecular weight
of a polymer. This system is confusing and complex. There are very well written
textbooks out there which clock in easily at hundreds of pages each full of
mathematical equations and statistics that attempt to explain these various
definitions of polymer molecular weight. When I train new employees at
PolySciTech, I skip the books and tell the following story. For the purpose of
this one, we will stick with the three most common molecular weight numbers
which are weight average (Mw) number average (Mn) and peak average (Mp).

The story begins:
“Once upon a time there was a census statistician tasked with defining the average weight
of people living in a certain city. He had neither time nor resources to weigh
every single person in the city, but wanted to get an estimate that he could
report back as the average ‘weight’ of the city. He thought of the best way to
do this and hit upon a simple plan. Letters were sent out at random informing a
representatively large numbered group of people to report to a nearby truck
weigh in station. The statistician watched as people lined up, young and old, heavy and thin,
and considered how brilliant his plan was. That is until a van pulled up. The statistician blanched as the people got out of the van
and he watched the tires pull away from the wheel-wells as they did so. They
all had on bibs featuring a pig holding a knife and fork and indeed one of them
was still gnawing away at a piece of pork-rib from their latest barbeque-fest.
From sight, he estimated each one to weigh at least 150 to 200 kilos if not
more and they all dutifully presented letters and hopped up on the scale. The statistician then began to feel faint as a school bus
pulled up and he realized that he was not tasked with defining the weight of ‘adults’,
but everyone in the city. The masses of squirrelly children, all with letters
in hand, were ushered up onto the scale by their teachers and the statistician
tried desperately to count them as they ran about and played with one another.
At last, everyone was on and the truck-station weigh in scale was activated and
the total mass of the group was recorded. The statistician sat down and counted
the number of people on the scale and divided the mass total by the number of
people to obtain the weight average of the people (this is polymer weight
average “Mw”). He came up with 100 kilos! He frowned at this. There was no way
the average was 100 kilos with all those little children on the scale and he
cast a wary glance at the barbeque fans. Each of them was contributing an
incredible amount of mass to the scale, but still only counted as one person.
He shook his head realizing that he could not report this average to his boss
since it was so badly skewed by a relatively small number of extra-heavy
people. The statistician thought harder and had everyone in the group simply
report their weight. He took the total ‘number’ at each weight and averaged
them together (this is polymer number average “Mn”). However, to his shock, he
came up with only 50 kilos! He cast a wary glance at the squirrelly children. Each
one was maybe less than 30 kilos but they heavily outnumbered the barbeque fans
so that when he averaged based on number the children now skewed the number too
low. Finally the statistician took
everyone out to a football field where they would have space. He laid out signs
and lines across the field and labeled each lined section in 5 kilo increments
and had each person stand in line behind the sign indicating their weight. He
then climbed up high into the stadiums and looked down on the group from above.
What he saw looked like a bell-curve. Although there were many children and
some barbeque fans, both were outnumbered by the large group of people which
were in between these two weight extremes. The statistician looked out and saw
that the peak of the curve occurred around 70 kilos and so he marked this down
as the peak average weight of the people (This is how we obtain peak average
molecular weight Mp). Finally everyone was dismissed and the statistician
reported back to his boss. His boss said “So what was the average weight of the
city?” to which the statistician replied “Which number do you want?”