## MWt Distribution Calculations

### Quick Start

Because polymer entanglement is so important and because it depends strongly on MWt it is worth exploring what "MWt" means for a polymer. Most of us know that "the" MWt of a polymer doesn't exist - there are different values depending on how you define or measure them.

1. Mn Number Average MWt
2. Mw Weight Average MWt
3. Mv Viscosity Average MWt
4. Polydispersity = Mw/Mn

Here we can create our own polymers by changing the MWt, the relative amount (height, h) and the width, w, of two components, and explore how these factors influence the different MWt values and the all-important Polydispersity value. Just play around with the sliders, then read in more detail about what is happening

### Distribution

MWt1
h1
w1
MWt2
h2
w2
MWt Max
Mn
Mw
Mv
Polydispersity

Our three different MWts are defined as a series of sums, shown as Σ. We go through the graph counting how many polymers, N, there are at each MWt and adding them together to give ΣN. Then we go through each MWt and multiply the N at that weight with the MWt, M, and sum them to give Σ(N.M). To get Mn, we divide Σ(N.M) by ΣN. The others follow a similar logic:

1. Mn Number Average MWt = Σ(N.M)/ΣN
2. Mw Weight Average MWt = Σ(N.M²)/Σ(N.M)
3. Mv Viscosity Average MWt = (ΣNM(1.5)/ΣNM)²
4. Polydispersity = Mw/Mn

The definitions aren't too obvious, so the app gives you a chance to play with whatever features of the MWt distribution interest you to see how that translates into the different definitions. You two MWt peaks to play with and you can change their heights and widths. They are deliberately skewed rather than Gaussian peaks to be more realistic. Clearly they are still idealised, but they give the general idea.

The curves show the cumulative values of the 3 different terms. So you see that the Number average is biased towards the largest peak, while the Weight average is biased towards the peak with the larger MWt values, with the Viscosity average in between.