Welcome back. In this lesson we're going to describe molecular weights with respect to a collection of polymers of different chain lengths. When we talk about the molecular weight, we're going to describe the number average molecular weight, the weight average molecular weight, and the polydispersity index. For example, let's consider a bin. In this particular bin, what we have are a number of BBs. And we know what the number is, and in addition to that, we know how much each of those BBs weighs and hence, we would be able to calculate, knowing the weight, we could calculate the total weight that's in bin number one. When we look at bin number two, we can do the same thing. We count the number of them and then each one of them has a weight. We multiply it by the number and we come up with a total weight in that bin. We now look at bin number three, and in the case of bin number three, we have a number of smaller BBs, all of which have lower weights, and what we would then have is the number of BBs and the total weight of all the BBs in the box. Continue on and with respect to bin four, we have a lot more smaller ones so we'll get a number, we get a weight. Our last bin has a lot of small ones and when we sum them all up we will wind up getting a certain weight. Now when you look at these boxes one of the things that you see, over on the left, although you don't have nearly as many, they're much heavier than what you have on the right. When we look at the right, we have a lot of little ones and so consequently the number average is going to be particularly affected by the number of small ones we have. Since we have more small ones than we do the big ones. So the number average is going to be very much affected by the small sizes. When we start looking at the left-hand side where we have a lot of the heavier spheres, then what will happen is the average that we're going to calculate, namely the weight average, is going to be determined primarily by those bins which have the largest BB's in the bin. So let's see how this works when we start calculating what goes on in a molecular weight distribution of polymers. So the problem that we're going to be looking at is a batch of polyvinyl chloride, and it's going to have a collection of polymer chains of different lengths. And what we want to do is we want to ultimately calculate the number averages, the weight averages, and the polydispersity. All right, so here are our data. Now what we have in each one of our bins, rather than having polymers of only one chain length, we have a distribution. They go basically from 5,000 to 10,000. Subsequently, as we go down the list what we see is a molecular weight range for each one of these bins and at the very bottom we have the heaviest collection associated with the molecules or the polymers that have molecular weights of 35,000 to 40,000. So rather than dealing with those individually, what we can do is take each one of the bins and we can come up with an average. So for each one of those bins, I have an average calculated based upon the distribution and we'll just assume that it is a simple mean. Now, when I start looking at the data, what was included was the number fraction that I have for each one of those bins. So for example, when we're looking at the bin that has the mean of 7500, the number that we have or the number fraction we have is .05. So each one of those bins then has a certain number fraction and of course this is part of the initial data. What you also know is that the sum of all those n sub i's ultimately has to equal to one. So let's consider how we're going to work the problem through. So what we do is, we start out with our raw set of data, which is on the left-hand side. So that includes the range, the mean and n sub i. So, here's our range. We then take that range, calculate a mean, and that's what's in this column here. Then we were also given how each fraction had a certain fraction number in each one of those bins, and that's what's given in the third column. Now, if we take the product of the second and third column, what we're going to get is the associated weight we have based upon number in each one of the bins. So that turns out to be the values that we have for each of the bin in terms of the bins themselves. And when we sum all of those up, what we will wind up getting is a number average, which is 21,150. Now, in order for us to calculate the distributions associated with the weight averages, what we're going to do is convert the n sub i's, that is the number we have in, or fraction we have in each bin, to a weight fraction in each bin and that turns out to be each one of those values in the niMi column. That is the average in each one of those bins, and what we wind up with is we take each of those bins divided by 21,150 and that's going to give us our weight fraction in each of the bins. And so those are what's given in the column that's presented here. Now if we take the product of the mean with the weight, or fraction per bin, what we're going to get is the column which we over here for number four. When we sum all of those up, that's going to give us the weight average associated with this collection of polymers. So if we take the weight average divided by the number average, it's going to give us the polydispersity index. And it turns out that number is 1.1. Now when we go through this calculation, what we see is the weights are skewed to the heavily weighted materials in the bins and the numbers are skewed which will be associated with the lower molecular weights. And if we look at a distribution of the number or the amount of polymers in each bin as a function of molecular weight, it's going to give us the distribution that looks something like the plot that I have on the screen right now. And you see that the weight average is to the right, the number average lies to the left. So this is the way for us to calculate those two variables, the weight average and the number average, of a collection of polymers. Thank you.