So guys, now we're finally going to get to the last piece of information that you can derive from a proton NMR and that's going to be what we call the integration. The integration describes the relative quantities of all the different hydrogens that are present and the integration will express them as relative ratios. Something that we've been talking about since the very beginning of proton NMR is that some of your peaks are going to be taller and some of them are going to be shorter. Now what's deceiving is to think that your integration depends only on height. It doesn't. It actually also depends on the width of the peak. As you can see, I have an example NMR here with 3 different peaks of different heights but they also have slightly different widths. We have to take that into account when we look at integration. In fact, integration is just a fancy way of saying that you're taking the area under a curve. You're not just looking at the exact height of it. You're looking at all the little slices underneath that curve and you're adding them up together. And then when you stack them up, you take the area. And that's really going to be what determines the relative ratio of these hydrogens.

For those of you who have already taken Calculus 2, this sounds very familiar to you. This is basically the concept of Riemann sums and taking integrals. But for this course, we're just going to let a computer do the work for us. So no fancy integrals that we're going to have to figure out.. The computer is going to figure out these Areas Under the Curve or these integrals, right? Basically, the integral of a function. It's going to do that work for us and it's going to spit out these different values. This red line at the top is actually what we call the integration. As you can see, they have different heights. One of them appears to be yay high. The next one seems to be about twice as tall as the other one. And the one after that seems to be about 3 times as tall as the other one.

What that means is that the computer actually looked at these functions, looked at this as a function and it added up all the little pieces underneath it and came up with a height of this. Now the important part for us isn't to be able to do the actual integral calculus to figure out the integration. It's to compare these heights and to say, well, how many of this type, let's say this is proton A, proton B, and proton C. How many of A are there? How many of B and how many of C and what are the ratios? In this case, the ratio would be a 1 to 2 to 3 ratio based on the integrations that the computer spit out for us. That's what we mean by relative ratio. Now does that mean that there's exactly 6 protons in this molecule? Actually in this case, it does mean that because it looks like I have 1, 3, 6 protons total. In this case, the ratio actually added up to the total number of protons. But just so you know, this ratio could also work for 12 protons or 18 protons or any multiple of 6. What it would basically tell us is that hey, you might have 18 protons but they're in a ratio of 1 to 2 to 3. And then that would basically tell you that you've got this many of type A, this many of type B, and this many of type C.

That said, we basically have all the information that we need now to draw our own NMR spectra. I know you're getting really excited at this point. What I'm going to do is introduce this cumulative practice problem and then I'll have you guys try it yourselves. I want you to draw the entire NMR spectrum for this molecule. Now what you're going to notice is that I did this is actually an easier way to draw it than just giving you a blank chart because I'm kind of giving you boxes to fill out. As long as you have those 9 boxes filled out, you get it right. Let me just show you what I'm looking for. First of all, notice that we have this PPM line right here. That means this is 0 and that means this is some high number. We're not going to worry about exact numbers here. We're just going to worry about the order that this should be more downfield than this. They should have higher numbers as they go along. What I want you to do is that the type of proton, H A H B H C goes here. You're going to order those protons in order of chemical shift. This is going to be based on basically the chemical shift. If it's very downfield, then that should be the one that goes furthest off to the left. Also, you are going to be responsible for so you don't have to tell me the exact value of the shift. You just have to put them in order. In terms of the splitting, you should be able to draw the types of splits that you would get here and we're going to assume n+1. So assume that the n+1 rule works here. Now by the way, notice that in this question, I didn't say assume n+1. Guess what? That's because it's up to you to determine that. If your teacher, if your professor has not made an explicit request to draw a tree diagram or to use a fancy formula, then you're always going to assume n+1. It's the simplest way to do splitting. I want you to draw the splits that would be predicted by Pascal's Triangle and n+1 there. Then finally the integrations. I want you to express the integrations as basically ratios or number of hydrogens. An example of an integration would be I'll just put it here. An example would be let's say, let me just give 3 H. 3 H would tell me that I have 3 hydrogens that are of that type, that are in that space. Now obviously we're going to go ahead and erase that because that's probably not the right answer for that box. Go ahead and what I would try to do, I'm trying to guide you through this so you can think of it one step at a time. Figure out what the chemical shifts are in terms of order of the protons. You can put that in these boxes here, 1, 2, 3. Then figure out what the splits are going to be. Figure out what types of splits they would be based on n plus 1. And then finally add up the number of hydrogens that would be of that type to get the final integrations. Then I'll go ahead and I'll solve the whole thing for you. Now it's your turn. Go for it.