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Spectrophotometry and the Beer–Lambert Law

Spectrophotometry is a technique that uses light absorption to measure the concentration of an analyte in solution. The amount of light absorbed by a solution is related to the analyte concentration by the Beer–Lambert law, which is expressed as follows: A = εbc, where ε is the molar absorptivity of the analyte, b is the path length (the distance the light travels through the solution), and c is the concentration of the analyte. Created by Sal Khan.

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Video transcript

- [Instructor] What I want to do in this video is to talk a little bit about spectrophotometry. Spectrophotometry, photometry, which sounds fairly sophisticated, but it's really based on a fairly simple principle. So if I have, let's say we have two solutions that contain some type of solute. So that is solution one, and then this is solution two. And let's just assume that our beakers have the same width. Now let's say solution, let me put it right here. Number one, and number two. Now let's say that solution one has less of the solute in it. So let me, let me make... So that's the water line right there. So this guy has less of it, and let's say it's yellow, or to our eyes, it looks yellow. So this has less of it. So this hasm actuallym let me do it this way. Let me shade it in like this. So it has less of it. And let's say solution number two has more of the solute, so it's more. So I'll just kind of represent that as more closely packed lines. So the concentration of the solute is higher here. So let me write higher concentration. Higher concentration. Concentration, and let's say, and this is a lower, lower concentration. Now let's think about what will happen if we shine some light through each of these beakers, and let's just assume that we are shining at a wavelength of light that is specific to the... That that is specifically sensitive to the solute that we have dissolved in here. But I'll just leave that pretty general right now. So let's say I have some light here of some intensity. So let's just call that... Let's call that the incident intensity. I'll just say that it's I0, so it's some intensity. What's going to happen as the light exits the other side of this beaker right here? Well, some of it is going to be absorbed by our little molecules inside the beaker, so you're going to have less light come out to the other side. I'll call this I1. Now, in this situation, if we shined the same amount of light into this beaker, so it's the same number that is, that is the same, the same intensity of light, what's going to happen? Well, more is going to be absorbed as the light travels through this beaker. It's just going to bump into more molecules, 'cause it's a higher concentration here. So the light that comes out when you have a higher concentration, I'll call that the intensity, I'll call that I2, this is going to have a lower intensity of light that's being transmitted than this one over here. In this case, I2 is going to have a lower intensity, is going to be less than I1. If you have another beaker that is maybe twice as wide, it's twice as wide, and let's say it has the same concentration as number two, we'll call this one number three, it has the same concentration as number two. So I'll try to make it look fairly similar, and you were to shine some light in here. Let's say you shine the same light in here, and you have some light that makes it through, that exits, and then this is actually what your eyes would see. So this is I3 right there. What do you think's going to happen? Well, it's the same concentration, but this light has to travel a further distance of that concentration. So once again, it's going to bump into more molecules and more of it will be absorbed, and so less light will be transmitted. So I2 is less than I1, and I3, I3 is actually going to be the least. And if you were looking at these, this has the least light, this has a little bit more light being transmitted, this has the most light being transmitted. So if you were to look at this, if you placed your eyeball right here, this one right here would have the lightest color. You're getting the most light into your eye. This would be a slightly darker color, and this would be the darkest color. That makes complete sense. If you dissolve something, if you dissolve a little bit of something in water, it will still be pretty transparent. If you dissolve a lot of something in water, it'll be more opaque. And if the cup that you're dissolving in, or the beaker that you're in, gets even longer, it'll get even more opaque. So hopefully, that gives you the intuition behind spectrophotometry. And so the next question is, well, what is it even good for? Why would I even care? Well, you could actually use this information. You could see how much light is transmitted versus how much you put in to actually figure out the concentration of a solution. That's why we're even talking about it in a chemistry context. So before we do that, and I'll show you an example of that in the next video, let me just define some, really, some terms of ways of measuring how concentrated this is, or ways of measuring how much light is transmitted versus how much was put in. So the first thing I will define is transmittance. And so when the people who defined it said, well, you know, what we care about is how much is transmitted versus how much went in, so let's just define transmittance as that ratio. So in this example, the transmittance of number one would be the amount that got through over the amount that you put in. Over here, the transmittance would be the amount that you got out over the amount that you put in. And as we see, this one right here will be a lower number. I2 is lower than I1, so this will have a lower transmittance then number one. So let's call this transmittance two. This is transmittance one. And transmittance three is the light that comes out, that gets through, over the light that goes in, and this is the smallest number, followed by that, followed by that. So this will have the least transmittance, it's the most opaque, followed by that, followed by that. Now another definition, which is really kind of a derivative of the transmittance, and not in the calculus sense, it's just derived from transmittance, and we'll see, it has pretty neat properties, is the notion of absorbance. And so here, we're trying to measure, how good is it at absorbing? This is measuring, how good are you at transmitting? A higher number says you're transmitting a lot, but absorbance is how good you're absorbing, so it's kind of the opposite. If you're good at transmitting, that means you're bad at absorbing, You don't have a lot to absorb. If you're good at absorbing, that means you're not transmitting much. So absorbance. Absorbance, right here, and absorbance is defined as the negative log of transmittance. And this logarithm is base 10, or you could view that if the transmittance, we've already defined as the negative, the negative log of the light that is transmitted over the light that is input, but the easiest way is the negative log of the transmittance. And so, if transmittance is a large number, absorbance is a small number, which makes sense. Now, what's also cool about this. is there something called the Beer-Lambert law, which you could verify, and this is... We'll actually use this in the next video, Beer-Lambert law. I actually don't know the history of where it came from, and I'm sure it's based on somebody named Beer, but I always imagined it's based on someone transmitting light through beer, the Beer-Lambert law. And this tells us, this tells us that the absorbance is proportional to the path length. So this would be, how far does the light have to go through the solution? So it's proportional to the path length times the concentration, times the concentration. Usually we use molarity for the concentration. Or another way to say it is that the absorbance is equal to some constant. It's usually a lowercase Epsilon like that. Some constant, and this is dependent on the solution, or the solute in question, what we actually have in here and the temperature and the pressure and all of that. It's equal to constant times the length it has to travel, times the concentration. Let me make it clear right here. This thing right here, this thing right here is concentration. Concentration. The reason why this is super useful, as you can imagine, so let's say we have an axis right here. That's axis, and over here, I'm measuring concentration. This is our concentration axis, and we're measuring it as molarity. And let's say the molarity starts at zero. It goes, you know, I don't know, .1, .2, .3, so on and so forth. And over here, you are measuring absorbance. In the vertical axis you measure absorbance. Now, let's say you have some solution, and you know the concentration, you know it is a .1 molar concentration. So let me write down M for molar. And you measure its absorbance and you just get some number here. So you measure its absorbance and you get its absorbance, so this is a low concentration, didn't absorb that much. You get, I don't know some number here, so let's say it's .25. And then let's say that you then take another known concentration, let's say .2 molar, and you say that, oh look, it has an absorbance right here at .5, and I should put a zero in front of these. 0.5 and 0.25. What this tells you, this is a linear relationship. For any concentration, the absorbance is going to be on a line. And if you want a little review of algebra, Epsilon times the length will be the slope. But the important thing to realize is that you have a line here. You have a line here, and why? And the reason that's useful is you could use a little bit of algebra, figure out the equation of a line, or you could just look at it graphically and say, okay, I had two known concentrations, and I was able to figure out the absorbance. You can then go the other way around. You could then measure for some unknown concentration. You could figure out its absorbance. So let's say there's some unknown concentration, and you figure out its absorbance is right over here. Let's say it's .4. Then you can just go on this line right here, and you say, okay, well then that must be, that must be a concentration of this, well, whatever number this is. And you could measure it, or you could actually figure it out algebraically. And so this will be pretty close to .2 molar, a little bit less than 0.2 molar. And we're gonna actually do an example of that in the next video.