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Spectrophotometry example

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Let's see if we can tackle this spectrophotometry example. I took this from the Kotz, Treichel and Townsend Chemistry & Chemical Reactivity book and did it with their permission. So let's see what the problem is. It says a solution of potassium permanganate-- let me underline that in a darker color-- potassium permanganate has an absorbance of 0.539 when measured at 540 nanometer in a 1 centimeter cell. So this 540 nanometers is the wavelength of light that we're measuring the absorbance of. And so this is probably a special wavelength of light for potassium permanganate, one that it tends to be good at absorbing. So it'll be pretty sensitive to how much solute we have in the solution. OK, and the beaker is 1 centimeter. So that's just the length right there. What is the concentration of potassium permanganate? Prior to determining the absorbance for the unknown solution, the following calibration data were collected for the spectrophotometer. The absorbances of these known concentrations were already measured. So what we're going to do is we're going to plot these. And then, essentially, this absorbance is going to sit on the line. We learned from the Beer-Lambert law, that is a linear relationship between absorbance and concentration. So this absorbance is going to sit some place on this line. And we're just going to have to read off where that concentration is. And that will be our unknown concentration. So let's plot this first. Let's plot our concentrations first. So this axis, the horizontal axis, will be our concentration axis. I'll draw the axis in blue right there. Let me scroll down a little bit more. I just need to make sure I have all this data here. So this is concentration in molarity. And let's see, it goes from 0.03 all the way to 0.15. So let's make this 0.03, then go three more. This over here is 0.06. One, two, three, then this over here is 0.09. This over here is 0.12. And then this over here is 0.15. And then the absorbances go-- well it's close to 0, or close to 0.1-- all the way up to close to 1. So let's make this right here 0.1. Let's make this 0.2, 0.3, 0.4, 0.5-- almost done-- 0.6, 0.7, 0.8, and then 0.9. And that, essentially, covers all of the values of absorbency that we have here. So let's plot the first one. When we had a concentration of potassium permanganate at 0.03 molar, our absorbance was 0.162. So 0.03, and then it goes to 0.16. This is 0.15, so 0.162 is going to be right over there. And then when we had 0.06 molarity of potassium permanganate our absorbance was 0.33. So 0.06, 0.33 which is right about-- this is 0.35, so 0.33 would be right about there. And we already see an interesting line form, but I'll plot all of these points. So at 0.09 molarity, we have 0.499. So almost 0.5 right over there. That's that value. And then at 0.12, we have 0.67 absorbance. So at 0.12, we have 0.67. So this is 0.12. This would be 0.65, so we have 0.67 absorbance right over there. And actually, what we're doing here, we're actually showing you that the Beer-Lambert law is true. At specific concentrations, we've measured the absorbance and you see that it's a linear relationship. Anyway, let's do this last one. At 0.15 molarity, we have absorbance of 0.84. So this right here is 0.15. I want to make sure I don't lose track of that line. And 0.84 is right over there. So you see the linear relationship? Let me draw the line. I don't have a line tool here, so I'm just going to try to freehand it. I'll draw a dotted line. Dotted lines are a little bit easier to adjust. I'm doing it in a slight green color, but I think you see this linear relationship. This is the Beer-Lambert law in effect. Now let's go back to our problem. We know that a solution, some mystery solution, has an absorbance of 0.539-- let me do our mystery solution in-- well, I've pretty much run out of colors. I'll do it in pink-- of 0.539. So our absorbance is 0.5-- this is 0.55, so 0.539 is going to be right over there. And we want to know the concentration of potassium permanganate. Well, if we just follow the Beer-Lambert law, it's got to sit on that line. So the concentration is going to be pretty darn close to this line right over here. And this over here looks like 0.10 molar. So this right here is 0, or at least just estimating it, looking at this, that looks like 0.10 molar, or 0.10 molarity for that solution. So that's the answer to our question just eyeballing it off of this chart. Let's try to get a little bit more exact. We know the Beer-Lambert law, and we can even figure out the constant. The Beer-Lambert law tells us that the absorbance is equal to some constant, times the length, times the concentration, where the length is measured in centimeters. So that is measured in centimeters. And the concentration is measured in moles per liter, or molarity. So we can figure out-- just based on one of these data points because we know that it's 0-- at 0 concentration the absorbance is going to be 0. So that's our other one. We can figure out what exactly this constant is right here. So we know all of these were measured at the same length, or at least that's what I'm assuming. They're all in a 1 centimeter cell. That's how far the light had to go through the solution. So in this example, our absorbance, our length, is equal to 1 centimeter. So let's see if we can figure out this constant right here for potassium permanganate at-- I guess this is probably standard temperature and pressure right here-- for this frequency of light. Which they told us up here it was 540 nanometers. So if we just take this first data point-- might as well take the first one, we get-- the absorbance was 0.162. That's going to be equal to this constant of proportionality times 1 centimeter. That's how wide the vial was. Times-- now what is the concentration? Well when the absorbance was 0.162, our concentration was 0.03 times 0.-- actually, I'll write all the significant digits there-- 0.0300. So if we want to solve for this epsilon, we can just divide both sides of this equation by 0.0300. So you divide both sides by 0.0300 and what do we get? These cancel out, this is just a 1. And so you get epsilon is equal to-- let's figure out what this number in blue is here. And I'll take out my calculator. And I have 0.162 divided by 0.03 is equal to 5.4. And actually more significant is, we could really say it's 5.40 since we have at least three significant digits in both situations. So 5.40 is our proportionality constant. And you would actually divide by 1 in both cases. We just want the number here. But if you wanted the units, you'd want to divide by that 1 centimeters as well. Now we can use this to figure out the exact answer to our problem without having to eyeball it like we just did. We know that for potassium permanganate at 540 nanometers, the absorbance is going to be equal to 5.4 times-- and I'll put the units here. The units of this proportionality constant right here is liters per centimeter mole. And you'll see it'll just cancel out with the distance which is in centimeters, or the length, and the molarity which is in moles per liter. And it just gives us a dimension list, absorbance. So times-- in our example the length is 1 centimeter-- times 1 centimeter, times the concentration. Now in our example they told us the absorbance was 0.539. That's going to be equal to 5.4 liters per centimeter mole, times 1 centimeter, times our concentration. Well this centimeter cancels out with that centimeter right over there. And then we can just divide both sides by 5.4 liters per mole. So let's do that. Let's divide both sides by 5.4 liters per mole, and what do we have? So on the right-hand side, all of this business is going to cancel out. We're just going to have this concentration left over. So our concentration is equal to-- let's figure out what this number is. So we have 0.539 divided by 5.4 gives us-- so we only have-- well this is actually 5.40. So we actually have three significant digits. So we could say 0.0998. So this is 0.0998. And then if you're dividing by liters per mole, that's the same thing as moles per liter. So we're able to get a much more exact answer by actually just going through the math. But this is pretty darn close. This exact answer's pretty darn close to what we estimated just by eyeballing it off the chart. 0.1 is only a little bit more than 0.0998. Anyway, hopefully you enjoyed that.