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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.