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Let's see if we can tackle this

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spectrophotometry example.

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I took this from the Kotz, Treichel and Townsend

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Chemistry & Chemical Reactivity book and did it

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with their permission.

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So let's see what the problem is.

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It says a solution of potassium permanganate-- let

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me underline that in a darker color-- potassium permanganate

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has an absorbance of 0.539 when measured at 540 nanometer

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in a 1 centimeter cell.

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So this 540 nanometers is the wavelength of light that we're

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measuring the absorbance of.

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And so this is probably a special wavelength of light

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for potassium permanganate, one that it tends to be good

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at absorbing.

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So it'll be pretty sensitive to how much solute we have in

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the solution.

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OK, and the beaker is 1 centimeter.

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So that's just the length right there.

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What is the concentration of potassium permanganate?

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Prior to determining the absorbance for the unknown

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solution, the following calibration data were

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collected for the spectrophotometer.

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The absorbances of these known concentrations

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were already measured.

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So what we're going to do is we're going to plot these.

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And then, essentially, this absorbance is going

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to sit on the line.

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We learned from the Beer-Lambert law, that is a

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linear relationship between absorbance and concentration.

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So this absorbance is going to sit some place on this line.

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And we're just going to have to read off where that

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concentration is.

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And that will be our unknown concentration.

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So let's plot this first. Let's plot our concentrations

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first. So this axis, the horizontal axis, will be our

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concentration axis.

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I'll draw the axis in blue right there.

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Let me scroll down a little bit more.

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I just need to make sure I have all this data here.

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So this is concentration in molarity.

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And let's see, it goes from 0.03 all the way to 0.15.

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So let's make this 0.03, then go three more.

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This over here is 0.06.

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One, two, three, then this over here is 0.09.

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This over here is 0.12.

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And then this over here is 0.15.

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And then the absorbances go-- well it's close to 0, or close

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to 0.1-- all the way up to close to 1.

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So let's make this right here 0.1.

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Let's make this 0.2, 0.3, 0.4, 0.5-- almost done-- 0.6, 0.7,

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0.8, and then 0.9.

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And that, essentially, covers all of the values of

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absorbency that we have here.

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So let's plot the first one.

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When we had a concentration of potassium permanganate at 0.03

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molar, our absorbance was 0.162.

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So 0.03, and then it goes to 0.16.

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This is 0.15, so 0.162 is going to be right over there.

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And then when we had 0.06 molarity of potassium

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permanganate our absorbance was 0.33.

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So 0.06, 0.33 which is right about-- this is 0.35, so 0.33

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would be right about there.

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And we already see an interesting line form, but

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I'll plot all of these points.

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So at 0.09 molarity, we have 0.499.

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So almost 0.5 right over there.

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That's that value.

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And then at 0.12, we have 0.67 absorbance.

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So at 0.12, we have 0.67.

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So this is 0.12.

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This would be 0.65, so we have 0.67

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absorbance right over there.

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And actually, what we're doing here, we're actually showing

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you that the Beer-Lambert law is true.

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At specific concentrations, we've measured the absorbance

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and you see that it's a linear relationship.

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Anyway, let's do this last one.

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At 0.15 molarity, we have absorbance of 0.84.

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So this right here is 0.15.

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I want to make sure I don't lose track of that line.

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And 0.84 is right over there.

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So you see the linear relationship?

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Let me draw the line.

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I don't have a line tool here, so I'm just going to try to

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freehand it.

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I'll draw a dotted line.

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Dotted lines are a little bit easier to adjust. I'm doing it

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in a slight green color, but I think you see this linear

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relationship.

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This is the Beer-Lambert law in effect.

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Now let's go back to our problem.

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We know that a solution, some mystery solution, has an

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absorbance of 0.539-- let me do our mystery solution in--

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well, I've pretty much run out of colors.

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I'll do it in pink-- of 0.539.

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So our absorbance is 0.5-- this is 0.55, so 0.539 is

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going to be right over there.

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And we want to know the concentration of potassium

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permanganate.

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Well, if we just follow the Beer-Lambert law, it's got to

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sit on that line.

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So the concentration is going to be pretty darn close to

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this line right over here.

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And this over here looks like 0.10 molar.

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So this right here is 0, or at least just estimating it,

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looking at this, that looks like 0.10 molar, or 0.10

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molarity for that solution.

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So that's the answer to our question just eyeballing it

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off of this chart.

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Let's try to get a little bit more exact.

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We know the Beer-Lambert law, and we can even

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figure out the constant.

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The Beer-Lambert law tells us that the absorbance is equal

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to some constant, times the length, times the

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concentration, where the length is measured in

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centimeters.

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So that is measured in centimeters.

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And the concentration is measured in moles per liter,

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or molarity.

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So we can figure out-- just based on one of these data

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points because we know that it's 0-- at 0 concentration

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the absorbance is going to be 0.

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So that's our other one.

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We can figure out what exactly this constant is right here.

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So we know all of these were measured at the same length,

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or at least that's what I'm assuming.

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They're all in a 1 centimeter cell.

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That's how far the light had to go through the solution.

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So in this example, our absorbance, our length, is

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equal to 1 centimeter.

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So let's see if we can figure out this constant right here

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for potassium permanganate at-- I guess this is probably

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standard temperature and pressure right here-- for this

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frequency of light.

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Which they told us up here it was 540 nanometers.

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So if we just take this first data point-- might as well

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take the first one, we get-- the absorbance was 0.162.

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That's going to be equal to this constant of

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proportionality times 1 centimeter.

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That's how wide the vial was.

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Times-- now what is the concentration?

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Well when the absorbance was 0.162, our concentration was

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0.03 times 0.-- actually, I'll write all the significant

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digits there-- 0.0300.

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So if we want to solve for this epsilon, we can just

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divide both sides of this equation by 0.0300.

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So you divide both sides by 0.0300 and what do we get?

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These cancel out, this is just a 1.

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And so you get epsilon is equal to-- let's figure out

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what this number in blue is here.

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And I'll take out my calculator.

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And I have 0.162 divided by 0.03 is equal to 5.4.

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And actually more significant is, we could really say it's

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5.40 since we have at least three significant digits in

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both situations.

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So 5.40 is our proportionality constant.

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And you would actually divide by 1 in both cases.

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We just want the number here.

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But if you wanted the units, you'd want to divide by that 1

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centimeters as well.

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Now we can use this to figure out the exact answer to our

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problem without having to eyeball it like we just did.

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We know that for potassium permanganate at 540

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nanometers, the absorbance is going to be equal to 5.4

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times-- and I'll put the units here.

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The units of this proportionality constant right

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here is liters per centimeter mole.

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And you'll see it'll just cancel out with the distance

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which is in centimeters, or the length, and the molarity

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which is in moles per liter.

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And it just gives us a dimension list, absorbance.

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So times-- in our example the length is 1 centimeter-- times

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1 centimeter, times the concentration.

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Now in our example they told us the absorbance was 0.539.

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That's going to be equal to 5.4 liters per centimeter

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mole, times 1 centimeter, times our concentration.

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Well this centimeter cancels out with that centimeter right

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over there.

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And then we can just divide both sides by

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5.4 liters per mole.

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So let's do that.

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Let's divide both sides by 5.4 liters per mole,

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and what do we have?

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So on the right-hand side, all of this business is going to

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cancel out.

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We're just going to have this concentration left over.

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So our concentration is equal to-- let's figure out what

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this number is.

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So we have 0.539 divided by 5.4 gives us-- so we only

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have-- well this is actually 5.40.

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So we actually have three significant digits.

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So we could say 0.0998.

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So this is 0.0998.

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And then if you're dividing by liters per mole, that's the

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same thing as moles per liter.

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So we're able to get a much more exact answer by actually

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just going through the math.

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But this is pretty darn close.

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This exact answer's pretty darn close to what we

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estimated just by eyeballing it off the chart.

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0.1 is only a little bit more than 0.0998.

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Anyway, hopefully you enjoyed that.

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