Uncertainty can be thought of as the range (+/-) that is associated with any given value.

Types of Uncertainty

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Types of Uncertainty

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So as we've said before, we said that with any calculation there lies a level of uncertainty which we call experimental error. Now here we can talk even more specifically about certain types of uncertainties, we're gonna first say that absolute uncertainty represents the plus or minus value associated with any numerical calculation. So for example, we say a student delivers 25.0 plus or minus 0.2 mL of water to a mixture. In this case, the plus or minus value of 0.2 mls would be the uncertainty that would represent our absolute uncertainty within this calculation. Now we're gonna say besides our absolute uncertainty, we have our relative uncertainty. The relative uncertainty is the absolute uncertainty divided by the associated measurement. So here in this case are absolute uncertainty, we said again, is the plus or minus 0.2 Ml So that would go on top divided by our measurement, which is the 25 mls that would give us our relative uncertainty of 250.1 from there. We could also calculate our percent relative uncertainty. Now our percent relative uncertainty is just our relative uncertainty, multiplied by 100. So, going and multiplying our relative uncertainty by 100 gives us 1000.1% as our percent relative uncertainty. As we delve deeper and deeper into calculations dealing with uncertainties, it's gonna become instrumental that you remember these three different types of uncertainties and their usefulness for different types of calculations will undergo Now that we've seen this, we can attempt to do the example question that's left here on the bottom. So you can temp it on your own at first, but if you get stuck, just come back and see how I approach that same question.

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example

Types of Uncertainty

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So here we have to calculate the relative and percent relative uncertainty from the given problem. Here we have as our question 3.25 plus or minus 0.3. Remember the plus or minus 0.3 represents our absolute uncertainty. So to figure out our relative uncertainty, remember it's our absolute uncertainty divided by the measurement itself. So when we do that we're gonna get initially .009231 here. Um it's customary to just rounded here to just one sig fig So we'll get 10.9 as our relative uncertainty. Rewriting that would become 3.25 plus or minus 0.9 as our complete relative uncertainty. Now if we wanted our percent relative uncertainty we take that value that we just got multiply it by 100. So you have .9%. So our percent relative uncertainty would be 3.25 plus or minus 0.9%. What these numbers are saying is that we expect our calculation the correct value, the true value to be somewhere within this range. So our answer should be 3.25 for the measurement plus or minus 0.9% of that. So our actual value would be in the range of minus 0.9% of this. 3.252 plus 0.9% of that 3.25. It would last somewhere within that range. Now that you've seen these two examples, the one that we did up above and then this new one here. See if you can attempt to this practice question here. I'm asking for the absolute uncertainty. When we're given the percent relative uncertainty from the very beginning, I'll give you guys a hint, try to work backwards, do the exact opposite of all the steps that we've done and see if you can figure out what the absolute uncertainty is once you've done that, come back and take a look at how I approach that same question.

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Problem

Problem

Calculate the absolute uncertainty from the given problem.

6.77 (± 5.6%)

A

0.38

B

0.056

C

0.83

D

0.038

Propagation of Uncertainty from Random Error

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Propagation of Uncertainty

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So here we're gonna talk about the calculations that we have to employ in order to deal with uncertainties when it comes to addition subtraction, multiplication and division. Now these are all related to random error. Of course when it comes to our calculations. So here we're gonna say with addition and subtraction or multiplication and division, we will use certain rules for propagation of our answer here. We're gonna use what's called the real rule. The real rule says that the first digit of the absolute uncertainty is the last significant digit in the answer. What does that say? So for example, let's say we had Absolute uncertainty of .0005 And we had a number a measurement of 7.95,. Here the first digit in the absolute uncertainty. That is significant. Is that five? What does it occur? It occurs in the fourth decimal place. That would mean that my measurement has to have four decimal places within it. So here are four decimal places here because there's a six year, I'd have to round this number up. So the measurement here would be 7.9588. So the 7.9588 plus or minus 0.5. As our measurement with its absolute uncertainty, we'll do another one. Let's say that our absolute uncertainty here was 0.1. And let's say that we had 13.2 35. So here our first digit in the absolute uncertainty. That's significant is that one? It's in the first decimal place that means our measurement has to have one decimal place. So here's our decimal place because this number here is a three. We wouldn't do anything to that too. So this comes out to 13.2 plus or minus 0.1. So when we're doing these propagation of our answers, we have to keep this in mind when we're dealing with addition, subtraction, multiplication or division when it comes to the measurement with the absolute uncertainty. Now here, if we take a look at addition and subtraction, we'll learn all the different ins and outs in terms of the calculations necessary to get our correct answer. But remember always rely on the real rule, click on the next video to see me go through this particular example when it comes to addition and subtraction.

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Propagation of Uncertainty

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So here for addition and subtraction. The uncertainty in our final answer is determined from each individual absolute uncertainty. So if we take a look here at section two, we have these three measurements. Now remember when we're adding or subtracting our final answer is the one with the least number of decimal places. So here this has two decimal places, Two decimal places, two decimal places. Which is why our answer here has two decimal places. But now we have to determine what the absolute uncertainty is right here. Which will be our uncertainty four because we already have these three initially. Now when it comes to addition and subtraction. So whether our units are adding the uncertainties are adding or whether the uncertainties are subtracting, it's the same basic method. We're gonna say that here, the uncertainty that we're looking forward equals the square root of each uncertainty squared added together. So for this example are three uncertainties are absolute uncertainties are 30.4 point 3.3. So we plug them into this formula. So 0.4 squared plus 0.3 squared plus 0.3 squared here right now would have the square root of 0.34. Initially When we plug that in we get what we get .058. Now this little eight here means that it is not a significant digit. Remember when it comes to the absolute uncertainty we're relying on the real rule. So we're looking at the first digit that's significant within the absolute uncertainty Here. I'm placing this eight here to let us know that the first significant digit in my absolute uncertainty is this five. But because that eight is there it's going to be rounded up to six. So we know that our measurement based on the least number of decimal places came out to 6.85. And then here we round up to six because this eight cause this five to get rounded up. So that's six. So going back on the real rule, we're gonna say here that this first significant digit within my absolute uncertainty happens in the second decimal place. That means that my measurement has to have two decimal places within it, which it does. So this is my final answer When we've done propagation of our answer, when it's dealing with addition and subtraction, let's say that that first digit happened in the third decimal place, That would mean that my answer would have to have three decimal places. So we'd have to put an additional number here. 6.850. But in this case it didn't so we didn't have to worry about it. But just remember the real rule, we look at the first significant digit in my absolute uncertainty to determine what decimal places in so we can determine how many decimal places my measurement will have at the end. So when it comes to addition and subtraction it's pretty straightforward. We take those uncertainties, we square them, we add them together, take the square root. Follow the real rule to get our final answer with multiplication and division, though it's a bit more strenuous more work involved in getting our final answer. So come back. Take a look at the next video and see how I go step by step to help us get the final answer when it comes to propagation of our answer, dealing with multiplication and division.

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example

Propagation of Uncertainty

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So when it comes to multiplication and division, like I said the steps much more in depth in terms of finding our final answer. So we're gonna say for multiplication and division we must first convert the absolute uncertainties into percent relative uncertainties. So here our values are 3.68 plus or minus 0.5 times 1.15 plus or minus 0.6, divided by 0.92 plus or minus 0.6. Here for our measurements, We see that the 3.68 has in it three significant digits or figures here. This one also has three significant figures. This one here has to at the moment our answer for our measurement will be 4.6 but again following the real rule that could change so it would be best to write out the whole answer for now and then round later on here we just put as 4.6 for simplicity sakes right now. Now here are uncertainty. We do not know. Remember when it comes to multiplication of our uncertainties or division of our uncertainties we have to change those absolute uncertainties into percent relative uncertainties. You would square them, add them together and take the square root of that answer. So here Remember to find our percent relative uncertainties. We first find our relative uncertainty. That would mean that we will take each absolute uncertainty and divided by its measurement. So we have .05 divided by 3.68 times 100 to make it a percentage. This absolute uncertainty divided by its measurement times 100. This uncertainty divided by its measurement times 100. As a result we have each one of these percent relative uncertainties again here um we're just keeping one sig fig in terms of our percent relative uncertainty. But here we put these little numbers here to say that there are not significant but based on what their values are, it could cause these numbers to round up around down. So here we bring those percentages down, we square them, Add them together. Doing that gives me 71.25% within this square root. Taking the square root of that gives me 8.4%. Now to find our absolute uncertainty, remember that would just be my percent relative uncertainty divided by 100 to get its decimal form then multiplied by its measurement. Doing that gives me my absolute uncertainty here. So here's my absolute uncertainty. Remember with the absolute uncertainty we're gonna say that the first significant digit here this night is not significant but it does cause this three to get rounded up to four. So remember following the real rule in the absolute uncertainty, the first significant digit in my absolute uncertainty determines the last significant figure within my measurement. So my first significant figure in my absolute uncertainty is in the first decimal place. So my final answer my measurement has to have one decimal place. So it's 4.6. So here this would be our measurement with our absolute uncertainty and our measurement with our percent relative uncertainty here. So again as before. When it comes to addition and subtraction, it's pretty straightforward. But when it comes to multiplication and division, we have to change our absolute uncertainties into percent relative uncertainties, then back to absolute uncertainties at the end. So it's quite a bit of work. But this is the method that we have to use for propagation of our answer depending on which operation we're doing. So as we delve deeper and deeper into these calculations, we'll move on from calculations to word problems. We have to keep in mind these different types of rules based on the operation being used.

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example

Propagation of Uncertainty Calculations

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determine the absolute and relative uncertainty to the following addition problem. Since we have just simply addition here or if we have subtraction we don't have to worry about extra steps needed to get our final answers. So what we're gonna do here first is realize with our measurements when you're adding subtracting it's gonna be the least number of decimal places. But following the real role, the first digit in the absolute uncertainty will determine the last significant figure or digit in my measurement for now. When I said when I add these three numbers together, it's gonna give me 5.0 to 8, make that eight small because as of right now this has three decimal places, Two decimal places, three decimal places. So we're technically supposed to have two decimal places at this point. But again we don't know that for sure because we don't know the absolute uncertainty yet. So I'm gonna put this little eight here as being not significant as of yet. Now at this point we have to figure out what our new absolute uncertainty will be. So to do that we take the square root, we're going to square each absolute uncertainty and add them together. So when I square them all add them together it's gonna be square root of .0006. Then I take the square root of that number. So I'm gonna get .024495. Following the real rule, we're gonna say that the first significant digit within my absolute uncertainty represents the last significant digit for my measurement. So here it's in the second decimal place. Which means my measurement needs to have two decimal places because of that. I'm gonna use this eight here to round this to up to three. So my measurement with the absolute uncertainty at this point is 5.03 Plus or -102. But we're not done yet. We have to do it for the relative uncertainty. Now remember your relative uncertainty equals your absolute uncertainty divided by your measurement. So my absolute uncertainty is .02 Divided by my measurement of 5.03. So that's gonna give me .003976. Here. Our first sigfig significant digit is that three? It's next to a nine here which means we'll round this up to four. So here my measurement With its relative uncertainty is 5.03 plus or -104. So that would be my two answers. Not that we've seen this attempt to do the one that's here on the bottom. This one here is a mixture of both addition and subtraction. But keep in mind the rules that we've done so far when it comes to addition and subtraction for the propagation of our answer. Once you've figured out what the answers are, come back and take a look and see if your answers match up with my answers

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Problem

Problem

Determine the absolute and relative uncertainty to the following addition and subtraction problem.

8.88 (± 0.03) - 3.29 (± 0.10) + 6.43 (± 0.001)

A

0.1 absolute; 0.012 relative

B

0.1 absolute; 0.010 relative

C

0.1 absolute; 0.008 relative

D

0.01 absolute; 0.0012 relative

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example

Propagation of Uncertainty Calculations

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So here at states determine the absolute and relative uncertainty to the following multiplication and division problem. So here we have a mixture of multiplication and division. Remember when it comes to determining the uncertainties, we must first figure out what the percent relative uncertainty will be. Before we do that though, realize here that we have these values multiplying and being divided by 9.17. So at this point let's just write down what that would be. So when we multiply the two numbers and then divide by 9.17. On the bottom, we get 7.5201. We won't worry about the number of significant figures because remember we're gonna base our final answer on the real role when it comes to looking at our absolute uncertainty. All right. So we have at this .7.5201. Now we're going to deal with the absolute uncertainties that we have here. So remember we're first gonna figure out what our percent relative uncertainty will be. So we take each of those uncertainty and divide them by their measurements, then multiplied by 100 Here for this one we're gonna get .258732%. Which we just round 2.2 and then little 6% here, remember that six. There is just a placeholder uh within our calculations to avoid any types of rounding errors. Now, next we do the next uncertainty. We divided by its measurement of 8.9 to 1 times 100 Here. That's gonna give me a .22419%. Which we recreate as 0.2, sub 2%. And then finally we have 20.3 uncertainty divided by the measurement of 9.17 multiply that by 100 that's gonna give me 1000.327154%. Which we just decreased down to 0.33%. So we just figured out the percent relative uncertainty for each one now that we have that we can figure out what the overall percent relative uncertainty will be. So square root we're gonna square each percentage and add them together. So then when we do that, we're gonna get inside the parentheses, I'm inside the square root function square root of .2249%. Since we're running out of room. Guys, let me take myself out of the image. So square root of that gives me .474236%. Which we can just round 2.4 sub 7%. So what we just found here is our percent relative uncertainty. Now remember if that's our percentage, that means that our relative uncertainty, we take that number and divide it by 100 that would give us this value here. This number here would represent our relative uncertainty. And then if you multiply the relative uncertainty by the overall measurement that we got in the very beginning, that'll give us our absolute uncertainty. So here are absolute uncertainty would come out to being .035344. Following the real rule, we're gonna keep one digit here 0.4. So this will give us our absolute uncertainty. And because our we're going to say here, remember based on the real rule, we're going to say that the first significant digit within our absolute uncertainty represents the last significant figure um within our measurement. So basically because this has two decimal places, our measurement at the end, we'll have two decimal places. So here, if we wanted the full answer, we'd say 7.52 plus or -104 represents our measurement with the absolute uncertainty and then we'd say 7.52 plus or minus here, we're gonna round this up to five plus or minus 50.5 represents our relative uncertainty. So those are the steps that we need to take in order to figure out what our answer is. Any time we're given uncertainties and we're dealing with operations of multiplication and division. Now that you've seen this one attempt to do the practice one that's left here on the bottom of the page attempted on your own. Don't worry. Just come back look at the next video and see how I approach that same exact practice problem for now guys, good luck

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Problem

Problem

Determine the absolute and relative uncertainty to the following multiplication and division problem.

1.12(±0.01) x 0.546 (±0.01) / 3.12(±0.02) x 1.12 (0.03)

A

0.003 absolute; 0.01 relative

B

0.03 absolute; 0.006 relative

C

0.006 absolute; 0.03 relative

D

0.006 absolute; 0.01 relative

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example

Propagation of Uncertainty Calculations

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here at States to students wish to prepare a stock solution for their lab experiment. Student A uses an un calibrated pipette that delivers 50.0 plus or minus 0.2 mls to deliver 200 mls to a container. Student B uses a calibrated pipette that delivers 40 plus or minus 400.1 mls to deliver 200 mls to a container. So we're asked to calculate the absolute uncertainty in each of their deliveries. All right. So the big thing here is one is using a pipette that is un calibrated while the other one is using one that is calibrated. This difference means that we're gonna have to take different approaches to get to our answer. So we're gonna say we have student a here and then we'll have students be here. All right. So if we're looking it's a student a we need to get to 200 mls. So that's our goal And we're doing it in multiples of 50. So we say to ourselves, okay, I would have to do it four times in order to get to 200 mls. So that's 50 plus or minus 500.2 mls And you add that four times. Okay? So then we add that four times this in. So adding that four times will get us to the 200 that we want plus last one. Alright? So we know that if we're adding The measurements that comes out to 200. And when it's uncanny braided, we don't do what we normally would to figure out the absolute uncertainty at the end because it's uncalculated. That means that my uncertainties are additive. That means I can just add them together. So we're just gonna do plus or minus 0.22 and add it to one another. So at the end that's gonna give me plus or minus 10.8 mls. So remember when it's uncalculated, we're gonna just add them together And student b though they are, it's a calibrated Pipat. So we're gonna have to do what we're normally going to do when adding up uncertainties with one another here. We're doing it in multiples of 40. Again, our goal is to get to 200. So we do 40 each time. That means we'd have to do it a total of five times in order to get to 200 mls, so it's 40 Plus or minus. And actually here for this answer following the real rules would actually be 200 because there's two decimal places here. Alright, so then we're adding all of these together, we have to do it five times. So adding it five times. Okay, so we have that. So we added that five times. Now, we know that In terms of the overall volume at the end, we know it's going to add up to 200 mls but then we have to determine what our overall absolute uncertainty will be. Remember when we're adding or subtracting to figure out our new absolute uncertainty. We're gonna take the square root. Take each one of these absolute uncertainties, square them and then add them all together. Okay, so adding them all up together gives me square root of 5.0 times 10 to the negative four, which just Comes down to .022361. As my new overall absolute uncertainty. And here it reduces down to .02. So because this has two decimal places in it, that means my final volume of 200 also has to have two decimal places in it. So here would be 200.0 plus or minus 0.2 mls. So what these two answers are showing us is that by calibrating your pipette, you decrease your absolute uncertainty, which means that you're going to get a final volume. That is closer to your desired amount of 200 mls when it's uncanny braided. There's much bigger variation in the final amount that you're going to get in that case would be plus or minus 2000.8. So it's four times of a difference in terms of your final volume. So again, out of these two students, student, b would be the more correct student because they they're using a calibrated pipette which helps us to get a final answer. That's closer to our true value of 200 mls. Now that we've seen this attempt to do the practice question that's left on the bottom, we've determined the volume of our two students but now we're asked to figure out the polarity. Remember polarity is just simply moles over leaders knowing that is the key to answering this question correctly. Also one more more more thing since its leaders and these are in middle leaders, you'd have to change these values into leaders. Remember that means you divide them by 1000. Both the measurement and the absolute uncertainty would be divided by 1000 so that this becomes leaders at the end. Put the put all those numbers together. Find out what the final answer will be if you're stuck. Don't worry, just come back and see how I approach this practice question.

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Problem

Problem

Based on the previous example calculate the molarity value for each student if they dissolve 0.300 (± 0.03) moles of analyte.

A

1.5 ± 0.2 M for both

B

1.5 ± 0.2 M for student A and 1.5 ± 0.1 M for student B

C

1.5 ± 0.1 M for student A and 1.5 ± 0.2 M for student B

D

1.5 ± 0.04 M for student A and 1.5 ± 0.01 M for student B

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example

Propagation of Uncertainty Calculations 4

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So class A 50 mil of your. It is certified by the manufacturer to deliver volumes within a tolerance I. E. Uncertainty of plus or minus 500.5 millimeters. They tell us the smallest graduations on the bureau are 0.1 mL. You use the bureau to titrate a solution adding five successive volumes to the solution. The following volumes were added. So here we have five different measurements for the amount of volume added to my total bure it by the bureau and it says what is the total volume added? And what is the uncertainty associated with this final volume? What we're gonna do here first is we're gonna add up all five of these measurements When we do that. That gives us a number of 30.80 ml for my total volume. But now we have to determine what the uncertainty is here. They're telling us that the manufacturer basically certifies that it's going to do it within this level of uncertainty. But here's the thing. The manufacturer certifying this does not mean that it is a calibrated bure it. Once you get the bureau in the lab you yourself have to calibrate it. So at this point because they're saying it's coming from the manufacturer we're assuming that this is an uncanny braided pipette or bure it. And remember when you're uncalculated that means you're gonna have systematic error. And when you have systematic error we find the uncertainty in a different way. All we do to find the uncertainty is we add up all the uncertainties from each one of the measurements. Each one of these measurements had an uncertainty of plus or minus 10.5 mls. Okay. Yeah. So here are all of our uncertainties and all we're doing here is we're just gonna add them all up. So that gives me a total of plus or minus 0.25 mL. So remember them saying that the manufacturer certifies this. Does that mean that it's calibrated yet? You get it in this form but then you yourself have to calibrate it later on because it's uncalculated. We have systematic error. And so we just add up all the uncertainties. So my final answer would be 30.80 plus or minus 0.25 middle leaders. If the bureau itself had been calibrated then we would have done we would have random error. And when we have random error we do the same process that we normally would do for addition and subtraction. We would have taking the square root and squared all of these uncertainties and added them up together and so on. And then that would have given us our total absolute uncertainty. But again, because the Bureau is on calibrated, it is not random error. It is systematic error. Which is pretty simple. We just add up all the uncertainties to get the total uncertainty at the end. So again, doing this gives us our final answer of 30.80 plus or minus 0.25 mL

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example

Propagation of Uncertainty Calculations 4

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So here we're told a class a 250 millimeter volumetric flask has an uncertainty of plus or minus 2500.15 mL and a 50 millimeter volumetric pipette has an uncertainty of plus or minus 500.5 mL here, we're told if I fill a 250 ml volumetric flask to the line and removed 50 and 4 15 millimeter alec watts with a volumetric pipette, I should have 50 ml of solution remaining in the flask. What is the absolute and relative uncertainty in the remaining volume? Alright, so we're starting out with a total of 250 plus or minus 0.15 mls and we're subtracting 50 mL four times. So that means I'm just subtracting out 50 Plus or -105 middle leaders. And I'm just gonna do that over and over again, four times minus 50 plus or minus 0.5 mL minus 50 plus or minus 500.5 mL. So in this operation here we have subtraction that's being undertaken. Remember when it's subtraction or addition? In order to find our overall absolute uncertainty, we're gonna take the square root, we're gonna take each one of these uncertainties, square them and then add them all together. So keep adding them together. So when we add them all together, our total at this point would be .03-5. We take the square root of that. That's gonna give me .180278. So here we're gonna use our first significant figure, which is this one here, and because there's an eight next to it, it rounds up to two. So my absolute uncertainty, my total absolute uncertainty is plus or -12. So here this would be 50 plus or -12 metal leaders. So that's my absolute uncertainty. Now. To find my relative uncertainty, I would take my absolute uncertainty and divided by my measurement. So that gives me a .004 for my relative uncertainty. So at the end it would be 50 plus or -104 middle leaders as my relative uncertainty. So just remember what are the operations that are being undertaken within the question and then apply the rules that we've learned previously. Addition. Subtraction follow one set of rules, whereas multiplication and division follow another set.

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Problem

Problem

I am making a 0.1 M KCl (molar mass 74.551) solution for an experiment. To measure the mass of the KCl, I will use an analytical balance that is only accurate to ± 0.01 g. I place a piece of paper on the balance and set the tare to read 0.00. I then put the KCl on the balance until it reads 6.79 g. What is the uncertainty in this mass?

A

0.010 g

B

0.014 g

C

0.0002 g

D

0.020 g

16

Problem

Problem

The volume of the solution I am making is 2.5 L. To measure this volume I will use a large graduated cylinder that can measure volume to ± 10 mL. What is the absolute uncertainty in my concentration?