Significant Figures: In Calculations - Video Tutorials & Practice Problems
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Significant Figures are often involved in chemical calculations.
Significant Figures In Calculations
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Significant Figures Calculations Concept 1
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So now we're gonna see how significant figures can be incorporated in different calculations that we'll be exposed to in chemistry. Now we're gonna start out with multiplication and division. We're gonna say when either multiplying or dividing different numbers, the final answer will contain the least significant figures. And if we take a look at this example, it says, perform the following calculation to the right number of sig figs. Here we have 3 values that are being multiplied together. We have 3.16 times 0.003027 times 5.7 times 10 to the negative 3. We just said that when you're multiplying or dividing its least number of sig figs for your final answer, so we need to determine the number of sig figs for each value. From our topic on significant figures, we know that if we have a decimal point, which all of them do, we move from left to right. Now remember, we're going to start counting once you get to our first non zero number. Here, 3 is our first non zero number, and once we start counting, we count all the way into the end. So 1, 2, 3, this has 3 sig figs. For the next one, skip, skip, skip, our first non zero is this 3. 1, 2, 3, 4. This has 4 sig figs. And then finally we have 5.7 times 10 to the negative 3 written in scientific notation. Remember, when it's written in scientific notation, just focus on the coefficient. We're going to say our not our first non zero number is this 5, and once we start counting, we count all the way into the end. So one, 2. This has 2 sig figs. Now based on our sig figs of 3, 4 and 2, we have to go with the least number of significant figures. That means our answer at the end can only have 2 significant figures. So when we first get our answer, what we see initially is 5.4522324 times 10 to the negative 5. We want 2 sig figs here. That 4 that we have though, we look to the right of it and see if we either keep it as 4 or we round up. Next to it, we have this long string of numbers, and we have a 5 there. Because that number is 5, that means we have to round up. So the 5.4 becomes now 5.5, and then times 10 to the negative 5. This represents our answer, which has the least number of significant figures based on the initial values given. We were given these three numbers initially and the one with the least number of sig figs was the one written in scientific notation. So that tells me that my final answer has to have that number of significant figures. Now that we've looked at multiplication and division, let's go on to our next video and let's see what happens when we incorporate addition and subtraction.
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Significant Figures Calculations Concept 2
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Now when either adding or subtracting different numbers, the final answer will contain the least decimal places. If we take a look at this example, it says perform the following calculation to the right number of sig figs. Now if our answer is based on the least number of decimal places, that's going to have a direct impact on the number of significant figures. If we take a look here, it says we have 402.09 minus 2.12.2 plus 2.671. If we look at these values, this one has 2 decimal places, this one here has 1 decimal place, and this one here has 3 decimal places. Based on that, we're going with the least number of decimal places, our answer can only have 1 decimal place at the end. When we punch all this in, we get 192.561. We can only have 1 decimal place. To the right of that 5, there's a 6 there. That means we have to round up. So this is 182.6 as our final answer, and if you wanted to talk about the number of sig figs, you'd move from left to right. Our first non zero number is this 1, and counting all the way through, we'd have 4 sig figs at the end. K. By following this least number of decimal places, it has a direct impact on the number of sig figs in our final answer. Up to this point, we've kept multiplication and division separate from and subtraction. But what happens when you mix them together? To find out what to do, click on the next video.
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Significant Figures Calculations Concept 3
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Now we're dealing with mixed operations. We have a combination of multiplication, division, subtraction, or addition. We're gonna say when dealing with this mixture of multiplication, division, addition, and subtraction, we must follow the order of operations. To help us remember the order of operations, we use PENDES, which stands for parenthesis, so what's in parenthesis is done first, exponents or powers, then we have multiplication slash division, and then addition/ subtraction. So that is our order of operations. Multiplication and division are grouped together, addition and subtraction are still grouped together. If we take a look here it says perform the following calculation to the right number of sig figs. We have in brackets 1.8 9 times 10 to the 6 times 3.005. Then we have 5.21 to the 3rd divided by 8.829 minus 6.5 plus 2.920. Alright. So we're following our order of operations, and in our order of operations, we're gonna do what's in here first because we have brackets and parentheses here. So we're gonna say, when we do everything inside of here, when we multiply everything, it comes out to 5679450. But when you're multiplying or dividing numbers, we have to look at the least number of sig figs. So here, this number has 3 sig figs, this number here has 4 sig figs. So our answer at the end when they multiply has to have 3 sig figs. So this initial answer that I got here becomes 5.6 8 times 10 to the 6, after I've changed it into scientific notation. Next, we have 5.21 to the 3rd, so that's exponent. So all this means is 5.21 times 5.21 times 5.21. 1. All of them are multiplying each other, all of them have 3 sig figs. What we would get initially from it is 141.4 20761. But again, when you're multiplying its least number of sig figs, so our answer would have to have 3 sig figs. So here, this will come out to be 141. Next, we have what's on the bottom here, 8.29 minuteus 6.5. If you're adding or subtracting its least number of decimal 3 decimal places, this one here has 3 decimal places, this one here has 1 decimal place. So our answer at the end must have 1 decimal place. So that will come out to be 2 point 3 as our number here. Next, we have plus 2.920. So we're looking at this portion down here now. We continue onward. Now the two numbers on the top are multiplying each other. Because they're multiplying, it's still Least Sigfigs. Here the coefficient has 3 sig figs, and here 141, oops, going the other way because it doesn't have a decimal place, 141 has 3 sig figs. So our answer at the end must have 3 sig figs. When When they multiply together it comes out as 8.01 times 10 to the 8. Notice I'm not putting everything all at once in my calculator. You have to do it piece by piece in order to isolate your final answer. Then on the bottom, these 2 are adding together, so when they add together initially, it comes up as, 5.220. But when you're adding or subtracting, it's least number of decimal places. This one here has 1, this one here has 3, so your answer at the end must have one decimal place. So that would be 5.2. Now we just have these two numbers that are dividing each other, so again it's least number of sig figs. This 8.01 has 3 sig figs in it. This 5.2 has 2 sig figs in it. So our answer at the end must have 2 sig figs. So this comes out as 1.5 times 10 to the 8. So this would be our final answer written to the correct number of significant figure based on this mix of operations. So just keep in mind the order of operations to guide you on what to do. And remember, multiplication and division is least sig figs, addition or subtraction is least decimal places.
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Problem
Problem
Perform the following calculation to the right number of sig figs:
[(1.7 × 106) ÷ (2.63 × 105)] + 6.96
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13.46
B
13.0
C
13.5
D
14.2
E
4.471
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Problem
Problem
What answer should be reported, with the correct number of significant figures, for the following calculation?