Lesson 3: Significant Digits

Lesson 3: Significant Digits Scientists take the ideas of precision and accuracy very seriously. You can actually take entire courses in University that show how to figure out the precision and accuracy of measurements. Guess what? I took 'em!... your opinion of me has probably slipped a few notches ; ) We need to know that when another scientist reports a finding to us, we can trust the accuracy and precision of all the measurements that have been done. A set of guidelines is needed while we do calculations so that we get rid of all those 4.243956528452940472 kind of answers you see on your calculator. The guidelines are there so we will know how many digits we should round off the final answer to show the correct precision. All of this boils down to something called Significant Digits, more commonly referred to as Sig Digs. To determine how many significant (important) digits a number has, follow these rules: 1. The numbers 1 to 9 are always sig digs. Zero ("0") is a sig dig if it comes to the right of a number between 1 and 9. Example 1: 13.869 -> Five sig digs. All the numbers are digits between 1 and 9. 1.304 -> Four sig digs. The zero counts because it appears to the right of the "3" 576.00 -> Five sig digs. The zeros count because they appear to the right of the "6" 0.0008 -> One sig dig. The zeros don t count, because they are to the left of the non-zero digits. 13 000 -> Five sig digs. Warning! Some text books will tell you that this number only has two sig digs, and that the zeros are ambiguous (a fancy way of saying iffy ). Sure, it is unlikely that your answer will come out to exactly 13000, but it could happen. And if those zeros were not significant we would have used a different way to write down the number (see the next lesson). We will simply follow the rule and say that it has five sig digs since those zeros are all appearing to the right of a non-zero digit. 2. When you add or subtract numbers, always check which of the numbers is the least precise (least numbers after the decimal). Use that many decimals in your final answer. Example 2: 11.623 + 2.0 + 0.14? If you type this on a calculator, you'll get 13.763. Round it off to a final answer of 13.8, since the number "2.0" is the least precise... it only has one sig dig after the decimal. We do this because we can t really trust how much rounding off might have been done for any decimals after this. 1/28/2007 studyphysics.ca Page 1 of 2

3. When you multiply or divide numbers, check which number has the fewest sig digs. Round off your answer so it has that many sig digs. Example 3: 4.56 x 13.8973? Putting this into a calculator you will get something like 63.371688. We round off our final answer to 63.4 (which has three sig digs), because "4.56" has the fewest sig digs... three sig digs. Your Data Sheet... You'll see that there are a bunch of constants on the back of your data sheet. These are all given as three sig digs, so treat them appropriately. You'll also notice that some formulas already have numbers on them. These numbers are considered perfect and can be considered to have an infinite number of sig digs. There are also situations where you might need to convert a number from one set of units to another. For example, 1 minute 60 seconds. This doesn't mean that 1 minute sort of equals 60 seconds... it is a perfect conversion, so it also has an infinite number of sig digs. 1/28/2007 studyphysics.ca Page 2 of 2

Significant figures worksheet 1. Round off the following numbers to three significant figures: a) 35.234 b) 2.34521 c) 0.035219 d) 2 533 521 e) 6 255 520 000 2. Give the largest and smallest value of the approximate number 35.21 ± 0.02 g 3. Five different voltmeters are used to measure the voltage in a circuit. Given that the following measurements are made, determine the average and uncertainty. 25.61 V, 25.63V, 25.65V, 25.64V, 25.63V 4. Complete the following computations: 26.215-0.3 10 + 0.1 25 x 3 (6.2 x 10 3 ) ( 3.55 x 10 12 ) 25.31 + 6.4 25.217 + 0.017 + 0.25-0.177 3.5 x 10 2 3.1 x 10 3 65.222 + 1.03 33.3 + 0.35 3.35 x 0.26 (6.3 x 10 7 ) (2.51 x 10-7 ) (3.214 x 10-5 ) 22.0 + 0.04 51.71 x 22.3 (2.00 x 10 23 )(3.51 x 10-22 )(3.5 x 10 3 ) (7.5 x 10-3 )(3.511 x 10 12 )(6.6 x 10-6 ) 22-0.01 29.39 + 0.2 799 x 877 (7.52 x 10 16 )(3.1 x 10 12 ) (2.5 x 10-7 ) 35.271 + 0.2 22.7 + 0.77 22 x 305

Significant figures worksheet (answers) 1. Round off the following numbers to three significant figures: a) 35.234 35.2 b) 2.34521 2.35 c) 0.035219 0.0352 d) 2 533 521 2.53 x 10 6 e) 6 255 520 000 6.26 x 10 2. Give the largest and smallest value of the approximate number 35.21 ± 0.02 g 35.19 to 35.23 3. Five different voltmeters are used to measure the voltage in a circuit. Given that the following measurements are made, determine the average and uncertainty. 25.61 V, 25.63V, 25.65V, 25.64V, 25.63V 25.63± 0.02 V 4. Complete the following computations: 26.215-0.3 65.222 + 1.03 25.9 66.25 22-0.01 22 10 + 0.1 10 33.3 + 0.35 33.7 29.39 + 0.2 29.6 25 x 3 3.35 x 0.26 8 x 10 1 0.87 (6.2 x 10 3 ) ( 3.55 x 10 12 ) (6.3 x 10 7 ) (2.51 x 10-7 ) 2.2 x 10 16 (3.214 x 10-5 ) 4.9 x 10 5 799 x 877 7.01 x 10 5 (7.52 x 10 16 )(3.1 x 10 12 ) (2.5 x 10-7 ) 9.3 x 10 35 25.31 + 6.4 31.7 22.0 + 0.04 22.0 35.271 + 0.2 35.5 25.217 + 0.017 + 0.25-0.177 25.31 51.71 x 22.3 22.7 + 0.77 1.15 x 10 3 23.5 3.5 x 10 2 3.1 x 10 3 0.11 (2.00 x 10 23 )(3.51 x 10-22 )(3.5 x 10 3 ) (7.5 x 10-3 )(3.511 x 10 12 )(6.6 x 10-6 ) 1.4 22 x 305 6.7 x 10 3

Lesson 4: Scientific Notation In the last section you learned how to use sig digs in your calculations. What do you do if you multiply numbers like 537 x 269 144 453... you are supposed to only have three sig digs, but your answer sure has more than three sig digs! We need a way to show the correct number of sig digs. What if you have a large number like 4 500 000 000 km (the distance from Neptune to the sun), or a small number like 0.000 000 010 cm (the diameter of an atom) and you don't want to be bothered with writing out all those zeros? We need a way to show really big and really small numbers. To get around these problems, we use Scientific Notation (sometimes called Exponential Notation). This system makes use of "powers of 10", raising 10 to whatever value you need. You can get either really big numbers by using positive powers like 105 100 000 You can also show really small numbers by using negative powers like 10-5 0.00001 Example 1: 10 5 10 x 10 x 10 x 10 x10 100 000 10-5 1/10 x 1/10 x 1/10 x 1/10 x 1/10 0.00001 Don't worry about spending half a minute using your calculator to figure out what 105 equals. Instead, notice that 105 written out has five zeros. Converting Numbers into Scientific Notation When you use scientific notation, follow these rules. We'll try them out on the numbers from the start of the lesson... 1. Move the decimal over so that only one non-zero number is to the left of the decimal. 4 500 000 000 > 4.500 000 000 0.000 000 010 > 000 000 01.0 2. Count how many spaces over you moved the decimal. If you moved it to the left it's positive, if you moved it to the right it's negative. 4.500 000 000 > moved 9 spaces left (+9) 000 000 01.0 > moved 8 spaces right (-8) 3. Get rid of any numbers that are not sig digs. This might depend on the numbers you used in your calculation. 4.500 000 000 > 4.5 I'm assuming that all those other zeros were probably just place holders, although I'd need a reason to do this in a real question. Warning! When you use sci not, you might be writing a number down with more or less sig digs than it actually has. Be very careful to pay attention to the number of sig digs you are really dealing with. 2/9/2013 studyphysics.ca Page 1 of 3

000 000 01.0 > 1.0 I'll keep this last zero. Since it was written in the original number for such a small number, it's probably significant. 4. Write down the number, multiplied by 10 to the power of however many spaces you found in step 2. 4 500 000 000 4.5 x 10 9 0.000 000 010 1.0 x 10-8 If you ever need to change a number in scientific notation back to regular form, do the reverse of the above. Scientific Notation on Your Calculator Most calculators now have a key on them for doing scientific notation. Look for one of the following... EXP (most Casio calculators) EE (TI 83, 84, and Nspire calculators. You have to use the 2nd function key to use it on the older 83's and 84's) On your calculator, type in the question as it's written. Remember, the calculator doesn't care about sig digs... it's up to you to round off the answer. Most of the time you'll be dealing with multiplying and dividing Scientific Notation, which makes it easier to figure out the sig digs. Use the rules we covered for sig digs, but don't look at the 10 to whatever power part... it does not count for sig digs. If you have to type in a negative number, use the (-) button on the calculator, not the subtraction button. Example 2: Determine the answer to the following question. Try doing it on your calculator to see if you get the same answer. 4.587 x 10 4 1.2 x 10-3 3.8 x 10 7 Warning! Do NOT use the "hat" symbol on your calculator to enter scientific notation (eg. 4.5 x 10^5). Your calculator will treat this as two separate numbers, and you will get some calculations wrong because of it (it screws up the proper order of operations). Do not make excuses like I'll use brackets, since you'll only screw up later. If you're using a TI graphing calculator, you'll notice that it shows scientific notation with an "E" (for exponent). Example 2 would look like this on your display... 2/9/2013 studyphysics.ca Page 2 of 3

Illustration 1: TI:83 Calculator To make typing scientific notation easier for me on this site, I will be using a style similar to the way a TI graphing calculator would show the numbers. For example, I'll be typing 4.587e4 instead of 4.587 x 10 4. Hope ya' don't mind. Video Killed the Radio Star! It's really easy to mix up typing a set of numbers on your calculator using scientific notation. Click on one of these links to watch how to do it on the model of calculator you have: Casio Calcultor TI 83, TI 84, or an older Nspire with the 84 Faceplate TI Nspire CX Casio Calculator TI 83 and TI 84 TI Nspire CX 2/9/2013 studyphysics.ca Page 3 of 3

Scientific Notation Worksheet Convert the following numbers into scientific notation: 1) 3,400 2) 0.000023 3) 101,000 4) 0.010 5) 45.01 6) 1,000,000 7) 0.00671 8) 4.50 Convert the following numbers into standard notation: 9) 2.30 x 10 4 10) 1.76 x 10-3 11) 1.901 x 10-7 12) 8.65 x 10-1 13) 9.11 x 10 3 14) 5.40 x 10 1 15) 1.76 x 10 0 16) 7.4 x 10-5 For chemistry help, visit www.chemfiesta.com 2003 Cavalcade Publishing All Rights Reserved

Scientific Notation Worksheet - Solutions Convert the following numbers into scientific notation: 1) 3,400 3.4 x 10 3 2) 0.000023 2.3 x 10-5 3) 101,000 1.01 x 10 5 4) 0.010 1.0 x 10-2 5) 45.01 4.501 x 10 1 6) 1,000,000 1 x 10 6 7) 0.00671 6.71 x 10-3 8) 4.50 4.50 x 10 0 Convert the following numbers into standard notation: 9) 2.30 x 10 4 23,000 10) 1.76 x 10-3 0.00176 11) 1.901 x 10-7 0.0000001901 12) 8.65 x 10-1 0.865 13) 9.11 x 10 3 9,110 14) 5.40 x 10 1 54.0 15) 1.76 x 10 0 1.76 16) 7.4 x 10-5 0.000074 For chemistry help, visit www.chemfiesta.com 2003 Cavalcade Publishing All Rights Reserved

Instructional Master Chemical Reactions Table 1 Chemical Reactions Reaction type formation simple decomposition Generalization elements compound compound elements complete combustion single replacement double replacement substance + oxygen most common oxides element + compound element + compound (metal + compound metal + compound nonmetal + compound nonmetal + compound) compound + compound compound + compound

Student Worksheet Classifying Chemical Reactions 1. (a) Classify each of the following reactions as formation, simple decomposition, single replacement, or double replacement reactions. (b) Balance each equation and add symbols to indicate states of matter for all reactants and products. (i) Cu + O 2 CuO (ii) Al + Fe 2 O 3 Al 2 O 3 + Fe (iii) Ag + S Ag 2 S (iv) H 2 O + electricity H 2 + O 2 (v) FeS + HCl FeCl 2 + H 2 S (vi) NaCl Na + Cl 2 (vii) NaOH + HCl H 2 O + NaCl (viii) Zn + HCl ZnCl 2 + H 2 2. Write balanced chemical equations for the following: (a) The decomposition reaction of hydrogen sulfide. (b) The single displacement reaction of copper metal and silver nitrate. (c) The synthesis reaction of sodium and fluorine. (d) The double displacement reaction of aluminium sulfate and calcium hydroxide.

Student Worksheet Solutions LSM 2.6C Solutions for Classifying Chemical Reactions, Extra Exercises 1. (a) 2 Cu(s) + O 2 (g) 2 CuO(s) formation (b) 2 Al(s) + Fe 2 O 3 (s) Al 2 O3(s) + 2 Fe(s) single replacement (c) 16 Ag(s) + S 8 (s) 8 Ag 2 S(s) formation (d) 2 H 2 O(l) + electricity 2 H 2 (g) + O 2 (g) simple decomposition (e) FeS(s) + 2 HCl(aq) FeCl 2 (aq) + H 2 S(g) double replacement (f) 2 NaCl(s) 2 Na(s) + Cl 2 (g) simple decomposition (g) NaOH(aq) + HCl(aq) H 2 O(l) + NaCl(aq) double replacement (h) Zn(s) + 2 HCl(aq) ZnCl 2 (aq) + H 2 (g) single replacement 2. (a) The decomposition reaction of hydrogen sulfide 8 H 2 S(g) 8 H 2 (g) + S 8 (s) (b) The single displacement reaction of copper metal and silver nitrate Cu(s) + 2 AgNO 3 (aq) Ag(s) + Cu(NO 3 ) 2 (aq) (c) The synthesis reaction of sodium and fluorine 2 Na(s) + F 2 (g) 2 NaF(aq) (d) The double replacement reaction of aluminium sulfate and calcium hydroxide Al 2 (SO 4 ) 3 (aq) + 3 Ca(OH) 2 (aq) 2 Al(OH) 3 (s) + 3 CaSO 4 (s) Copyright 2007 Thomson Nelson Review Unit Lab and Study Masters 63

Student Worksheet Predicting Chemical Reactions For each of the following questions, classify the reaction type (formation, simple decomposition, combustion, single replacement, double replacement, or other), and predict the balanced chemical equation. Provide a word equation as well. 1. Al(s) + O 2 (g) 2. Ag 2 O(s) 3. Br 2 (l) + KI(aq) 4. A strip of zinc metal is placed into a copper(ii) nitrate solution. 5. BaCl 2 (aq) + Na 2 SO 4 (aq) 6. Sulfuric acid is neutralized by aqueous sodium hydroxide. 7. CuS(s) + NaCH 3 COO(aq) 8. CuS(s) + O 2 (g) 9. Propane burns in air. 10. Na 2 CO 3 (aq) + HCl(aq) NaCl(aq) + CO 2 (g) + H 2 O(l)

Student Worksheet Solutions LSM 2.6E Solutions for Predicting Chemical Reactions, Extra Exercises For each of the following questions, classify the reaction type (synthesis, decomposition, combustion, single replacement, double replacement, or other), and predict the balanced chemical equation. Provide a word equation as well. 1. 4 Al(s) + 3 O 2 (g) 2 Al 2 O 3 (s) formation or combustion aluminium + oxygen aluminium oxide 2. 2 Ag 2 O(s) 4 Ag(s) + O 2 (g) simple decomposition silver oxide silver + oxygen 3. Br 2 (l) + 2 KI(aq) I 2 (s) + 2 KBr(aq) single replacement bromine + potassium iodide iodine + potassium bromide 4. A strip of zinc metal is placed into a copper(ii) nitrate solution. Zn(s) + Cu(NO 3 ) 2 (aq) Cu(s) + Zn(NO 3 ) 2 (aq) single replacement zinc + copper(ii) nitrate copper + zinc nitrate 5. BaCl 2 (aq) + Na 2 SO 4 (aq) BaSO 4 (s) + 2 NaCl(aq) double replacement barium chloride + sodium sulfate barium sulfate + sodium chloride 6. Sulfuric acid is neutralized by aqueous sodium hydroxide. H 2 SO 4 (aq) + 2 NaOH(aq) 2 HOH(l) + Na 2 SO 4 (aq) double replacement sulfuric acid + sodium hydroxide water + sodium sulfate 7. Na 2 S(aq) + Cu(CH 3 COO) 2 (aq) CuS(s) + 2 NaCH 3 COO(aq) double replacement sodium sulfide + copper(ii) acetate copper(ii) sulfide + sodium acetate 8. 2 CuS(s) + 3 O 2 (g) 2 CuO(s) + 2 SO 2 (g) combustion copper(ii) sulfide + oxygen copper(ii) oxide + sulfur dioxide 9. Propane burns in air. C 3 H 8 (g) + 5 O 2 (g) 3 CO 2 (g) + 4 H 2 O(g) combustion propane + oxygen carbon dioxide + water 10. Na 2 CO 3 (aq) + 2 HCl(aq) 2 NaCl(aq) + CO 2 (g) + H 2 O(l) other sodium carbonate + hydrochloric acid sodium chloride + carbon dioxide + water Copyright 2007 Thomson Nelson Review Unit Lab and Study Masters 65