First Assignment Scientific Notation This First Assignment may take between 1 hours depending on your background and prior knowledgeplease follow the instructions carefully. a. Read all the information provided. b. Complete the Practice and Self Check assignments. Mark and correct your Practice questions. (As an online student it is an important skill to learn to self assess throughout your work.) c. Complete the last page and submit it to your teacher for marking. d. You may use a calculator for any part of the assignment.
Students should be able to demonstrate the following Prescribed Learning Outcomes: Correctly determine the number of significant digits in a number Perform calculations using the rules for significant digits Combine significant digits and scientific notation in calculations Significant Figures Not all numbers are created equal. In fact even numbers which represent the same value may not be equal to one another, based on our level of accuracy in measuring them. As scientists, and science students, it is important to be able to express to those reading our work the level of accuracy portrayed in our measurement. Significant digits are the rules followed to show this accuracy. There are two kinds of numbers in the world exact: o Example: There are exactly 12 eggs in a dozen. o Example: Most people have exactly 10 fingers and 10 toes. inexact numbers: o Example: Any measurement. If I quickly measure the width of a piece of notebook paper, I might get 220 mm (2 significant figures). If I am more precise, I might get 216 mm (3 significant figures). An even more precise measurement would be 215.6 mm (4 significant figures). o Measurements of any physical information are only as good as the instrument used to measure the information and the level of sensitivity to those measurements. The inaccuracy stems from the precision of the measuring device and any error introduced by the person measuring. Because a measurement contains some degree of inaccuracy, the numbers of digits that are valid for the measurement are also limited. So when using any instrument for measurement we are able to determine some place values easily. But there is one place value we must estimate. This is the last place value we have any confidence in, and even that is a best guess. Precision versus Accuracy Accuracy refers to how closely a measured value agrees with the correct value. Precision refers to how closely individual measurements agree with each other. Page 2 of 10
accurate (the average is accurate) not precise precise not accurate accurate and precise In any measurement, the number of significant figures is critical. The number of significant figures is the number of digits believed to be correct by the person doing the measuring. It includes one estimated digit. How can we determine how many significant numbers a measurement has? The following rules apply to determining the number of significant figures in a measured quantity: 1. All nonzero digits are significant. 457 cm (three significant figures) 0.25 g (two significant figures). An example can be seen measuring the fish below using the meter stick. We can see that it is bigger than the 40 cm mark, counting shows that it is just a bit longer than 46 cm. From the angle we are viewing it is very difficult to tell but it looks to be fractionally larger than 46 cm, so we are able to estimate a single decimal place beyond the ones to 46.1 cm. As the 4 and 6 are known, and the 1 an estimate we say there are 3 significant digits in this measurement. Any digit that is NOT ZERO is significant. Page 3 of 10
2. Zeros between nonzero digits (Captive Zeros) are significant. 1005 kg (four significant figures) 1.03 cm (three significant figures). In the number 10.02 the zeros are captured between the 1 and the 2. Since 1 and 2 are significant any digit between them is significant.. 3. Zeros to the left of the first nonzero digits in a number are not significant; they merely indicate the position of the decimal point. 0.02 g (one significant figure) 0.0026 cm (two significant figures). Red blood cells are about 0.000 006 meters in diameter. All the zeros in the front of the 6 represent decimal places and are not significant. 4. When a number ends in zeros that are to the right of the decimal point, they are significant. 0.0200 g (three significant figures) 3.0 cm (two significant figures). Occasionally a scientist will express a number like 10.00 this is very different from the number 10. When a scientist takes the effort to purposely record place values with zeros after a decimal the scientist is implying that those values were measured. Otherwise the number 10 would have been recorded instead. Zeros at the end of a number, after a decimal, are significant. 5. When a number ends in zeros that are to the left of a decimal point, the zeros are not necessarily significant. 130 cm (two or three significant figures) Page 4 of 10
10,300 g (three, four, or five significant figures) The population of the earth is 6 446 000 000 people according to the CIA factbook dated 20 April, 2006. How is this number arrived at, and how accurate is it? did the CIA go out and count everyone? There is some ambiguity in this number. It seems reasonable to assume there to be greater than 6 billion but less than 7 billion people so the first 6 is significant. If the CIA collected its data carefully from governments, and simply added these together we might assume that the 4, 4, and 6 to also be relatively accurate, but how many people will be living in India and China who do not fill out their countries' census? Can we really predict the world population to the nearest million? Perhaps 6 400 000 000 is a more accurate estimate. But are all those zeros significant? We are unsure and so as scientists we have to assume they ARE NOT! That means the only digits we have confidence in is the 6 and the 4 is an estimate. Any zero digit at the end of a number greater than one is assumed to not be significant. So how do scientists record really large numbers like the population of earth, and really small ones like the diameter of a blood cell without including many insignificant zeros? Use of standard exponential notation avoids the potential ambiguity of whether the zeros at the end of a number are significant. For example, a mass of 10,300 g can be written in exponential notation showing three, four, or five significant figures: 1.03 x 10 4 g (three significant figures) 1.030 x 10 4 g (four significant figures) 1.0300 x 10 4 g (five significant figures) In these numbers all the zeros to the right of the decimal point are significant (rules 2 and 4). Practice and Self-Check A State the number of significant digits in each measurement: a) 3804 m b) 0.0046 m c) 0.003 089 d) 4.06 x 10 5 m Page 5 of 10
If you need more information about Scientific Notation read the following information. Then complete the Practice and Self Check B. If you are proficient with Scientific Notation go directly to the Practice and Self Check B. Scientific Notation Scientific Notation is a method of writing numbers which are very large, and numbers which are very small. In Science it is common to have to work with masses of planets and stars (very large numbers), charges on electrons and protons (very small numbers). Scientific notation is a method of expressing those numbers in a short way rather than having to write them out in full. For example, one ampere of electric current is the same as 6 240 000 000 000 000 000 electrons passing a point in a wire in one second. This same number can be written, in scientific notation, as 6.24 x 10 18 electrons/second. This means 624 followed by 16 zeros. Any number can be expressed in scientific notation. Here are some examples: 0.10 = 1.0 x 10-1 1.0 = 1.0 x 10 0 10.0 = 1.00 x 10 1 100.0 = 1.000 x 10 2 1 000.0 = 1.0000 x 10 3 Power-of-Ten Notation 0.000001 = 10-6 1 = 10 0.00001 = 10-5 10 = 10 1 0.0001 = 10-4 100 = 10 2 0.001 = 10-3 1 000 = 10 3 0.01 = 10-2 10 000 = 10 4 0.1 = 10-1 100 000 = 10 5 1 = 10 0 1 000 000 = 10 6 Page 6 of 10
Practice and Self-Check B Write the following measurements in scientific notation. (a) 0.00572 kg (b) 520 000 000 000 km (c) 300 000 000 m/s (d) 0.000 000 000 000 000 000 16 C (e) 118.70004 g Rule for Multiplication and Division In multiplication and division of measurements, the result must be reported as having the same number of significant figures as the least precise measurement. If you multiply 21.3 cm by 9.80 cm, the answer is 209, not 208.74. Since the less precise measurement, 21.3 cm, has only three significant figures, the product has three significant digits. Example 1 A rectangle is measured to be 3.22 cm by 2.1 cm. What is the area of the rectangle shown in the appropriate significant figures? Multiply 3.22 cm by 2.1 cm. Solution: 3.22 cm 2.1 cm = 6.762 cm 2 The less precise factor, 2.1 cm, contains two significant digits. Therefore the product has only two significant digits. The answer 2 is then best stated as 6.8 cm. Example 2 A cyclists rides a distance of 36.5 m over a time of 3.414 s. How fast did he ride? Divide 36.5 m by 3.414 s. Page 7 of 10
Solution: 36.5 m 3.414 s = 10.6913 m/s The less precise factor, 36.5 m, contains three significant digits. Therefore the division has only three significant digits. The answer is then best stated as 10.7 m/s. Rule for Addition and Subtraction In addition and subtraction, the result of your calculation can never be more precise than the least precise measurement. The result of the calculation must be reported to the same number of decimal places as that of the term with the least number of decimal places. Example 1 Add 24.686 m + 2.343 m + 3.21 m Solution: 24.686 m 2.343 m 3.21 m 30.239 m =30.24 m Note that 3.21 m is the least precise measurement. Therefore round off the result to the nearest hundredth on one metre and the answer is best stated as 30.24 m. You will follow the same rule for subtraction. Rounding Rules 1. Look at the leftmost digit to be removed. If less than 5: truncate (preceding number is unchanged) If 5 or greater: increase by 1(round up) 2. When a calculation involves two or more steps, retain at least two additional digits beyond the correct number of significant figures for intermediate calculations. Round off to the correct number of significant figures only for the answer you want to report (which is usually the final answer). This helps to avoid accumulated round-off error. Page 8 of 10
Practice and Self-Check C Answer the following questions, and mark your practice work before completing the First Assignment on the last page. 1. Express the following measurements in scientific notation. a) 156.90 a) b) 12 000 b) c) 0.0345 c) d) 0.008 90 d) 2. Give the number of significant digits in the following measurements. a) 2.09910 m a) b) 5600 km b) c) 0.006 70 kg c) d) 809 g d) 3. Solve the following problems and give the answer in the correct number of significant digits. a) 2.674 m 2.0 m = a) b) 5.25 L x 1.3 L = b) c) 9.0 cm + 7.67 cm + 5.44 cm = c) d) 10.07 g 3.1 g = d) Answers: Self Check A (a) 4 (b) 2 (c) 4 (d) 3 Self Check B(a) 0.00572 kg = 5.72 x 10-3 kg (b) 520 000 000 000 km = 5.2 x 10 11 km (c) 300 000 000 m/s = 3.0 x 10 8 m/s (d) 0.00000000000000000016 C = 1.6x10-19 C (e) 118.70004 g = 1.1870004 x 10 2 g Self Check C1a)1.569 x 10 2 b)1.2 x 10 4 c)3.45 x 10-2 d) 8.9 x 10-3 2a)6 b)2 c)3 d)3 3a)1.3 b)6.8 L 2 c)22.1 cm d)7.0 Page 9 of 10
Unit 1 Hand-In Assignment Total /43 Name: Date: Note: You MUST SHOW ALL work, correct units & sig. figs for each question to receive full marks! 1. Express the following measurements in scientific notation. (3) a) 68000 kg ANSWERS b) 401 000 m c) 0.0036 kg 2. State the number of significant digits in each measurement. (3) a) 3803.0 m b) 0.003 067 m c) 4.0 x10^4 m 3. Perform the following calculations, and express your answers in scientific notation with the correct number of significant digits.(4) a) 12.5 x 2.00 b) 10.0 2.5 c) (4.11 x 10^7 )+ (5.7 x 10^7 ) d) (4.536 x 10^-3) (0.347 x 10^-3)
4. Describe the difference between chemical and physical change in terms of what occurs with the atoms involved.(2) 5. A sample of an element has a mass of 146.28 g and will displace 27.50 ml of water. What is the density and identity of the element?(2) 6. Draw the heating curve for zinc, as it is heated from a solid to a gas. Indicate its m.p. and b.p. and use correctly labeled axes.(2) 7. State the element that has a:(3) a) m.p. of 1064 C b) density of 6.15 g / 3 cm c) a freezing point of 631 C 8. Describe each of the following as a physical or chemical change:(4) a) snow changes to water on a hot day b) an egg is fried on a frying pan c) wood is burned in a fire place d) ice cubes shrink in a freezer 9. What volume does 852.6 g of lead take up?(2) 10. What mass does 17.358 ml of mercury have?(2)
11. Draw the cooling curve for iodine. Indicate the mp, bp and label each axis.(2) 12. State what WHMIS and MSDS stand for?(2) 13. What word describes the change from a: (3) a) gas to a liquid b) liquid to solid c) liquid to gas 14. Complete the relationships. (4) a) 1 Mg = g b) 100 cl = L c) 100 m = hm d) 1000000 µl= L 15. Use unit analysis to perform the following conversions: (3) a) 6.372 hl to ml b) 4.9 x 10 15 µg to Mg c) 8.774 x 10 3 cm 3 to m 3 17. Given the following relationships, determine how many zings can be obtained when you trade 20.6 balls. (2) 4 clangs = 3 dangs, 7 dangs = 3 jars, 2 balls = 5 clangs, 6 jars = 1 zing