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Math and Measurement Unit 2 Numbers…which ones are important? What is 13/7? Is it 1.8571428? Or…is it 1.86? Or 1.9? Or 2? Where do we round? Significant Digits Rules for Counting Significant Figures - Details • Exact numbers have an infinite number of significant figures. 1 inch = 2.54 cm, exactly Rules for Counting Significant Figures - Details • Nonzero integers always count as significant figures. 3456 has 4 sig figs. Rules for Counting Significant Figures - Details • Zeros - Leading zeros do not count as significant figures. • 0.0486 has 3 sig figs. Rules for Counting Significant Figures - Details • Zeros - Captive zeros always count as significant figures. • 16.07 has 4 sig figs. Rules for Counting Significant Figures - Details • Zeros Trailing zeros are significant only if the number contains a decimal point. 9.300 has 4 sig figs. Sig Fig Practice #1 How many significant figures in each of the following? 1.0070 m 5 sig figs 17.10 kg 4 sig figs 100,890 L 5 sig figs 3.29 x 103 s 3 sig figs 0.0054 cm 2 sig figs 3,200,000 2 sig figs Rules for Significant Figures in Mathematical Operations • Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation. 6.38 x 2.0 = 12.76 13 (2 sig figs) Sig Fig Practice #2 Calculation Calculator says: Answer 3.24 m x 7.0 m 22.68 m2 23 m2 100.0 g ÷ 23.7 cm3 4.219409283 g/cm3 4.22 g/cm3 0.02 cm x 2.371 cm 0.04742 cm2 0.05 cm2 710 m ÷ 3.0 s 236.6666667 m/s 240 m/s 1818.2 lb x 3.23 ft 5872.786 lb·ft 5870 lb·ft 1.030 g ÷ 2.87 mL 2.9561 g/mL 2.96 g/mL Rules for Significant Figures in Mathematical Operations • Addition and Subtraction: The number of decimal places in the result equals the number of decimal places in the least precise measurement. 6.8 + 11.934 = 18.734 18.7 (3 sig figs) Sig Fig Practice #3 Calculation Calculator says: Answer 3.24 m + 7.0 m 10.24 m 10.2 m 100.0 g - 23.73 g 76.27 g 76.3 g 0.02 cm + 2.371 cm 2.391 cm 2.39 cm 713.1 L - 3.872 L 709.228 L 709.2 L 1818.2 lb + 3.37 lb 1821.57 lb 1821.6 lb 2.030 mL - 1.870 mL 0.16 mL 0.160 mL Practice Question Questions 1-2 refer to the following sets of numbers. A.1.023 g B.0.0030 mL C.40,500 m 1.Is a number containing three significant figures 2.Is a measure of mass Scientific Notation In science, we deal with some very LARGE numbers: 1 mole = 602000000000000000000000 In science, we deal with some very SMALL numbers: Mass of an electron = 0.000000000000000000000000000000091 kg Imagine the difficulty of calculating the mass of 1 mole of electrons! 0.000000000000000000000000000000091 kg x 602000000000000000000000 ??????????????????????????????????? Scientific Notation: A method of representing very large or very small numbers in the form: M x 10n M is a number between 1 and 10 n is an integer 2 500 000 000. 9 8 7 6 5 4 3 2 1 Step #1: Insert an understood decimal point Step #2: Decide where the decimal must end up so that one number is to its left Step #3: Count how many places you bounce the decimal point Step #4: Re-write in the form M x 10n 2.5 x 10 9 The exponent is the number of places we moved the decimal. 0.0000579 1 2 3 4 5 Step #2: Decide where the decimal must end up so that one number is to its left Step #3: Count how many places you bounce the decimal point Step #4: Re-write in the form M x 10n 5.79 x 10 -5 The exponent is negative because the number we started with was less than 1. When you move the decimal to the left, the exponent is positive. When you move the decimal to the right, the exponent is negative You Try! Convert the following numbers to scientific Answers: notation 1. 21.9 1. 2.19 X 101 2. 6022 2. 6.022 X 103 3. 0.12011 3. 1.2011 X 10-1 Convert the following into expanded form 4. 1800 4. 1.8 X 103 5. .000081 5. 8.1 X 10-5 6. 720 6. 7.2 X 102 Multiplying and Dividing in Scientific Notation 1. Multiply or divide the Example #1 “M” values 2. If multiplying, add the (1.35 x 104) x (2.35 x 105) exponents 3. If dividing, subtract the exponents. 4. If necessary, move decimal and adjust Example #2 exponent to get numbers back into (2.6 x 104) / (4.6 x 103) scientific notation. You try 1. (6 X 103) x (4 X 10-3) Answers: 1. 2.4 x 101 2. (3 x 104) x (4.5 x 105) 2. 1.35 x 1010 3. 5 x 10-3 3. ( 4.5 x 10-5) / (9 x 10-3) Nature of Measurement Measurement – quantitative observation consisting of 2 parts Part 1 – number Part 2 – unit Examples: 20 grams 6.63 x 10-34 Joule seconds Accurate or Precise? Accurate measurements are close to the actual or accepted value. Precise measurements are close to one another. More than one measurement must be taken to determine if the measurements are precise. The Fundamental SI Units (le Système International, SI) SI Prefixes Prefix Abbr. Meaning Exponent Giga G 1,000,000,000 109 Mega M 1,000,000 106 Kilo K 1,000 103 Hecto h 100 102 Deca D 10 101 Base unit 1 deci d 1/10 10-1 Centi c 1/100 10-2 Milli m 1/1000 10-3 Micro 1/1,000,000 10-6 Nano n 1/ 1,000,000,000 10-9 Converting Among SI Units Convert 2.6 grams to Convert 5.25 decigrams to milligrams micrograms You Try! • Convert the following Answer: 1. 3.4 liters to milliliters 1. 3,400 mL 2. 7,899 milligrams to 2. 7.899 g grams 3. .0277 cg 3. 277 kilograms to 4. 2,000,000 ug centigrams 4. 2 meters to micrometers Derived SI Units • Produced by multiplying or dividing standard units. 5m For Example: Area = (Length)(Width) Width 2.5 m Length Area = (2.5 m)(5 m) = 12.5 m2 Density The ratio of mass to volume, or mass divided by volume. mass m Density = D= volume V Density • A measure of how closely matter is packed into a volume. • Unique for each compound. – Density of water is 1.00 g/mL at 25˚C. – Increasing temperature decreases the density, so densities are given with temperatures. • An intensive property. • Substances that are less dense float in substances more dense. Density Problems A sample of aluminum metal has a mass of 8.4 g. The volume of the sample is 3.1 cm3. Calculate the density of aluminum. Given: m = 8.4 g 8.4 g Don’t V = 3.1 cm3 D= forget 3.1 cm3 units! Unknown: D = ? Equation: Box D = 2.7 g/cm3 m answer! D= v Don’t forget units!!! You try! An unknown liquid is discovered at a crime scene. A volume of 2.3 mL has a mass of 4.1 grams, what the liquid’s density? Density Problems Diamond has a density of 3.26 g/cm3. What is the mass of a diamond that has a volume of 0.350 cm3? Density Problems A sample of metal is found to have a mass of 4.56 g and a density of 1.98 g/mL. What is the volume of this metal? Density Problem (No calculator) The typical battery in a car is filled with a solution of sulfuric acid, which is approximately 39.9% sulfuric acid. If the density of this solution is 1.3 g/mL, determine the number of grams of acid present in 500.0 mL of battery solution. 1. What is the volume of 5 grams of this substance? 2. What is the approximate density of the substance? Converting Temperatures C = 5/9 (F-32) F= K = C + 273 C= You Try! Convert the following Answers temperatures 1. 293 K to Celsius 1. 20 K 2. Room temperature to 2. About 24 C Celsius 3. About 309 K 3. Internal body temperature to Kelvin Steps to complete these problems Step 1: Read the problem CAREFULLY. Step 2: Determine the unit for the answer Step 3: Write down all values given in the problem and retrieve any needed conversion factors Step 4: Set up the problem (watch carefully as teacher does this step) Step 5: Calculate—Multiply by numbers on the top and divide by those on the bottom Practice #1 The record long jump is 349.5 inches. Convert this to meters. There are 2.54 cm in an inch. Practice #2 A car is traveling 55.0 miles per hour. Convert this to meters per second. One mile is equal to 1.61 km. Practice #3 How many mg are there is a 5.00 grain aspirin tablet? 1 grain = 0.00229 oz. There are 454 grams/lb. There are 16 oz./lb Practice #4 Convert 24 km/h to m/s (write out all steps before using calculator). Practice #5 In 1980, the US produced 18.4 billion (18.4 X 109) pounds of phosphoric acid to be used in the manufacture of fertilizer. The average cost of the acid is $318/ton. (1 ton = 2000 lbs). What was the total value of the phosphoric acid produced?