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Numbers in Science Chapter 2 Measurement What is measurement? A measurement has 2 Parts – the Number and the Unit Quantitative Observation Based on a comparison to an accepted scale. Number Tells Comparison Unit Tells Scale There are two common unit scales English Metric The Unit The measurement System units English (US) Metric (rest of the world) Length – inches/feet Distance – mile Volume – gallon/quart Mass- pound Length – meter Distance – kilometer Volume – liter Mass - gram Related Units in the Metric System All units in the metric system are related to the fundamental unit by a power of 10 The power of 10 is indicated by a prefix The prefixes are always the same, regardless of the fundamental unit Fundamental Unit 100 Fundamental SI Units Established in 1960 by an international agreement to Name of Unit Abbreviation standardize science units These metric system Mass units are in theKilogram kg Physical Quantity Length Meter m Time Second s Temperature Kelvin K Energy Joules J Pressure Pascal Pa Volume Cubic meters m3 Length….. SI unit = meter (m) About 3½ inches longer than a yard 1 meter = distance between marks on standard metal rod in a Paris vault or distance covered by a certain number of wavelengths of a special color of light Commonly use centimeters (cm) 1 inch (English Units) = 2.54 cm (exactly) Figure 2.1: Comparison of English and metric units for length on a ruler. Volume Measure of the amount of three-dimensional space occupied by a substance SI unit = cubic meter (m3) Commonly measure solid volume in cubic centimeters (cm3) Commonly measure liquid or gas volume in milliliters (mL) ◦ ◦ 1 L is slightly larger than 1 quart 1 mL = 1 cm3 Mass Measure of the amount of matter present in an object SI unit = kilogram (kg) Commonly measure mass in grams (g) or milligrams (mg) 1 kg = 2.2046 pounds, 1 lbs.. = 453.59 g Temperature Scales Any idea what the three most common temperature scales are? Fahrenheit Scale, °F ◦ Celsius Scale, °C ◦ ◦ Water’s freezing point = 32°F, boiling point = 212°F Temperature unit larger than the Fahrenheit Water’s freezing point = 0°C, boiling point = 100°C Kelvin Scale, K (SI unit) ◦ ◦ Temperature unit same size as Celsius Water’s freezing point = 273 K, boiling point = 373 K Thermometers based on the three temperature scales in (a) ice water and (b) boiling water. The number Scientific Notation Technique Used to Express Very Large or Very Small Numbers 135,000,000,000,000,000,000 meters 0.00000000000465 liters Based on Powers of 10 What is power of 10 Big? 0,10, 100, 1000, 10,000 100, 101, 102, 103, 104 What is the power of 10 Small? 0.1, 0.01, 0.001, 0.0001 10-1, 10-2, 10-3, 10-4 Writing Numbers in Scientific Notation 1. Locate the Decimal Point : 1,438. 2. Move the decimal point to the right of the non-zero digit in the largest place - The new number is now between 1 and 10 - 1.438 3. Now, multiply this number by a power of 10 (10n), where n is the number of places you moved the decimal point - In our case, we moved 3 spaces, so n = 3 (103) The final step for the number…… 4. Determine the sign on the exponent n If the decimal point was moved left, n is + If the decimal point was moved right, n is – If the decimal point was not moved, n is 0 - We moved left, so 3 is positive - 1.438 x 103 Writing Numbers in Standard Form We reverse the process and go from a number in scientific notation to standard form….. 1 Determine the sign of n of 10n 2 Determine the value of the exponent of 10 3 If n is + the decimal point will move to the right If n is – the decimal point will move to the left Tells the number of places to move the decimal point Move the decimal point and rewrite the number Try it for these numbers: 2.687 x 106 and 9.8 x 10-2 Let’s Practice….. Change these numbers to Scientific Notation: 1,340,000,000,000 697, 000 0.00000000000912 1.34 x 1012 6.97 x 105 9.12 x 10-12 Change these numbers to Standard Form: 3.76 x 10-5 8.2 x 108 1.0 x 101 0.0000376 820,000,000 10 Are you sure about that number? Uncertainty in Measured Numbers cm A measurement always has some amount of uncertainty, you always seem to be guessing what the smallest division is… To indicate the uncertainty of a single measurement scientists use a system called significant figures The last digit written in a measurement is the number that is considered to be uncertain Rules, Rules, Rules…. We follow guidelines (i.e. rules) to determine what numbers are significant Nonzero integers are always significant 2753 89.659 .281 Zeros Captive zeros are always significant (zero sandwich) 1001.4 55.0702 4780.012 Significant Figures – Tricky Zeros Zeros Leading zeros never count as significant figures 0.00048 0.0037009 0.0000000802 Trailing zeros are significant if the number has a decimal point 22,000 63,850. 0.00630100 2.70900 100,000 Significant Figures Scientific Notation All numbers before the “x” are significant. Don’t worry about any other rules. 7.0 x 10-4 g has 2 significant figures 2.010 x 108 m has 4 significant figures How many significant figures are in these numbers? 102,340 1.0 x 107 1,908,021.0 0.01796 1,200.00 0.000002 92,017 0.1192 8.01010 x 1014 Have a little fun remembering sig figs http://www.youtube.com/watch?v=ZuVPkBb-z2I Exact Numbers Exact Numbers are numbers known with certainty Unlimited number of significant figures They are either counting numbers number of sides on a square or defined 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm 1 kg = 1000 g, 1 LB = 16 oz 1000 mL = 1 L; 1 gal = 4 qts. 1 minute = 60 seconds Calculations with Significant Figures Exact numbers do not affect the number of significant figures in an answer Answers to calculations must be rounded to the proper number of significant figures For addition and subtraction, the last digit to the right is the uncertain digit. round at the end of the calculation Use the least number of decimal places For multiplication, count the number of sig figs in each number in the calculation, then go with the smallest number of sig figs Use the least number of significant figures Rules for Rounding Off If the digit to be removed • • is less than 5, the preceding digit stays the same Round 87.482 to 4 sig figs. is equal to or greater than 5, the preceding digit is increased by 1 Round 0.00649710 to 3 sig figs. In a series of calculations, carry the extra digits to the final result and then round off Don’t forget to add place-holding zeros if necessary to keep value the same!! Round 80,150,000 to 3 sig figs. Examples of Sig Figs in Math 1) 5.18 x 0.0208 2) 21 + 13.8 + 130.36 3) 116.8 – 0.33 Answers must be in the proper number of significant digits!!! Solutions: 1) 0.107744 round to proper # sig fig 1) 2) 165.47 1) 3) 5.18 has 3 sig figs, 0.0208 has 3 sig figs so answer is 0.108 Limiting number of sig figs in addition is the smallest number of decimal places = 12 (no decimals) answer is 165 116.47 1) Same rule as above so answer is 116.5 Moving unit to unit: Conversion Exact Numbers Exact Numbers are numbers known with certainty They are either counting numbers number of sides on a square or defined 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm 1 kg = 1000 g, 1 LB = 16 oz 1000 mL = 1 L; 1 gal = 4 qts. 1 minute = 60 seconds The Metric System Fundamental Unit 100 Movement in the Metric system In the metric system, it is easy it is to convert numbers to different units. Let’s convert 113 cm to meters Figure out what you have to begin with and where you need to go.. How many cm in 1 meter? 100 cm in 1 meter Set up the math sentence, and check that the units cancel properly. 113 cm [1 m/100 cm] = 1.13 m Let’s Practice converting metric units 250 mL to Liters 0.250 mL 475 cg to kg 47,500,000 or 4.75 x 107 1.75 kg to grams 1,750 grams 328 mm to dm 3.28 dm 88 µL to mL 0.088 mL 0.00075 nL to µL 0.75 µL Converting Between Metric and non-Metric (English) units Converting non-Metric Units Many problems involve using equivalence statements to convert one unit of measurement to another Conversion factors are relationships between two units Conversion factors are generated from equivalence statements e.g. 1 inch = 2.54 cm can give 2.54 cm 1in or 1in 2.54 cm Converting non-Metric Units Arrange conversion factor so starting unit is on the bottom of the conversion factor You may string conversion factors together for problems that involve more than one conversion factor. Convert kilometers to miles Convert kilometers to inches Find the relationship(s) between the starting and final units. Write an equivalence statement and a conversion factor for each relationship. Arrange the conversion factor(s) to cancel starting unit and result in goal unit. Practice Convert 1.89 km to miles Convert 5.6 lbs to grams Find equivalence statement 1mile = 1.609 km 1.89 km (1 mile/1.609 km) 1.17 miles Find equivalence statement 454 grams = 1 lb 5.6 lbs(454 grams/1 lb) 2500 grams Convert 2.3 L to pints Find equivalence statements: 1L = 1.06 qts, 1 qt = 2 pints 2.3 L(1.06 qts/1L)(2 pints/1 qt) 4.9 pints Temperature Conversions To find Celsius from Fahrenheit oC To find Fahrenheit from Celsius oF = 1.8(oC) +32 Celsius to Kelvin = (oF -32)/1.8 K = oC + 273 Kelvin to Celsius oC = K – 273 Temperature Conversion Examples 1) 180°C to Kelvin 1) 2) 2) 23°C to Fahrenheit 1) 2) 3) 3) Use the conversion factor: F = (1.80)C + 32 F = (1.80)23 + 32 F=73.4 or 73°F 87°F to Celsius 1) 2) 3) 4) To convert Celsius to Kelvin add 273 180+ 273 = 453 K Use the conversion factor C=5/9(F-32) C = 5/9(87-32) C = 30.5555555… or 31°C 694 K to Celsius 1) 2) To convert K to C, subtract 273 694-273= 421°C Measurements and Calculations Density Density is a physical property of matter representing the mass per unit volume For equal volumes, denser object has larger mass For equal masses, denser object has small volume Solids = g/cm3 Mass Liquids = g/mL Density Gases = g/L Volume Volume of a solid can be determined by water displacement Density : solids > liquids >>> gases In a heterogeneous mixture, denser object sinks Using Density in Calculations Mass Density Volume Mass Volume Density Mass Density Volume Density Example Problems What is the density of a metal with a mass of 11.76 g whose volume occupies 6.30 cm3? What volume of ethanol (density = 0.785 g/mL) has a mass of 2.04 lbs? What is the mass (in mg) of a gas that has a density of 0.0125 g/L in a 500. mL container? How could you find your density? Volume by displacement To determine the volume to insert into the density equation, you must find out the difference between the initial volume and the final volume. A student attempting to find the density of copper records a mass of 75.2 g. When the copper is inserted into a graduated cylinder, the volume of the cylinder increases from 50.0 mL to 58.5 mL. What is the density of the copper in g/mL? A student masses a piece of unusually shaped metal and determines the mass to be 187.7 grams. After placing the metal in a graduated cylinder, the water level rose from 50.0 mL to 60.2 mL. What is the density of the metal? A piece of lead (density = 11.34 g/cm3) has a mass of 162.4 g. If a student places the piece of lead in a graduated cylinder, what is the final volume of the graduated cylinder if the initial volume is 10.0 mL? Percent Error Percent error – absolute value of the error divided by the accepted value, multiplied by 100%. % error = measured value – accepted value x 100% accepted value Accepted value – correct value based on reliable sources. Experimental (measured) value – value physically measured in the lab. Percent Error Example In the lab, you determined the density of ethanol to be 1.04 g/mL. The accepted density of ethanol is 0.785 g/mL. What is the percent error? The accepted value for the density of lead is 11.34 g/cm3. When you experimentally determined the density of a sample of lead, you found that a 85.2 gram sample of lead displaced 7.35 mL of water. What is the percent error in this experiment? Joe measured the boiling point of hexane to be 66.9 °C. If the actual boiling point of hexane is 69 °C , what is the percent error? A student calculated the volume of a cube to be 68.98 cm3. If the true volume is 71.08 cm3, what is the student’s percent error? Tom used the density of copper and the volume of water displaced to measure the mass of a copper pipe to be 145.67 g. When he actually weighed the sample, he found a mass of 146.82 g. What was his percent error?