Measurement and Calculations

Transcription

1 Measurement and Calculations Quantitative Observation How much? Need Measurement Measurement is the comparison of a physical quantity to be measured with a unit of measurement-that is a fixed standard of measurement consisting of 2 parts Part 1 number Part 2 scale (unit) Examples: 20 grams : consist a number and a unit The Metric System (SI) The metric system and SI (international system) are decimal systems based on 10 used in most of the world used everywhere by scientists 3 1

2 Units of Measurement The Fundamental SI Units Physical Quantity Name of Unit Abbreviation Mass kilogram kg Length meter m Time second s Temperature kelvin K Electric current ampere A Amount of substance mole mol Luminous intensity candela cd A prefix Prefixes in front of a unit increases or decreases the size of that unit makes units larger or smaller that the initial unit by one or more factors of 10 indicates a numerical value prefix = value 1 kilometer = 1000 meters 1 kilogram = 1000 grams 5 Metric and SI Prefixes Prefix Symbol Meaning Decimal Multiple giga G billion mega M million kilo k thousand (unit) meter, gram, liter, second 1 deci d tenth centi c hundredth milli m thousandth micro millionth nano n billionth

3 Fundamental Units Length Mass, Length, Time, Temperature is measured using a meterstick has the unit meter (m) in both the metric and SI systems 7 Mass The mass of an object measures the quantity of matter An object has a fixed mass (m), which is independent of where or how the mass is measured has the unit gram(g) in the metric system and kilogram(kg) in the SI system Relation between mass and weight Weight is the force of gravity on an object. It is directly proportional to mass. W = g m Mass is a physical property that measures the amount of matter in an object, whereas weight measures the pull of gravity on an object. 8 Temperature is a measure of how hot or cold an object is compared to another object indicates that heat flows from the object with a higher temperature to the object with a lower temperature is measured using a thermometer units are degrees Celsius ( C) and kelvins (K) T K = T C T F = 1.8(T C )

4 Temperature Scales Temperature scales are Fahrenheit, Celsius(metric), and Kelvin (SI) have reference points for the boiling and freezing points of water A comparison of the Fahrenheit, Celsius, and Kelvin temperature scales between the freezing and boiling points of water. 10 Celsius and Fahrenheit Formula On the Fahrenheit scale, there are 180 F between the freezing and boiling points and on the Celsius scale, there are 100 C. 180 F = 9 F = 1.8 F 100 C 5 C 1 C In the formula for calculating the Fahrenheit temperature, adding 32 adjusts the zero point of water from 0 C to 32 F. T F = 1.8(T C ) Kelvin Temperature Scale The SI temperature scale, called Kelvin scale, assigns a value of zero to the coldest possible temperature, C, sometime called absolute zero. The Kelvin temperature scale has 100 units between freezing and boiling points 100 K = 100 C or 1 K = 1 C adds 273 to the Celsius temperature of 0 K (absolute zero) is the lowest possible temperature 0 K = C K = T C

5 Question 1.6: When converting from Celsius to Fahrenheit temperature scales, the fraction 9/5 is often encountered. This indicates that A. Five degrees on the Celsius scale is actually the same change in temperature as nine degrees on the Fahrenheit scale. B. Celsius degrees are smaller than Fahrenheit degrees. C. Zero degrees Fahrenheit would then correspond to 1.8 degrees Celsius. D. I do not know how to convert from Celsius to Fahrenheit (will I need to?). Question 1.7 On a cold winter day, the temperature is 15 C. What is that temperature in F? 1. 1) 19 F 2. 2) 5 F F F 0% 0% 0% 0% ) 5 F T F = 1.8 T C + 32 Solution T F = 1.8( 15 C) + 32 = = 5 F Note: Be sure to use the change sign key on your calculator to enter the minus sign. 1.8 x 15 +/ =

6 Derived Units Area (m 2 ), volume (m 3 ), density (kg/m 3 ), speed (m/s) etc are some of the derived quantities that you are familiar with. They are derived quantities rather than fundamental quantities because they can be expressed using one or more of the seven base units. Volume is the space occupied by a matter has the unit liter (L) in the metric system 1 L = qt Volume Measurement has the unit m 3 (cubic meter) in the SI system For liguid it can be measured using a graduated cylinder 17 For each of the following, indicate whether the unit describes 1) length 2) mass or 3) volume. A. Learning Check A bag of onions has a mass of 2.6 kg. B. C. D. A person is 2.0 m tall. A medication contains 0.50 g of aspirin. A bottle contains 1.5 L of water. 18 6

7 Solution For each of the following, indicate whether the unit describes 1) length 2) mass or 3) volume. 2 A. A bag of onions has a mass of 2.6 kg. 1 B. A person is 2.0 m tall. 2 C. A medication contains 0.50 g of aspirin. 3 D. A bottle contains 1.5 L of water. 19 Density compares the mass of an object to its volume is the mass of a substance divided by its volume Density expression: D = mass = g or g = g/cm 3 volume ml cm 3 Note: 1 ml = 1 cm 3 Density 20 Sink or Float Objects that sink in water are more dense than water. Ice floats in water because the density of ice is less than the density of water. Aluminum sinks because its density is greater than the density of water. 21 7

8 Learning Check Which diagram correctly represents the liquid layers in the cylinder? Karo (K) syrup (1.4 g/ml), vegetable (V) oil (0.91 g/ml), water (W) (1.0 g/ml) V W K K W V W V K 22 Solution 1) V W K vegetable oil 0.91 g/ml water 1.0 g/ml Karo syrup 1.4 g/ml 23 Question 1.8 Osmium is a very dense metal. What is its density, in g/cm 3, if 50.0 g of osmium has a volume of 2.22 cm 3? 1. 1) 2.25 g/cm ) 22.5 g/cm ) 111 g/cm 3 33% 33% 33%

9 Solution STEP 1 Given 50.0 g; 2.22 cm 3 Need density, in g/cm 3 STEP 2 Plan Write the density expression. D = mass volume STEP 3 Express mass in grams and volume in cm 3 mass = 50.0 g volume = 2.22 cm 3 STEP 4 Set up problem D = 50.0 g = g/cm cm 3 = 22.5 g/cm 3 (3 SF) 25 Writing Numbers Provide some numbers written below One 1 1*10 0 Ten 10 1*10 1 Four thousand *10 3 Fiften million *10 7 One tenth 0.1 1*10-1 One hundredth *10-2 Now write One trillionth 1*10-12 Which way is a more effective way to write a number? Do you see anything in common on how those numbers were written? Scientific Notation Scientific notation is used to write very large or very small numbers for the width of a human hair ( m) is written as 8 x 10-6 m for a large number such as s is written as 4.5 x 10 6 s 27 9

10 Scientific Notation A number written in scientific notation contains a coefficient (C) and a power of 10. C 10 n where C is a number greater than 1 and less than 10 coefficient power of ten coefficient power of ten 1.5 x x 10-4 To write a number in scientific notation, the decimal point is moved after the first digit. The spaces moved are shown as a power of ten = 5.2 x = 3.78 x spaces left 3 spaces right 28 Some Powers of Ten 29 Question 1.9 Select the correct scientific notation for each ) 8 x ) 8 x ) 0.8 x % 33% 33%

11 Question 1.10 Write 1.8 x 10 5 as a standard number % % 9% Practice Normal Notation 9,000,000, ,800, Fill in the above table. Scientific Notation 1-32 Solutions: Normal Notation Scientific Notation 9,000,000, ,800,

12 Scientific Notation You will use your calculator to perform mathematical operations in chemistry. Use these simple rules for mathematical operations involving exponents to confirm that your answer makes sense. Three operations are most common: Multiplication ( )( ) = ( ) 10 6+( 3) = When multiplying numbers in scientific notation, multiply the coefficients and add the exponents Division Scientific Notation ( )/( ) = (4.0/2.0) 10 ( 2) 3 = When dividing numbers in scientific notation, divide the coefficients and subtract the exponents. Raising the exponent to a power ( ) 2 = (3.0) = When raising numbers in scientific notation to a power, raise the coefficient to that power, then multiply the exponents Question 1.11 Multiplication and Division of Exponents 1. ( )( ) = 2. ( ) / ( ) = 3. ( ) 4 = Solutions: 1. ( )( ) = ( ) / ( ) = ( ) 4 =

13 Measured Numbers A measuring tool is used to determine a quantity such as the length or the mass of an object provides numbers for a measurement called measured numbers 37 Reading a Meterstick. l l.... l l.... l 4.. cm The markings on the meterstick at the end of the orange line are read as The first digit 2 plus the second digit 2.7 The last digit is obtained by estimating. The end of the line might be estimated between as half-way (0.05) or a little more (0.06), which gives a reported length of 2.75 cm or 2.76 cm. 38 Known + Estimated Digits In the length reported as 2.76 cm, the digits 2 and 7 are certain (known) the final digit 6 was estimated (uncertain) all three digits (2.76) are significant including the estimated digit 39 13

14 Learning Check. l l.... l l.... l 10.. cm What is the length of the red line? 1) 9.0 cm 2) 9.03 cm 3) 9.04 cm Solution The length of the red line could be reported as 2) 9.03 cm or 3) 9.04 cm The estimated digit may be slightly different. Both readings are acceptable. 40 Zero as a Measured Number. l l.... l l.... l 5.. cm For this measurement, the first and second known digits are 4.5. Because the line ends on a mark, the estimated digit in the hundredths place is 0. This measurement is reported as 4.50 cm. 41 Uncertainty in Measurement When report a measurement a digit that must be estimated is called an uncertain digit. A measurement always has some degree of uncertainty. All measurements are subject to error. In talking about the degree of uncertainty in a measurement, we use the words accuracy and precision. 14

15 Precision vs. Accuracy The precision of a measured number is the extent of the agreement between repeated measurements of its value. If repeated measurements are close in value, then the number is precise, but not necessarily accurate. Accuracy is the difference between the value of a measured number and its expected or correct value. The number is accurate if it is close to its true value (much like hitting a bulls-eye on a dart board) Precision vs. Accuracy Figures from p Error and Percent Error Error = experimental value accepted value. %Error = error /accepted value X

16 Significant Figures in Measured Numbers Significant figures The minimum number of digits required to report a value without loss of accuracy is the number of significant figures obtained from a measurement include all of the known digits plus the estimated digit. The number of significant figures reported in a measurement depends on the measuring tool. 46 Significant Figures 47 Exact Numbers An exact number is obtained when objects are counted Example: counting objects 2 baseballs 4 pizzas from numbers in a defined relationship Example: defined relationships 1 foot = 12 inches 1 meter = 100 cm Exact numbers can be assumed to have unlimited number of significant figures 48 16

17 Learning Check State the number of significant figures in each of the following measurements: A m B L Solution 2 4 C g D m Solution Question 1.12 In which set(s) do both numbers contain the same number of significant figures? A and B and 4.00 x 10 2 C and D and 1.02 Question1.13: Exact and Measured Numbers A. Exact numbers are obtained by 1. using a measuring tool 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool 2. counting 3. definition Solution:.A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool 51 17

18 Rounding Off Answers In calculations answers must have the same number of significant figures as the measured numbers calculated answers are usually rounded off rounding rules are used to obtain the correct number of significant figures 52 Rounding Off Calculated Answers When the first digit dropped is 4 or less, the retained numbers remain the same. to round off to 3 significant figures drop the digits 32 = 45.8 When the first digit dropped is 5 or greater, the last retained digit is increased by 1. to round off to 2 significant figures drop the digits 884 = 2.5 (increase by 0.1) 53 Adding Significant Zeros Sometimes a calculated answer requires more significant digits. Then one or more zeros are added. Calculated Zeros Added to Answer Give 3 Significant Figures

19 Calculations with Measured Numbers In calculations with measured numbers, significant figures or decimal places are used to determine the correct number of figures in the final answer. 55 When multiplying or dividing, use the same number of significant figures (SFs) as in the measurement with the fewest significant figures rounding rules to obtain the correct number of significant figures Example Multiplication and Division x = = 5.3 (rounded) 4 SFs 2 SFs calculator 2 SFs 56 Addition and Subtraction When adding or subtracting, use the same number of decimal places as the measurement with the fewest decimal places rounding rules to adjust the number of digits in the answer 25.2 one decimal place two decimal places calculated answer 26.5 final answer (one decimal place) 57 19

20 Give an answer for each with the correct number of significant figures. A x 4.2 = Learning Check 1) 9 2) 9.2 3) B = 1) ) 62 3) 60 C x = x ) ) 11 3) A x 4.2 = 2) 9.2 B = 3) 60 C x = 2) x Solution On a calculator, enter each number followed by the operation key x = = 11 (rounded off) 59 Learning Check For each calculation, round the answer to give the correct number of digits. A = 1) 257 2) ) B = 1) ) )

21 Solution A hundredths place (2 decimal places) tenths place (1 decimal place) + 2 ones place rounds to 257 answer (1) B thousandths place (3 decimal places) tenths place (1 decimal place) round to 40.7 answer (3) 61 Learning Check Indicate the unit that matches the description. 1. A mass that is 1000 times greater than 1 gram. 1) kilogram 2) milligram 3) megagram 2. A length that is 1/100 of 1 meter. 1) decimeter 2) centimeter 3) millimeter 3. A unit of time that is 1/1000 of a second. 1) nanosecond 2) microsecond 3) millisecond 62 Solution Indicate the unit that matches the description. 1. A mass that is 1000 times greater than 1 gram. 1) kilogram 2. A length that is 1/100 of 1 meter. 2) centimeter 3. A unit of time that is 1/1000 of a second. 3) millisecond 63 21

22 Metric Equalities An equality states the same measurement in two different units can be written using the relationships between two metric units Example: 1 meter is the same length as 100 cm and 1000 mm. 1 m = 100 cm 1 m = 1000 mm 100 cm = 1000 mm 64 Metric Equalities for Volume 65 Metric Equalities for Mass Several equalities can be written for mass in the metric (SI) system. Example: 1 gram is the same mass as 1000 mg and kg. 1 kg = 1000 g 1 g = 1000 mg 1 mg = g 1 mg = 1000 µg 66 22

23 Equalities Equalities use two different units to describe the same measured amount are written for relationships between units of the metric system, U.S. units, or between metric and U.S. units Examples: 1 m = 1000 mm 1 lb = 16 oz lb = 1 kg 67 Exact and Measured Numbers in Equalities Equalities between units of the same system are definitions and are exact numbers different systems (metric and U.S.) use measured numbers and count as significant figures 68 Some Common Equalities 69 23

24 Indicate the unit that completes each of the following equalities: A m = 1) 1 mm 2) 1 km 3) 1dm B g = 1) 1 mg 2) 1 kg 3) 1dg C m = Learning Check 1) 1 mm 2) 1 cm 3) 1dm 70 Solution Indicate the unit that completes each of the following equalities: A. 2) 1000 m = 1 km B. 1) g = 1 mg C. 2) 0.01 m = 1 cm 71 Conversion Factors A conversion factor is a fraction obtained from an equality Equality: 1 in. = 2.54 cm written as a ratio with a numerator and denominator is inverted to give two conversion factors for every equality 1 in. and 2.54 cm 2.54 cm 1 in

25 Factors with Powers A conversion factor can be squared or cubed on both sides of the equality. Equality 1 in. = 2.54 cm 1 in. and 2.54 cm 2.54 cm 1 in. Equality squared (1 in.) 2 = (2.54 cm) 2 (1 in.) 2 and (2.54 cm) 2 (2.54 cm) 2 (1 in.) 2 Equality cubed (1 in.) 3 = (2.54 cm) 3 (1 in.) 3 and (2.54 cm) 3 (2.54 cm) 3 (1 in.) 3 73 Conversion Factors in a Problem An equality and conversion factors may be obtained from information in a word problem are for that problem only Example: The price of one pound (1 lb) of red peppers is $2.39. Equality: 1 lb peppers = $2.49 Conversion factors: 1 lb red peppers and $2.39 $ lb red peppers 74 Percent as a Conversion Factor A percent factor gives the ratio of the part to the whole. Percent(%) = _part x 100 whole uses matching units to express the percent uses the value 100 and a unit for the whole Example: A food contains 30% (by mass) fat. Equality: 100 g food = 30 g fat Conversion factors: 30 g fat and 100 g food 100 g food 30 g fat 75 25

26 Learning Check Write the equality and conversion factors for each of the following: A. square meters and square centimeters 1 m 2 = (100 cm) 2 (1m) 2 and (100 cm) 2 (100 cm) 2 (1m) 2 B. jewelry that contains 18% gold C. one gallon of gas costing $3.29 D. a water sample with 55 ppb of chromium (Cr) 76 Solution B. 100 g of jewelry = 18 g of gold 18 g gold and 100 g jewelry 100 g jewelry 18 g gold C. 1 gal of gas = $ gal and $3.29 $ gal D. 1 kg of water = 55 g of chromium 1 kg water and 55 g chromium 55 g chromium 1 kg water 77 Problem Solving: Use Dimensional Analysis Dimensional analysis is the method of calculation in which one carries along the units for quantities-use conversion factor. To solve a problem first identify the given unit identify the needed unit Example: A person has a height of 2.0 m. What is that height in inches? The given unit is the initial unit of height. given unit = meters (m) The needed unit is the unit for the answer. needed unit = inches (in.) 78 26

27 Guide to Problem Solving Using Conversion Factors STEP 1 State the given and needed quantities. STEP 2 Write a plan to convert the given unit to the needed unit. STEP 3 State the equalities and conversion factors needed to cancel units. STEP 4 Set up problem to cancel units and calculate answer. Unit 1 x Unit 2 = Unit 2 Unit 1 Given Conversion Needed unit factor unit 79 Setting up a Problem How many minutes are 2.5 hours? STEP 1 Given 2.5 h Need min STEP 2 Plan hours minutes STEP 3 Equalities 1 h = 60 min STEP 4 Set up problem 2.5 h x 60 min = 150 min 1 h (2 SFs) given conversion needed unit factor unit 80 Using Two or More Factors Often, two or more conversion factors are required to obtain the unit needed for the answer. Unit 1 Unit 2 Unit 3 Additional conversion factors are placed in the setup to cancel each preceding unit. Given unit x factor 1 x factor 2 = needed unit Unit 1 x Unit 2 x Unit 3 = Unit 3 Unit 1 Unit

28 Example How many minutes are in 1.4 days? STEP 1 Given 1.4 days Need minutes STEP 2 Plan days hours minutes STEP 3 Equalties: 1 day = 24 h 1 h = 60 min STEP 4 Set up problem: 1.4 days x 24 h x 60 min = 2.0 x 10 3 min 1 day 1 h 2 SFs Exact Exact = 2 SFs 82 Check the Unit Cancellation Be sure to check your unit cancellation in the setup. The units in the conversion factors must cancel to give the correct unit for the answer. Example: What is wrong with the following setup? 1.4 day x 1 day x 1 h 24 h 60 min Units = day 2 /min, which is not the unit needed. Units do not cancel properly; the setup is wrong. 83 Percent Factor in a Problem If the thickness of the skin fold at the waist indicates 11% of body fat, how much fat, in kg, does a person have with a body mass of 86 kg? STEP 1 Given 86 kg; 11% fat Need kg fat STEP 2 Plan kg body mass kg fat STEP 3 Equalities/Conversion factors 100 kg of body mass = 11 kg of fat 100 kg body mass and 11 kg fat 11 kg fat 100 kg body mass STEP 4 Set Up Problem 86 kg mass x 11 kg fat = 9.5 kg of fat 100 kg mass 84 28

29 Question 1.14 How many lb of sugar are in 120 g of candy if the candy is 25% (by mass) sugar? 1 lb= 453.6g 85 STEP 1 Given 120 g of candy Need lb of sugar STEP 2 Plan g lb of candy lb of sugar STEP 3 Equalities 1 lb = g STEP 4 Set Up Problem 100 g of candy = 25 g of sugar % factor 120 g candy x 1 lb candy x 25 lb sugar g candy 100 lb candy = lb of sugar Solution 86 Density as a Conversion Factor For a density of 3.8 g/ml, an equality is written as 3.8 g = 1 ml and two conversion factors are written as Conversion 3.8 g and 1 ml factors 1 ml 3.8 g 87 29

30 Question 1.15 The density of octane, a component of gasoline, is g/ml. What is the mass, in kg, of 875 ml of octane? 1. 1) kg 2. 2) 614 kg 3. 3) 1.25 kg STEP 1 Given D = g/ml; Need mass in kg of octane STEP 2 Plan ml g kg STEP 3 Equalities Solution V= 875 ml g = 1 ml 1 kg = 1000 g STEP 4 Set Up Problem 875 ml x g x 1 kg = kg (1) 1 ml 1000 g 89 A solid completely submerged in water displaces its own volume of water. The volume of the solid is calculated from the volume difference ml ml = 9.5 ml = 9.5 cm 3 Volume by Displacement The density of a solid is determined from its mass and the volume it displaces when submerged in water

31 Density Using Volume Displacement The density of the object is calculated from its mass and volume. mass_ = g = 7.2 g/cm 3 volume 9.5 cm 3 91 Question 1.16 What is the density (g/cm 3 ) of 48.0 g of a metal if the level of water in a graduated cylinder rises from 25.0 ml to 33.0 ml after the metal is added? g/cm g/cm g/cm ml 33.0 ml Solution STEP 1 Given 48.0 g volume of water = 25.0 ml volume of water + metal = 33.0 ml Need Density (g/cm 3 ) STEP 2 Plan Calculate the volume difference, change unit to cm 3, and place in density expression. STEP 3 Express mass in grams and volume in cm 3 volume of solid = 33.0 ml ml = 8.0 ml 8.0 ml x 1 cm 3 = 8.0 cm 3 1 ml STEP 4 Set Up Problem Density = 48.0 g = 6.0 g = 6.0 g/cm cm 3 1 cm

32 Question 1.17 Which of the following samples of metals will displace the greatest volume of water? A. 25 g of aluminum 2.70 g/ml 89% B. 45 g of gold 19.3 g/ml C. 75 g of lead 11.3 g/ml 8% 0% 3% A. B. C. D. A) 25 g of aluminum 2.70 g/ml Plan: Calculate the volume for each metal and select the metal sample with the greatest volume. A) 25 g x 1 ml = 9.3 ml aluminum 2.70 g B) 45 g x 1 ml = 2.3 ml gold 19.3 g C) 75 g x 1 ml = 6.6 ml lead 11.3 g Solution 95 Question 1.18: The normal temperature of a chickadee is F. What is that temperature on the Celsius scale? C C C C 0% 0% 0% 0%

33 Solution 3) T F = 1.8T C + 32 T C = (T F 32 ) 1.8 = ( ) 1.8 = 73.8 = 41.0 C Circle Graph PERCENT CHLORINE IN THE STRATOSPHERE Hydrogen Chloride (HCl) Methyl Chloride (CH3Cl) Carbon tetrachloride (CCl4) Methyl Chloroform (C2H3Cl3) CFC-11 CFC-12 CFC-13 HCFC-22 6% 3% 3% 15% 28% 12% 10% 23% Bar Graph Food Serving Halibut (3 oz) Almonds (1 oz) Cashews (1 oz) Spinach (1/2 Baked Potato Bran Flakes Kidney Beans Raisins (1/4 cup) (medium) (3/4 cup) (1/2 cup) cup) Food Serving 33

34 Mass, g 10/2/2016 Line Graph Density of Aluminum Volume, cm 3 34