3.1 Introduction to Measurement Measurement Measurement of a quantity is made by comparison with a standard that serves to represent the magnitude of a unit. Result of a measurement = number x unit
3.1 Introduction to Measurement Measurements in the U.S. are made in theUnited States Customary System. Unit of mass is 1 pound Unit of length is 1 inch Measurements in science are made in the International System (SI). Unit of mass is 1 kilogram Unit of length is 1 meter
3.1 Introduction to Measurement Mass of a notebook = 2.4 kg The mass of the notebook is 2.4 times the unit of mass, which is 1 kilogram
3.1 Introduction to Measurement The SI system is defined by seven base units. Examples of base units include: QuantityBase Unit Mass Kilogram Length Meter Temperature Kelvin Time Second Other measurement units are derived from the base units; accordingly, they are called derived units.
3.2 Exponential Notation Exponentials BE B is the base E is the power or exponent 104 = 10 × 10 × 10 × 10 = 10,000 10–4 = = ××× = = 0.0001
3.2 Exponential Notation Exponential Notation a.bcd × 10E Coefficient: a.bcd Usually 1 ≤ coefficient < 10 Exponent: E A whole number
3.2 Exponential Notation Conversion Between Decimal Numbers and Standard Exponential Notation Example: Convert 724,000 to standard exponential notation 7.24 × 10e 7 2 4 0 0 0 . Five places 7.24 × 105
3.2 Exponential Notation Conversion Between Decimal Numbers and Standard Exponential Notation Example: Convert 0.000427 to standard exponential notation 4.27 × 10e 0 . 0 0 0 4 2 7 Four places 4.27 × 10–4
3.3 Dimensional Analysis Dimensional Analysis A problem-solving technique when quantities are directly proportional. A conversion factor is a mathematical statement that two quantities are directly proportional to one another Examples: 7 Days = 1 week 7 Days/1 week 7 Days per week
3.3 Dimensional Analysis Direct Proportionality Two variables, X and Y, are directly proportional if there is a nonzero constant m that relates them in the form Y= m X This relationship can also be expressed as a proportionality: YX
3.3 Dimensional Analysis How to Solve a Problem by Dimensional Analysis Sample Problem: How many days are in 23 weeks? Step 1: Identify and write down the GIVEN quantity, including GIVEN: 23 weeks units.
3.3 Dimensional Analysis How to Solve a Problem by Dimensional Analysis Sample Problem: How many days are in 23 weeks? Step 2: Identify and write down the GIVEN: 23 weeks units of the WANTED WANTED: days quantity.
3.3 Dimensional Analysis How to Solve a Problem by Dimensional Analysis Sample Problem: How many days are in 23 weeks? Step 3:GIVEN: 23 weeks Write down the WANTED: days PER/PATH.PER: 7 days/week PATH: wk days
3.3 Dimensional Analysis How to Solve a Problem by Dimensional Analysis Sample Problem: How many days are in 23 weeks? Step 4:GIVEN: 23 weeks Write the calculation WANTED: days setup. Include units. PER: 7 days/week PATH: wk days 23 weeks × =
3.3 Dimensional Analysis How to Solve a Problem by Dimensional Analysis Sample Problem: How many days are in 23 weeks? Step 5:GIVEN: 23 weeks Calculate the answer. WANTED: days PER: 7 days/week PATH: wk days 23 weeks × = 161 days
3.3 Dimensional Analysis How to Solve a Problem by Dimensional Analysis Sample Problem: How many days are in 23 weeks? Step 6:GIVEN: 23 weeks Check the answer to be sure WANTED: days both the number and the PER: 7 days/week units make sense. PATH: wk days 23 weeks × = 161 days More days (smaller unit) than weeks (larger unit). OK.
3.4 Metric Units: Mass and Weight Mass is a measure of quantity of matter. Unit for mass is 1 kilogram Weight is a measure of the force of gravitational attraction. Unit for force is 1 Neuton. Mass and weight are directly proportional to each other. Same weight, same mass. Weight = mass x acceleration of gravity Acceleration of gravity on the earth = 9.80 m/s2 Weight of one apple (0.1 kg) is about 1 Neuton
The SI unit of mass is the kilogram, kg. It is defined as the mass of a platinum-iridium cylinder stored in a vault in France. 1 lb 0.45359237 kg (definition) A kilogram is about 2.2 pounds. 3.4 Metric Units. Kilogram as unit
3.4 Metric Units For most laboratory work, the basic metric mass unit is used: the gram, g.
3.4 Metric Units In the metric system, units that are larger than the basic unit are larger by multiples of 10. For example, the kilo- unit is 1000 times larger than the basic unit. Units that are smaller than the basic unit are smaller by fractions that are also multiples of 10. For example, the milli- unit is 1/1000 times smaller than the basic unit.
3.4 Metric Units: Metric Prefixes Large UnitsSmall Units Metric MetricMetricMetric PrefixSymbolMultiplePrefixSymbolMultiple tera- T 1012 Unit 1 giga- G 109deci- d 0.1 mega- M 106centi- c 0.01 kilo- k 1000 milli- m 0.001 hecto- h 100 micro- µ 10–6 deca- da 10 nano- n 10–9 Unit 1 pico- p 10–12
3.4 Metric Units. Length The SI unit of length is the meter, m. It is defined as the distance light travels in a vacuum in 1/299,792,458 second One inch is defined as 2.54 centimeters.(1 in. ≡ 2.54 cm) One meter is about forty inches:
Metric Units One inch is defined as 2.54 centimeters.
3.4 Metric Units. Volume The SI unit of volume is the cubic meter, m3. A more practical unit for laboratory work is the cubic centimeter, cm3.
3.4 Metric Units One liter (L) is defined as exactly 1000 cubic centimeters. 1 mL ≡ 0.001 L ≡ 1 cm3
3.4 Metric Units Metric RelationshipExample 1000 units per kilounit 1000 meters per kilometer 1000 m/km 100 centiunits per unit 100 centigrams per gram 100 cg/g 1000 milliunits per unit 1000 milliliters per liter 1000 mL/L
3.4 Metric Units Example: How many centigrams are in 0.87 gram? Solution: Use dimensional analysis. GIVEN: 0.87 g WANTED: cg PER: 100 cg/g PATH: g cg 0.87 g × = 87 cg More centigrams (smaller unit) than grams (larger unit). OK.
3.4 Metric Units Example: How many kilometers are in 2,335 meters? Solution: Use dimensional analysis. GIVEN: 2335 m WANTED: km PER: 1000 m/km PATH: m km 2335 m × = 2.335 km More meters (smaller unit) than kilometers (larger unit). OK.
3.4 Metric Units Example: How many milliliters are in 0.00339 liter? Solution: Use dimensional analysis. GIVEN: 0.00339 L WANTED: mL PER: 1000 mL/L PATH: L mL 0.00339 L × = 3.39 mL More milliliters (smaller unit) than liters (larger unit). OK.
3.5 Significant Figures Uncertainty in Measurement No measurement is exact. In scientific writing, the uncertainty associated with a measured quantity is always included. By convention, a measured quantity is expressed by stating all digits known accurately plus one uncertain digit.
3.5 Significant Figures The bottom board is one meter long. How long is the top board? More than half as long as the meter stick, but less than one meter—about 6/10 of a meter. The uncertain digit is the last digit written: 0.6 m
3.5 Significant Figures Now the meter stick has marks every 0.1 m, numbered in centimeters. How long is the board? Between 0.6 m and 0.7 m with certainty, and the uncertain digit must be estimated—the board is about 4/10 of the way between 0.6 m and 0.7 m: 0.64 m.
3.5 Significant Figures The measuring instrument now has centimeter marks. How long is the board? Between 0.64 m and 0.65 m with certainty. It is about 3/10 of the way between the two marks, so we record 0.643 m as the length of the board.
3.5 Significant Figures The measuring instrument now has millimeter marks. We could estimate between the millimeter marks, but the alignment of the board and the meter stick has an uncertainty of a millimeter or so. We have reached the limit of this measuring instrument: 0.643 m.
3.5 Significant Figures Significant Figures Significant figures are applied to measurements and quantities calculated from measurements. They do not apply to exact numbers. An exact number has no uncertainty. Types of exact numbers: Counting numbers Numbers fixed by definition
3.5 Significant Figures Significant Figures The number of significant figures in a quantity is the number of digits that are known accurately plus the one that is uncertain—the uncertain digit. The uncertain digit is the last digit written when expressing a scientific measurement.
3.5 Significant Figures Significant Figures The measurement process, not the unit in which the result is expressed, determines the number of significant figures in a quantity. The length of the board in the previous illustrations was 0.643 m. Expressed in centimeters, it is 64.3 cm. They are the same measurement with the same uncertainty. Both must have the same number of significant figures.
3.5 Significant Figures Significant Figures The location of the decimal point has nothing to do with significant figures. The same 0.643 m board is 0.000643 km. The three zeros before the decimal point are not significant. Begin counting significant figures at the first nonzero digit, not at the decimal point.
3.5 Significant Figures Significant Figures The uncertain digit is the last digit written. If the uncertain digit is a zero to the right of the decimal point, that zero must be written. If the mass of a sample on a triple beam balance is 15.10 g, and the balance is accurate to ±0.01 g, the last digit recorded must be zero to indicate the correct uncertainty.
3.5 Significant Figures Significant Figures Exponential notation must be used for very large numbers to show if final zeros are significant. If the length of the 0.643 m board is expressed in micrometers, its length is 643,000 µm. The uncertainty is ±1,000 µm. The ordinary decimal number makes this ambiguous. Writing 6.43 × 105 µm shows clearly the correct location of the uncertain digit.
3.5 Significant Figures Rounding a Calculated Number If the first digit to be dropped is less than 5, leave the digit before it unchanged. Examples: Round to three significant figures. Answers 1.743 m 1.74 m 0.041239 kg 0.0412 kg
3.5 Significant Figures Rounding a Calculated Number If the first digit to be dropped is 5 or more, increase the digit before it by 1. Examples: Round to three significant figures. Answers 32.88 mL 32.9 mL 0.0097761 km 0.00978 km
3.5 Significant Figures Significant Figure Rule for Addition and Subtraction Round off the answer to the first column that has an uncertain digit.
3.5 Significant Figures Example: The following is a list of masses of items to be shipped. What is the total mass of the package? Carton: 226 g; Item 1: 33.5 g; Item 2: 589 g; Packaging: 11.88 g Answer: 2 2 6 g 3 3 . 5 g 5 8 9 g 1 1 . 8 8 g 8 6 0 . 3 8 g = 860 g = 8.60 × 102 g
3.5 Significant Figures Significant Figure Rule For Multiplication and Division Round off the answer to the same number of significant figures as the smallest number of significant figures in any factor.
3.5 Significant Figures Example: What is the volume of a cube that is 34.49 cm long, 23.0 cm wide and 15 cm high? Solution: Use algebra because the GIVENS and WANTED are related by a formula, volume = length × width × height. V = l × w × h = 34.49 cm × 23.0 cm × 15 cm 4 sf3 sf 2 sf = 11,899.05 cm3 (unrounded) The answer is rounded to 2 sf, 1.2 × 104 cm3
3.6 Metric–USCS Conversions Conversions between the United States Customary System (USCS) and the metric system are made by applying dimensional analysis. Length 1 in. 2.54 cm (definition of an inch) Mass 1 lb 453.59237 g (definition of a pound) Volume 1 gal 3.785411784 L (definition of a gallon)
3.6 Metric–USCS Conversions Some Useful Definitions Length 1 ft 12 in. 1 yd 3 ft 1 mi 5280 ft Mass 1 lb 16 oz Volume 1 qt 32 fl oz 1 gal 4 qt