1

Measurement

• Quantitative Observation

• Comparison Based on an Accepted Scale– e.g. Meter Stick

• Has 2 Parts – the Number and the Unit– Number Tells Comparison– Unit Tells Scale

2

Scientific Notation

• Technique Used to Express Very Large or Very Small Numbers

• Based on Powers of 10

• To Compare Numbers Written in Scientific Notation– First Compare Exponents of 10– Then Compare Numbers

3

Writing Numbers in Scientific Notation1 Locate the Decimal Point2 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

3 Multiply the new number by 10n

– where n is the number of places you moved the decimal point

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

4

Writing Numbers in Standard Form

1 Determine the sign of n of 10n

– If n is + the decimal point will move to the right– If n is – the decimal point will move to the left

2 Determine the value of the exponent of 10– Tells the number of places to move the decimal

point

3 Move the decimal point and rewrite the number

5

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

6

Table 2.2: The Commonly used Prefixes in the Metric System

7

Length• SI unit = meter (m)

– About 3½ inches longer than a yard• 1 meter = one ten-millionth the distance from the North Pole to the

Equator = distance between marks on standard metal rod in a Paris vault = distance covered by a certain number of wavelengths of a special color of light

• Commonly use centimeters (cm)

– 1 m = 100 cm

– 1 cm = 0.01 m = 10 mm

– 1 inch = 2.54 cm (exactly)

8

Table 2.3: The Metric System for Measuring Length

9

Figure 2.1: Comparison of English and metric units for length on a ruler

10

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)

– 1 m3 = 106 cm3 – 1 cm3 = 10-6 m3 = 0.000001 m3

• Commonly measure liquid or gas volume in milliliters (mL)– 1 L is slightly larger than 1 quart– 1 L = 1 dm3 = 1000 mL = 103 mL – 1 mL = 0.001 L = 10-3 L– 1 mL = 1 cm3

11

Figure 2.2: Cubes

12

Figure 2.3: A 100-ml Graduated Cylinder

13

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– 1 kg = 1000 g = 103 g, 1 g = 1000 mg = 103 mg– 1 g = 0.001 kg = 10-3 kg, 1 mg = 0.001 g = 10-3 g

14

Figure 2.4: An electronic analytical

balance used in chemistry labs

15

Uncertainty in Measured Numbers

• A measurement always has some amount of uncertainty

• Uncertainty comes from limitations of the techniques used for comparison

• To understand how reliable a measurement is, we need to understand the limitations of the measurement

16

Reporting Measurements

• 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

• Unless stated otherwise, the uncertainty in the last digit is ±1

17

Figure 2.5: Measuring a Pin

18

Rules for Counting Significant Figures

• Nonzero integers are always significant• Zeros

– Leading zeros never count as significant figures– Captive zeros are always significant– Trailing zeros are significant if the number has

a decimal point

• Exact numbers have an unlimited number of significant figures

19

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

20

Calculations with Significant Figures

• Calculators/computers do not know about 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– round at the end of the calculation

21

Rules for Rounding Off

• If the digit to be removed• is less than 5, the preceding digit stays the

same• is equal to or greater than 5, the preceding

digit is increased by 1

• 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!!

22

Multiplication/Division with Significant Figures

• Result has the same number of significant figures as the measurement with the smallest number of significant figures

• Count the number of significant figures in each measurement

• Round the result so it has the same number of significant figures as the measurement with the smallest number of significant figures

4.5 cm x 0.200 cm = 0.90 cm2

2 sig figs 3 sig figs 2 sig figs

23

Adding/Subtracting Numbers with Significant Figures

• Result is limited by the number with the smallest number of significant decimal places

• Find last significant figure in each measurement

• Find which one is “left-most”• Round answer to the same decimal place

450 mL + 27.5 mL = 480 mLprecise to 10’s place precise to 0.1’s place precise to 10’s place

24

Problem Solving and Dimensional Analysis

• Many problems in chemistry involve using equivalence statements to convert one unit of measurement to another

• Conversion factors are relationships between two units– May be exact or measured– Both parts of the conversion factor should have the same

number of significant figures

• Conversion factors generated from equivalence statements– e.g. 1 inch = 2.54 cm can give or

in1cm54.2

cm54.2in1

25

• Arrange conversion factors so starting unit cancels– Arrange conversion factor so starting unit is on

the bottom of the conversion factor

• May string conversion factors

Problem Solving and Dimensional Analysis

26

Converting One Unit to Another

• Find the relationship(s) between the starting and goal units. Write an equivalence statement for each relationship.

• Write a conversion factor for each equivalence statement.

• Arrange the conversion factor(s) to cancel starting unit and result in goal unit.

27

Converting One Unit to Another

• Check that the units cancel properly

• Multiply and Divide the numbers to give the answer with the proper unit.

• Check your significant figures

• Check that your answer makes sense!

28

Temperature Scales• Fahrenheit Scale, °F

– Water’s freezing point = 32°F, boiling point = 212°F

• Celsius Scale, °C– Temperature unit larger than the Fahrenheit

– Water’s freezing point = 0°C, boiling point = 100°C

• Kelvin Scale, K– Temperature unit same size as Celsius

– Water’s freezing point = 273 K, boiling point = 373 K

29

Figure 2.6: Thermometers based on the three temperature scales in (a) ice water and (b) boiling water

30

Figure 2.7: The three major temperature scales

31

Figure 2.8: Converting 70°C to units measured on the Kelvin scale

32

Figure 2.9: Comparison of the Celsius and Fahrenheit scales

33

Density• Density is a 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

– 1 cm3 = 1 mL

• Liquids = g/mL

• Gases = g/L

• Volume of a solid can be determined by water displacement

• Density : solids > liquids >>> gases

• In a heterogeneous mixture, denser object sinks

VolumeMass

Density

34

Using Density in Calculations

VolumeMass

Density

DensityMass

Volume

Volume Density Mass

35

Figure 2.10: (a) Tank of water. (b) Person submerged in the tank, raising the level of the water.