Measurements and Calculations

Chapter 2

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

1 2 3 4

Writing Numbers in Scientific Notation

Locate the Decimal Point 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 Multiply the new number by 10

n

– where

n

is the number of places you moved the decimal point 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 10

n

– If

n

– If

n

is + the decimal point will move to the right 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

7

8

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) 9

10

Artist's conception of the lost Mars Climate Orbiter.

11

Volume

• Measure of the amount of three-dimensional space occupied by a substance • SI unit = cubic meter (m 3 ) • Commonly measure solid volume in cubic centimeters (cm 3 ) – 1 m 3 = 10 6 cm 3 – 1 cm 3 = 10 -6 m 3 = 0.000001 m 3 • Commonly measure liquid or gas volume in milliliters (mL) – 1 L is slightly larger than 1 quart – 1 L = 1 dL 3 = 1000 mL = 10 – 1 mL = 0.001 L = 10 -3 L 3 – 1 mL = 1 cm 3 mL 12

Figure 2.2: The largest drawing represents a cube that has 1 m in length and a volume of 1 m 3 . The middle-size cube has sides 1 dm in length and a volume of 1 dm 3 . The smalles cube has sides 1 cm in length and a volume of 1cm 3 .

Figure 2.3: A 100-mL graduated cylinder

.

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 = 10 3 mg g, 1 g = 1000 mg = 10 3 – 1 g = 0.001 kg = 10 -3 10 -3 g kg, 1 mg = 0.001 g = 15

16

17

18

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 19

Temperature

20

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

21

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 22

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 23

Rules for Counting Significant Figures

• Nonzero integers are always significant • Zeros – Leading zeros never count as significant figures – Captive (embedded) zeros are always significant – Trailing zeros are significant if the number has a decimal point • Exact numbers have an unlimited number of significant figures 24

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!!

25

Exact Numbers

•

Exact Numbers

certainty are numbers known with • 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 26

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 27

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 cm 2 2 sig figs 3 sig figs 2 sig figs 28

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 mL

precise to 10’s place precise to 0.1’s place precise to 10’s place 29

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 2 .

54 cm 1 in – e.g. 1 inch = 2.54 cm can give 1 in 2 .

54 30

Problem Solving and Dimensional Analysis

• 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 31

• • •

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.

32

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!

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/cm 3 – 1 cm 3 = 1 mL • Liquids = g/mL • Gases = g/L

Density



Mass Volume

• Volume of a solid can be determined by water displacement • Density : solids > liquids >>> gases • In a heterogeneous mixture, denser object sinks 34

Using Density in Calculations

Density



Mass Volume Volume



Mass Density Mass



Density



Volume

35

36

Figure 2.11: A hydrometer being used to determine the density of the antifreeze solution in a car’s radiator .