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2

Measurement and

Significant Figures

Measurement and

Significant Figures

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Measurements

• Experiments are performed.

• Numerical values or data are obtained from these measurements.

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Significant Figures

• The number of digits that are known plus one estimated digit are considered significant in a measured quantity

estimated5.16143

known

5

estimated6.06320

Significant Figures

• The number of digits that are known plus one estimated digit are considered significant in a measured quantity

known

6

12 inches = 1 foot100 centimeters = 1 meter

• Exact numbers have an infinite numbers of significant figures.

• Exact numbers occur in simple counting operations

Exact Numbers

• Defined numbers are exact.

12345

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461

All nonzero numbers are significant.

Significant Figures

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461

All nonzero numbers are significant.

Significant Figures

9

461

All nonzero numbers are significant.

Significant Figures

10

461

3 Significant Figures

All nonzero numbers are significant.

Significant Figures

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401

3 Significant Figures

A zero is significant when it is between nonzero digits.

Significant Figures

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A zero is significant when it is between nonzero digits.

5 Significant Figures

600.39

Significant Figures

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3 Significant Figures

30.9

A zero is significant when it is between nonzero digits.

Significant Figures

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A zero is significant at the end of a number that includes a decimal point.

5 Significant Figures

000.55

Significant Figures

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A zero is significant at the end of a number that includes a decimal point.

5 Significant Figures

0391.2

Significant Figures

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A zero is not significant when it is before the first nonzero digit.

1 Significant Figure

600.0

Significant Figures

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A zero is not significant when it is before the first nonzero digit.

3 Significant Figures

907.0

Significant Figures

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A zero is not significant when it is at the end of a number without a decimal point.

1 Significant Figure

00005

Significant Figures

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A zero is not significant when it is at the end of a number without a decimal point.

4 Significant Figures

01786

Significant Figures

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Roundingoff Numbers

Roundingoff Numbers

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Rounding Off Numbers

• Often when calculations are performed extra digits are present in the results.

• It is necessary to drop these extra digits so as to express the answer to the correct number of significant figures.

• When digits are dropped the value of the last digit retained is determined by a process known as rounding off numbers.

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80.873

Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed.

4 or less

Rounding Off Numbers

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1.875377

Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed.

4 or less

Rounding Off Numbers

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5 or greater

5.459672

Rule 2. When the first digit after those you want to retain is 5 or greater, that digit and all others to its right are dropped. The last digit retained is increased by 1.

drop these figuresincrease by 1

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Rounding Off Numbers

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Scientific Notation of Numbers

Scientific Notation of Numbers

Scientific notationIf a If a numbernumber is is largerlarger than 1than 1

Move decimal point XX places leftleft to get a number between 1 and 10.

1 2 3 , 0 0 0 , 0 0 0.

The resulting number is multiplied by 10XX.

= 1.23 x 108

Scientific notationIf a number is If a number is smallersmaller than 1than 1

Move decimal point XX places rightright to get a number between 1 and 10.

0. 0 0 0 0 0 0 1 2 3 = 1.23 x 10-7-7

The resulting number is multiplied by 10-X-X.

ExamplesWrite in Scientific Notation:

25 =

8931.5 =

0.000593 =

0.0000004 =

3,210. =

2.5 x 101

8.9315 x 103

5.93 x 10-4

4 x 10-7

3.210 x 103

×10

1.44939 × 10-2 =

Scientific notation

0.0144939

On Calculator

1.44939 (-) 2EE

Means ×10 Change

Sign

1.44939E -2

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Significant Figures in Calculations

Significant Figures in Calculations

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The results of a calculation cannot be more precise than the least precise measurement.

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Multiplication or DivisionMultiplication or Division

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In multiplication or division, the answer must contain the same number of significant figures as in the measurement that has the least number of significant figures.

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(190.6)(2.3) = 438.38

438.38

Answer given by calculator.

2.3 has two significant figures.

190.6 has four significant figures.

The answer should have two significant figures because 2.3 is the number with the fewest significant figures.

Drop these three digits.

Round off this digit to four.

The correct answer is 440 or 4.4 x 102

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Addition or SubtractionAddition or Subtraction

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The results of an addition or a subtraction must be expressed to the same precision as the least precise measurement.

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The result must be rounded to the same number of decimal places as the value with the fewest decimal places.

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Add 125.17, 129 and 52.2

125.17129.

52.2306.37

Answer given by calculator.

Least precise number.

Round off to the nearest unit.

306.37

Correct answer.

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1.039 - 1.020Calculate

1.039

1.039 - 1.020 = 0.018286814

1.039

Answer given by calculator.

1.039 - 1.020 = 0.019

0.019 = 0.018286814

1.039

The answer should have two significant figures because 0.019 is the number with the fewest significant figures.

2 80.018 6814

Two significant figures.

Drop these 6 digits.

0.018286814

Correct answer.

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The Metric System

The Metric System

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• The metric or International System (SI, Systeme International) is a decimal system of units.

• It is built around standard units.

• It uses prefixes representing powers of 10 to express quantities that are larger or smaller than the standard units.

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Standard Units of Measurement

Quantity Metric Unit (abbr.) SI Unit (abbr.)

Length meter (m) meter (m)

Mass gram (g) kilogram (kg) Volume liter (L) cubic meter (m3)Temperature Celsius (ºC) Kelvin (K)

Energy calorie (cal) Joule (J)

Pressure atmosphere (atm) pascal (Pa)

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Prefixes and Numerical Values for SI Units Power of 10

Prefix Symbol Numerical Value Equivalent

exa E 1,000,000,000,000,000,000 1018

peta P 1,000,000,000,000,000 1015

tera T 1,000,000,000,000 1012

giga G 1,000,000,000 109

mega M 1,000,000 106

kilo k 1,000 103

hecto h 100 102

deca da 10 101

— — 1 100

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Prefixes and Numerical Values for SI Units

deci d 0.1 10-1

centi c 0.01 10-2

milli m 0.001 10-3

micro 0.000001 10-6

nano n 0.000000001 10-9

pico p 0.000000000001 10-12

femto f 0.00000000000001 10-15

atto a 0.000000000000000001 10-18

Power of 10Prefix Symbol Numerical Value Equivalent

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Problem SolvingProblem Solving

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Dimensional Analysis

Dimensional analysis converts one unit to another by using conversion factors.

unit1 x conversion factor = unit2

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Basic Steps

1. Read the problem carefully. Determine what is to be solved for and write it down.

2. Tabulate the data given in the problem.– Label all factors and measurements with

the proper units.

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3. Determine which principles are involved and which unit relationships are needed to solve the problem.

– You may need to refer to tables for needed data.

4. Set up the problem in a neat, organized and logical fashion.

– Make sure unwanted units cancel. – Use sample problems in the text as

guides for setting up the problem.

Basic Steps

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5. Proceed with the necessary mathematical operations.

– Make certain that your answer contains the proper number of significant figures.

6. Check the answer to make sure it is reasonable.

Basic Steps

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Measurement of LengthMeasurement of Length

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The standard unit of length in the SI system is the meter. 1 meter is the distance that light travels in a vacuum during

of a second.1299,792,458

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• 1 meter is a little longer than a yard

• 1 meter = 39.37 inches

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Metric Units of Length Exponential

Unit Abbreviation Metric Equivalent Equivalent

kilometer km 1,000 m 103 m

meter m 1 m 100 m

decimeter dm 0.1 m 10-1 m

centimeter cm 0.01 m 10-2 m

millimeter mm 0.001 m 10-3 m

micrometer m 0.000001 m 10-6 m

nanometer nm 0.000000001 m 10-9 m

angstrom Å 0.0000000001 m 10-10 m

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How many feet are there in 22.5 inches?

• It must cancel inches.

• It must introduce feet

unit1 x conversion factor = unit2

in x conversion factor = ft

The conversion factor must accomplish two things:

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The conversion factor takes a fractional

form.

ftin = ft

in

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Putting in the measured value and the

ratio of feet to inches produces:

1 ft22.5 in = 1.875 ft

12 in

= 1.88 ft

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Convert 3.7×1015 in to miles.

153.7 10 in 1 ftx

12 in

1 milex

5280 ft10 = 5.8 10 miles

Inches can be converted to miles by writing down conversion factors in succession.

in ft miles

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Convert 4.51030 cm to kilometers.

304.5 10 cm 1 mx

100 cm

1 kmx

1000 m25 = 4.5 10 km

Centimeters can be converted to kilometers by writing down conversion factors in succession.

cm m km

Conversion of unitsExamples:

10.7 T = ? fl oz

62.04 mi = ? in

5.5 kg = ? mg

9.3 ft = ? cm

5.7 g/ml = ? lbs/qt

3.18 in2 = ? cm2

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Measurement of MassMeasurement of Mass

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The standard unit of mass in the SI system is the kilogram. 1 kilogram is equal to the mass of a platinum-iridium cylinder kept in a vault at Sevres, France.

1 kg = 2.205 pounds

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Measurement of VolumeMeasurement of Volume

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• In the SI system the standard unit of volume is the cubic meter (m3).

• The liter (L) and milliliter (mL) are the standard units of volume used in most chemical laboratories. 1 mL = 1 cm3 = 1cc

• Volume is the amount of space occupied by matter.

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Measurement of Temperature

Measurement of Temperature

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Heat

• A form of energy that is associated with the motion of small particles of matter.

• Heat refers to the quantity of this energy associated with the system.

• System is the entity that is being heated or cooled.

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Temperature

• A measure of the intensity of heat.

• It does not depend on the size of the system.

• Heat always flows from a region of higher temperature to a region of colder temperature.

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Temperature Measurement

• The SI unit of temperature is the Kelvin.

• There are three temperature scales: Kelvin, Celsius and Fahrenheit.

• In the laboratory temperature is commonly measured with a thermometer.

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Gabriel Daniel Fahrenheit Anders Celsius William Thomson

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Degree Symbols

degrees Celsius = oC

Kelvin (absolute) = K

degrees Fahrenheit = oF

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o oF - 32 = 1.8 C

To convert between the scales use the following relationships.

o oF = 1.8 C + 32

oK = C + 273.15

oo ( F - 32)C =

1.8

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It is not uncommon for temperatures in the Canadian planes to reach –60.oF and below during the winter. What is this temperature in oC and K?

oo F - 32C =

1.8

o o60. - 32C = = -51 C

1.8

–

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It is not uncommon for temperatures in the Canadian planes to reach –60.oF and below during the winter. What is this temperature in oC and K?

oK = C + 273.15

oK = -51 C + 273.15 = 222 K

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DensityDensity

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Density is the ratio of the mass of a substance to the volume occupied by that substance.

massd =

volume

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Mass is usually expressed in grams and volume in ml or cm3.

gd =

mL3

gd =

cm

The density of gases is expressed in grams per liter.

gd =

L

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ExamplesExamples

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A 13.5 mL sample of an unknown liquid has a mass of 12.4 g. What is the density of the liquid?

MD

V 0.919 g/mL12.4g

13.5mL

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46.0 mL

98.1 g

A graduated cylinder is filled to the 35.0 mL mark with water. A copper nugget weighing 98.1 grams is immersed into the cylinder and the water level rises to the 46.0 mL. What is the volume of the copper nugget? What is the density of copper?

35.0 mL

copper nugget final initialV = V -V = 46.0mL - 35.0mL = 11.0mL

g/mL8.92mL11.0g98.1

VM

D

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The density of ether is 0.714 g/mL. What is the mass of 25.0 milliliters of ether?

Method 1 (a) Solve the density equation for mass.

massd =

volume

(b) Substitute the data and calculate.

mass = density x volume

0.714 g25.0 mL x = 17.9 g

mL

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The density of ether is 0.714 g/mL. What is the mass of 25.0 milliliters of ether?

Method 2 Dimensional Analysis. Use density as a conversion factor. Convert:

0.714 g25.0 ml x = 17.9 g

mL

mL → g

gmL x = g

mLThe conversion of units is

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The density of oxygen at 0oC is 1.429 g/L. What is the volume of 32.00 grams of oxygen at this temperature?

Method 1

(a) Solve the density equation for volume.

massd =

volume

(b) Substitute the data and calculate.

massvolume =

density

2

2

32.00 g Ovolume = = 22.39 L

1.429 g O /L

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The density of oxygen at 0oC is 1.429 g/L. What is the volume of 32.00 grams of oxygen at this temperature?

Method 2 Dimensional Analysis. Use density as a conversion factor. Convert:

2 22

1 L32.00 g O x = 22.39 L O

1.429 g O

g → L

Lg x = L

gThe conversion of units is

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