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Standards for Measurement Chapter 2

Standards for Measurement Chapter 2

Hein and Arena Eugene Passer Chemistry Department Bronx Community College© John Wiley and Sons, Inc

Version 2.0

12th Edition

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Chapter Outline

2.1 Scientific Notation

2.2 Measurement and Uncertainty

2.6 Problem Solving

2.4 Significant Figures in Calculations

2.5 The Metric System

2.8 Measurement of Temperature

2.9 Density

2.3 Significant Figures2.7 Measuring Mass and Volume

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2.12.1

Scientific Notation2.12.1

Scientific Notation

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6022000000000000000000000.00000000000000000000625

• Very large and very small numbers like these are awkward and difficult to work with.

• Very large and very small numbers are often encountered in science.

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602200000000000000000000

A method for representing these numbers in a simpler form is called scientific notation.

0.00000000000000000000625

6.022 x 1023

6.25 x 10-21

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Scientific Notation

• Move the decimal point in the original number so that it is located after the first nonzero digit.

• Follow the new number by a multiplication sign and 10 with an exponent (power).

• The exponent is equal to the number of places that the decimal point was shifted.

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Write 6419 in scientific notation.

64196419.641.9x10164.19x1026.419 x 103

decimal after first nonzero

digitpower of 10

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Write 0.000654 in scientific notation.

0.0006540.00654 x 10-10.0654 x 10-20.654 x 10-3 6.54 x 10-4

decimal after first nonzero

digitpower of 10

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2.22.2

Measurementand Uncertainty

2.22.2

Measurementand Uncertainty

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Measurements

• Experiments are performed.

• Measurements are made.

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Form of a Measurement

70.0 kilograms = 154 pounds

numerical value

unit

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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

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uncertain6.06320

Significant Figures

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

certain

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Reading a ThermometerReading a Thermometer

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Temperature is estimated to be 21.2oC. The last 2 is uncertain.

The temperature 21.2oC is expressed to 3 significant figures.

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Temperature is estimated to be 22.0oC. The last 0 is uncertain.

The temperature 22.0oC is expressed to 3 significant figures.

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Temperature is estimated to be 22.11oC. The last 1 is uncertain.

The temperature 22.11oC is expressed to 4 significant figures.

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12 inches = 1 foot100 centimeters = 1 meter

• Exact numbers have an infinite number of significant figures.

• Exact numbers occur in simple counting operations

Exact Numbers

• Defined numbers are exact.

12345

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2.32.3

Significant Figures2.32.3

Significant Figures

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461

All nonzero numbers are significant.

Significant Figures

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461

All nonzero numbers are significant.

Significant Figures

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461

All nonzero numbers are significant.

Significant Figures

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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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Rounding Off NumbersRounding Off Numbers

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• Often when calculations are performed on a calculator 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

Rules for Rounding Off

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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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2.42.4

Significant Figures in Calculations

2.42.4

Significant Figures in Calculations

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The results of a calculation based on measurements 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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2.52.5

The Metric System2.52.5

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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International System’s Standard Units of Measurement

Quantity Name of Unit Abbreviation

Length meter m

Mass kilogram kg Temperature Kelvin K

Time second s

Amount of substance m mole

Electric Current ampere A

Luminous Intensity candela cd

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Common Prefixes and Numerical Values for SI Units Power of 10

Prefix Symbol Numerical Value Equivalent

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

Power of 10Prefix Symbol Numerical Value Equivalent

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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.1

299,792,458

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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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2.62.6

Problem Solving2.62.6

Problem 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 known and what is to be solved for and write it down.

It is important to label all factors and units with the proper labels.

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2. 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.

3. 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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4. Proceed with the necessary mathematical operations.

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

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

Basic Steps

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Length ConversionLength Conversion

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How many millimeters are there in 2.5 meters?

• It must cancel meters.

• It must introduce millimeters

unit1 x conversion factor = unit2

m x conversion factor = mm

The conversion factor must accomplish two things:

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The conversion factor takes a fractional

form.

mmm x = mm

m

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conversion factor

conversion factor

The conversion factor is derived from the equality.

1 m = 1000 mm

Divide both sides by 1000 mm

Divide both sides by 1 m

1 m 1000 mm = 1

1 m 1 m

1 m 1000 mm = 1

1000m 1000 mm

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Use the conversion factor with millimeters in the numerator and meters in the denominator.

1000 mmx

1 m2.5 m = 2500 mm

32.5 x 10 mm

How many millimeters are there in 2.5 meters?

1000 mm

1 m

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16.0 in2.54 cm

x 1 in

= 40.6 cm

2.54 cm1 in

Use this conversion factor

Convert 16.0 inches to centimeters.

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Convert 16.0 inches to centimeters.

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Convert 3.7 x 103 cm to micrometers.

33.7 x 10 cm1 m

x 100 cm

610 μmx

1 m7 = 3.7 x 10 μm

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

cm m meters

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Centimeters can be converted to micrometers by a series of two conversion factors.

cm m meters

33.7 x 10 cm1 m

x 100 cm

1 = 3.7 x 10 m

610 μmx

1 m7 = 3.7 x 10 μm13.7 x 10 m

Convert 3.7 x 103 cm to micrometers.

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2.72.7

MeasuringMass and Volume

2.72.7

MeasuringMass and Volume

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MassMass

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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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Metric Units of mass Exponential

Unit Abbreviation Gram Equivalent Equivalent

kilogram kg 1,000 g 103 g

gram g 1 g 100 g

decigram dg 0.1 g 10-1 g

centigram cg 0.01 g 10-2 g

milligram mg 0.001 g 10-3 g

microgram g 0.000001 g 10-6 g

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Convert 45 decigrams to grams.

45 dg1 g

x 10 dg

= 4.5 g

1 g = 10 dg

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An atom of hydrogen weighs 1.674 x 10-24 g. How many ounces does the atom weigh?

1 lbx

454 g-241.674 x 10 g -27 3.69 x 10 lb

16 ozx

1 lb-26 5.90 x 10 oz-273.69 x 10 lb

1 lb = 454 g

16 oz = 1 lb

Grams can be converted to ounces using a series of two conversion factors.

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An atom of hydrogen weighs 1.674 x 10-24 g. How many ounces does the atom weigh?

-241.674 x 10 g1 lb

x454 g

16 ozx

1 lb-26 5.90 x 10 oz

Grams can be converted to ounces using a single linear expression by writing down conversion factors in succession.

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VolumeVolume

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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.

• Volume is the amount of space occupied by matter.

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Convert 4.61 x 102 microliters to milliliters.

Microliters can be converted to milliliters using a series of two conversion factors.

L L mL

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1 Lx

10 μL24.61x10 μL -4 4.61x10 L

-1 = 4.61 x 10 mL-44.61x10 L1000 mL

x1 L

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Microliters can be converted to milliliters using a linear expression by writing down conversion factors in succession.

L L mL

24.61x10 μL 6

1 Lx

10 μL1000 mL

x1 L

-1= 4.61 x 10 mL

Convert 4.61 x 102 microliters to milliliters.

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2.82.8

Measurement of Temperature

2.82.8

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.

• The 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 lower 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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Degree Symbols

degrees Celsius = oC

Kelvin (absolute) = K

degrees Fahrenheit = oF

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o o oF - 32 = 1.8 x C

To convert between the scales, use the following relationships:

o o oF = 1.8 x C + 32

oK = C + 273.15

oo F - 32C =

1.8

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180 Farenheit Degrees

= 100 Celcius degrees

180 =1.8

100

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It is not uncommon for temperatures in the Canadian plains to reach –60oF 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 –60oF 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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2.92.9

Density2.92.9

Density

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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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Density varies with temperature

o

2

4 CH O

1.0000 g gd = = 1.0000

1.0000 mL mL

o

2

80 CH O

1.0000 g gd = = 0.97182

1.0290 mL mL

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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 = = 22.40 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 = 22.40 L O

1.429 g O

g → L

Lg x = L

gThe conversion of units is

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