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1
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
2
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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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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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
15
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
24
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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• 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
6
Rounding Off Numbers
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The results of a calculation based on measurements cannot be more precise than the least precise measurement.
41
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.
42
(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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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.
46
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 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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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
56
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
58
Dimensional Analysis
Dimensional analysis converts one unit to another by using conversion factors.
unit1 x conversion factor = unit2
59
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.
60
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
61
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
63
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:
65
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
67
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.
69
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.
73
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
74
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
76
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.
79
• 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.
81
Convert 4.61 x 102 microliters to milliliters.
Microliters can be converted to milliliters using a series of two conversion factors.
L L mL
6
1 Lx
10 μL24.61x10 μL -4 4.61x10 L
-1 = 4.61 x 10 mL-44.61x10 L1000 mL
x1 L
82
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.
84
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.
85
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.
86
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.
88
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
90
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
91
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
93
Density is the ratio of the mass of a substance to the volume occupied by that substance.
massd =
volume
94
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
95
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
99
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
100
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
101
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
102
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
103
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
104
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