1. 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 2.5 The Metric System
2.2 Measurement and Uncertainty
2.6 Problem Solving
2.7 Measuring Mass and Volume
2.3 Significant Figures
2.8 Measurement of Temperature
2.4 Significant Figures in Calculations
2.9 Density

3. 2.1 Scientific Notation

4. Very large and very small numbers are often encountered in science.
Very large and very small numbers like these are awkward and difficult to work with.

5. A method for representing these numbers in a simpler form is called scientific notation.
6.022 x 1023
6.25 x 10-21

6. 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.

7. Write 6419 in scientific notation.
decimal after first nonzero digit
power of 10
64.19x102
6.419 x 103
641.9x101
6419.
6419

8. Write 0.000654 in scientific notation.
decimal after first nonzero digit
power of 10
x 10-1
x 10-2
0.654 x 10-3
6.54 x 10-4

9. 2.2 Measurement and Uncertainty

10. Measurements
Experiments are performed.
Measurements are made.

11. Form of a Measurement
numerical value
70.0 kilograms =
154 pounds
unit

12. Significant Figures
The number of digits that are known plus one estimated digit are considered significant in a measured quantity
known
estimated

13. Significant Figures
The number of digits that are known plus one estimated digit are considered significant in a measured quantity
certain
uncertain

14. Reading a Thermometer

15. The temperature 21.2oC is expressed to 3 significant figures.
Temperature is estimated to be 21.2oC. The last 2 is uncertain.

16. The temperature 22.0oC is expressed to 3 significant figures.
Temperature is estimated to be 22.0oC. The last 0 is uncertain.

17. The temperature 22.11oC is expressed to 4 significant figures.
Temperature is estimated to be 22.11oC. The last 1 is uncertain.

18. Exact Numbers
Exact numbers have an infinite number of significant figures.
Exact numbers occur in simple counting operations 2 1 3 5 4
Defined numbers are exact.
100 centimeters = 1 meter
12 inches = 1 foot

19. 2.3 Significant Figures

20. Significant Figures
All nonzero numbers are significant.
461

21. Significant Figures
All nonzero numbers are significant.
461

22. Significant Figures
All nonzero numbers are significant.
461

23. 461 Significant Figures All nonzero numbers are significant.

24. Significant Figures
A zero is significant when it is between nonzero digits.
3 Significant Figures
401

25. Significant Figures
A zero is significant when it is between nonzero digits.
5 Significant Figures 9 3 . 6

26. Significant Figures
A zero is significant when it is between nonzero digits.
3 Significant Figures 9 . 3

27. Significant Figures
A zero is significant at the end of a number that includes a decimal point.
5 Significant Figures 5 5 .

28. Significant Figures
A zero is significant at the end of a number that includes a decimal point.
5 Significant Figures 2 . 1 9 3

29. Significant Figures
A zero is not significant when it is before the first nonzero digit.
1 Significant Figure . 6

30. Significant Figures
A zero is not significant when it is before the first nonzero digit.
3 Significant Figures . 7 9

31. Significant Figures
A zero is not significant when it is at the end of a number without a decimal point.
1 Significant Figure 5

32. Significant Figures
A zero is not significant when it is at the end of a number without a decimal point.
4 Significant Figures 6 8 7 1

33. Rounding Off Numbers

34. 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.

35. Rules for Rounding Off
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
80.873

36. Rounding Off Numbers
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

37. Rounding Off Numbers
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.
5 or greater
drop these figures
increase by 1 6

38. 2.4 Significant Figures in Calculations

39. The results of a calculation based on measurements cannot be more precise than the least precise measurement.

40. Multiplication or Division

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. The correct answer is 440 or 4.4 x 102
2.3 has two significant figures.
(190.6)(2.3) =
190.6 has four significant figures.
Answer given by calculator.
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.
438.38
The correct answer is 440 or 4.4 x 102

43. Addition or Subtraction

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

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

46. Round off to the nearest unit.
Add , 129 and 52.2
Least precise number.
125.17
129.
52.2
Answer given by calculator.
306.37
Correct answer.
Round off to the nearest unit.
306.37

47. 0.018286814 Answer given by calculator. Two significant figures.
Drop these 6 digits.
Correct answer.
The answer should have two significant figures because is the number with the fewest significant figures.

48. 2.5 The Metric System

49. 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.

50. 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

51. Common Prefixes and Numerical Values for SI Units
Power of 10
Prefix Symbol Numerical Value Equivalent
giga G 1,000,000,
mega M 1,000,
kilo k 1,
hecto h
deca da
— — 1 100

52. Prefixes and Numerical Values for SI Units
Power of 10
Prefix Symbol Numerical Value Equivalent
deci d
centi c
milli m
micro 
nano n
pico p
femto f

53. Measurement of Length

54. The standard unit of length in the SI system is the meter
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.

55. 1 meter = inches
1 meter is a little longer than a yard

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 m 10-3 m
micrometer m m 10-6 m
nanometer nm m 10-9 m
angstrom Å m m

57. 2.6 Problem Solving

58. unit1 x conversion factor = unit2
Dimensional Analysis
Dimensional analysis converts one unit to another by using conversion factors.
unit1 x conversion factor = unit2

59. Basic Steps
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. Basic Steps
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.
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.

61. Basic Steps Proceed with the necessary mathematical operations.
Make certain that your answer contains the proper number of significant figures.
Check the answer to make sure it is reasonable.

62. Length Conversion

63. How many millimeters are there in 2.5 meters?
The conversion factor must accomplish two things:
unit1 x conversion factor = unit2
m x conversion factor = mm
It must cancel meters.
It must introduce millimeters

64. The conversion factor takes a fractional form.

65. The conversion factor is derived from the equality. 1 m = 1000 mm
Divide both sides by 1000 mm
conversion factor
Divide both sides by 1 m
conversion factor

66. How many millimeters are there in 2.5 meters?
Use the conversion factor with millimeters in the numerator and meters in the denominator.

67. Convert 16.0 inches to centimeters.
Use this conversion factor

68. Convert 16.0 inches to centimeters.

69. Convert 3.7 x 103 cm to micrometers.
Centimeters can be converted to micrometers by writing down conversion factors in succession.
cm  m  meters

70. Convert 3.7 x 103 cm to micrometers.
Centimeters can be converted to micrometers by a series of two conversion factors.
cm  m  meters

71. 2.7 Measuring Mass and Volume

72. Mass

73. The standard unit of mass in the SI system is the kilogram
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 = 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 g 10-3 g
microgram g g 10-6 g

75. Convert 45 decigrams to grams.
1 g = 10 dg

76. An atom of hydrogen weighs 1. 674 x 10-24 g
An atom of hydrogen weighs x g. How many ounces does the atom weigh?
Grams can be converted to ounces using a series of two conversion factors.
1 lb = 454 g
16 oz = 1 lb

77. An atom of hydrogen weighs 1. 674 x 10-24 g
An atom of hydrogen weighs x g. How many ounces does the atom weigh?
Grams can be converted to ounces using a single linear expression by writing down conversion factors in succession.

78. Volume

79. Volume is the amount of space occupied by matter.
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.

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

82. Convert 4.61 x 102 microliters to milliliters.
Microliters can be converted to milliliters using a linear expression by writing down conversion factors in succession.
L  L  mL

83. 2.8 Measurement of Temperature

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.

87. degrees Fahrenheit = oF
Degree Symbols
degrees Celsius = oC
Kelvin (absolute) = K
degrees Fahrenheit = oF

88. To convert between the scales, use the following relationships:

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?

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?

92. 2.9 Density

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

94. The density of gases is expressed in grams per liter.
Mass is usually expressed in grams and volume in mL or cm3.

95. Density varies with temperature

98. Examples

99. A 13. 5 mL sample of an unknown liquid has a mass of 12. 4 g
A 13.5 mL sample of an unknown liquid has a mass of 12.4 g. What is the density of the liquid?

100. A graduated cylinder is filled to the 35. 0 mL mark with water
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
46.0 mL
98.1 g

101. The density of ether is 0. 714 g/mL. What is the mass of 25
The density of ether is g/mL. What is the mass of 25.0 milliliters of ether?
Method 1
(a) Solve the density equation for mass.
(b) Substitute the data and calculate.

102. The density of ether is 0. 714 g/mL. What is the mass of 25
The density of ether is g/mL. What is the mass of 25.0 milliliters of ether?
Method 2 Dimensional Analysis. Use density as a conversion factor. Convert:
mL → g
The conversion of units is

103. (b) Substitute the data and calculate.
The density of oxygen at 0oC is g/L. What is the volume of grams of oxygen at this temperature?
Method 1
(a) Solve the density equation for volume.
(b) Substitute the data and calculate.

104. The conversion of units is
The density of oxygen at 0oC is g/L. What is the volume of grams of oxygen at this temperature?
Method 2 Dimensional Analysis. Use density as a conversion factor. Convert:
g → L
The conversion of units is

105. The End