1. Measurement and Calculations
CHAPTER #2
Measurement and Calculations

2. Measurement A measurement is a quantitative observation.
Measurements have 2 parts a number and unit.
Number is a comparison known found on a measuring device.
Unit tells the type of measurement and a scale.

3. Measurment
The number comes from a measuring device, such as a ruler, clock, or speedometer, to name a few examples of measuring devices. The unit is a word or abbreviated word describing the measurement and the scale used.
All measuring devices contain a scale. Scales contain space between the lines. The last number of a measurement, called a significant figure, is a guess as to the number between the lines.

4. Since measurements contain a guess, they cannot be exact.

5. Since measurements contain a guess, they cannot be exact.
11.64 cm

6. Measurements
11.64 cm
Since the last number is a guess most observers would agree between cm. This being the case is usually expressed as ±0.01 cm

7. Measurements
When we make a measurement the last recorded number is always an estimate due to reading between the lines. If the object being measured appears to be on the line, then a zero is used to describe the fact that the object is on the line.

8. Measurements
This means that the last recorded number will usually vary depending on who is estimating the last number. This produces uncertainty, or error in the measurement.

9. Significant Figure Definition
Significant figures are the number of numbers read from a measuring device.

10. What Are Numbers?
Numbers are any integers from 1- ∞, and sometimes zero.
Zero serves two purposes, it is used as a decimal place holder, a number, or both. How do we determine if a zero is a number or a position holder when determining the number of significant figures for a measurement?

11. The Zero Test
To determine if a zero is a number or a decimal spacer, consider dropping one or more of the zero digits. If dropping a zero changes the value of the measurement, then the zero is a decimal position holder and is not considered to be a number and therefore cannot be counted as a significant figure.

12. For Example
Consider the measurement 100 cm. Dropping the last two zeros changes the value to 1, so the zeros are position holders and not numbers. Since significant figures are numbers by definition, then they are not counted in the significant figure count, thus 100 cm has only one significant figure.
Now consider the measurement cm. If the last zero is dropped the value of the measurement remains the same. Here the last zero does not space the decimal in this measurement. Since zeros are either decimal position holders, or numbers, then the zero in this case must be a number and counted in the significant figure count since is not a decimal spacer.

13. Sandwiched Zeros
What about the zeros in the center of the measurement of cm? Since the last zero is a number and the one at the beginning is a number then the center zeros are sandwiched by two numbers. Sandwiched zeros are always counted as significant figures, thus giving cm four significant figures

14. Zeros Both Numbers and Spacers?
For a zero to be counted as a spacer and a number additional information must be given:
Common sense to be gained in the laboratory
A measuring device so states that they are significant.

15. Examples
Consider the following list of measurements and determine how many significant figures each measurement contains.
Measurements
SigFigs
10 cm
10.0 cm
101 cm
101.0 cm
1.00 X 10-3 cm

16. Examples
Consider the following list of measurements and determine how many significant figures each measurement contains.
Measurements
SigFigs
Reason
10 cm
10.0 cm
101 cm
101.0 cm
1.00 X 10-3 cm 1
Zero is a spacer for sure. Additional information required to see if it is a number

17. Examples
Consider the following list of measurements and determine how many significant figures each measurement contains.
Measurements
SigFigs
Reason
10 cm
10.0 cm
101 cm
101.0 cm
1.00 X 10-3 cm 1
Zero is a spacer for sure. Additional information required to see if it is a number 3
The last number is not a spacer, since dropping it the value is unchanged. The other zero is sandwiched.

18. Examples
Consider the following list of measurements and determine how many significant figures each measurement contains.
Measurements
SigFigs
Reason
10 cm
10.0 cm
101 cm
101.0 cm
1.00 X 10-3 cm 1
Zero is a spacer for sure. Additional information required to see if it is a number 3 3
Zero is sandwiched here

19. Examples
Consider the following list of measurements and determine how many significant figures each measurement contains.
Measurements
SigFigs
Reason
10 cm
10.0 cm
101 cm
101.0 cm
1.00 X 10-3 cm 1
Zero is a spacer for sure. Additional information required to see if it is a number 3 3
Zero is sandwiched here 4
Zero is not a spacer. The other zero is sandwiched.

20. Examples
Consider the following list of measurements and determine how many significant figures each measurement contains.
Measurements
SigFigs
Reason
10 cm
10.0 cm
101 cm
101.0 cm
1.00 X 10-3 cm 1
Zero is a spacer for sure. Additional information required to see if it is a number 3 3
Zero is sandwiched here 4
The last zero is not a spacer . The other zero is sandwiched. 3
Only look at the coefficient, these zeros are not spacers

21. Example
How can we express 100 cm to three significant figures?

22. Example
How can we express 100 cm to three significant figures? Use Scientific Notation!

23. Scientific Notation
A way to abbreviate large or small numbers

24. Scientific Notation Examples
Convert the following into scientific notation.
454,000 mi

25. Scientific Notation Examples
Convert the following into scientific notation.
454,000 mi
Step 1, place a decimal to the right of the first non-zero number.

26. Scientific Notation Examples
Convert the following into scientific notation.
454,000 mi
Step 1, place a decimal to the right of the first non-zero number.
4.54

27. Scientific Notation Examples
Convert the following into scientific notation.
454,000 mi
Step 1, place a decimal to the right of the first non-zero number.
Step 2, place X 10 after the number.
4.54

28. Scientific Notation Examples
Convert the following into scientific notation.
454,000 mi
Step 1, place a decimal to the right of the first non-zero number.
Step 2, place X 10 after the number.
4.54 X 10

29. Scientific Notation Examples
Convert the following into scientific notation.
454,000 mi
Step 1, place a decimal to the right of the first non-zero number.
Step 2, place X 10 after the number.
Step 3, count from the old decimal location to the new decimal location
4.54 X 10

30. Scientific Notation Examples
Convert the following into scientific notation.
454,000 mi
Step 1, place a decimal to the right of the first non-zero number.
Step 2, place X 10 after the number.
Step 3, count from the old decimal location to the new decimal location, this number of places becomes the power of 10.
4.54 X 10

31. Scientific Notation Examples
Convert the following into scientific notation.
454,000 mi
Step 1, place a decimal to the right of the first non-zero number.
Step 2, place X 10 after the number.
Step 3, count from the old decimal location to the new decimal location, this number of places becomes the power of 10.
4.54 X 105 mi

32. Scientific Notation Examples
Convert the following into scientific notation.
454,000 mi
Step 1, place a decimal to the right of the first non-zero number.
Step 2, place X 10 after the number.
Step 3, count from the old decimal location to the new decimal location, this number of places becomes the power of 10.
Note: Be sure that the answer contains the same number of significant figures as the starting measurement
4.54 X 105 mi

33. Scientific Notation Examples
Convert the following into scientific notation.
b mi
Step 1, place a decimal to the right of the first non-zero number.

34. Scientific Notation Examples
Convert the following into scientific notation.
b mi
Step 1, place a decimal to the right of the first non-zero number.
Note: Be sure that the answer contains the same number of significant figures as the starting measurement
2.83 mi

35. Scientific Notation Examples
Convert the following into scientific notation.
b mi
Step 1, place a decimal to the right of the first non-zero number.
Step 2, place X 10 after the number.
Note: Be sure that the answer contains the same number of significant figures as the starting measurement
2.83 X 10 mi

36. Scientific Notation Examples
Convert the following into scientific notation.
b mi
Step 1, place a decimal to the right of the first non-zero number.
Step 2, place X 10 after the number.
Step 3, count from the old decimal location to the new decimal location, this number of places becomes the power of 10, unless the number is less than one, if so, then negative power
Note: Be sure that the answer contains the same number of significant figures as the starting measurement
2.83 X 10-3 mi

37. Example
How can we express 100 cm to three significant figures? Use Scientific Notation!

38. Example
How can we express 100 cm to three significant figures? Use Scientific Notation!
1.00 X 102

39. Review
If there are 37 students in this room, then how many significant figures are there?

40. Review
If there are 37 students in this room, then how many significant figures are there? None

41. Review
If there are 37 students in this room, then how many significant figures are there? None
How many significant figures are there in 100 mL?

42. Review
If there are 37 students in this room, then how many significant figures are there? None
How many significant figures are there in 100 mL? None

43. Review
If there are 37 students in this room, then how many significant figures are there? None
How many significant figures are there in 100 mL? None
There are 5280 ft in a mile; how many significant figures?

44. Review
If there are 37 students in this room, then how many significant figures are there? None
How many significant figures are there in 100 mL? None
There are 5280 ft in a mile; how many significant figures? None

45. Review
If there are 37 students in this room, then how many significant figures are there? None
How many significant figures are there in 100 mL? None
There are 5280 ft in a mile; how many significant figures? None

46. QUALITY OF MEASUREMNTS
Accuracy-How close a measurement is to the true value.
Precision-How close multiple measurements of the same objects are to each other.

47. Examples of Accuracy and Precision

48. ROUNDING
When measurements are combined to provide information, can the resultant information be of a higher quality than the measurements?

49. ROUNDING
When measurements are combined to provide information, can the information be of a higher quality than the measurements? No, information provide by combining measurements cannot have an accuracy, or precision greater than the least precise measurement that provided the information.

50. Why Round After a Calculation
Since information provided by combining measurements cannot have a higher quality than the measurements providing the information, then answers to problems must be rounded to give the same quality as the measurement with the least quality.
Rounding rules are designed to give answers the desired quality. They are posted on the course web site and restated on the following slide.

51. ROUNDING RULES
Rounding is the process of providing results that have the same quality as measurements with the least quality. Since there are different mathematical methods of combining measurements, then different rounding rules are required to provide sensible results of measurement combinations.

52. Addition and Subtraction
Round the calculated answer so that it contains the same number of decimal places as the measurement with the least number of decimal places.

53. Multiplication and Division
Round the calculated answer so that it contains the same number of significant figures as the measurement with the least number of significant figures. In other words, if the measurement with the least number of significant figures contains two significant figures, then the rounded answer should contain two significant figures.

54. Logarithms
Round the calculated answer so that it contains the same number of decimal places as the measurement with the least number of significant figures. In other words, if the measurement with the least number of significant figures contains two significant figures, then the rounded answer should contain two decimal places.

55. Anti-logarithms
Round answer so that the number of significant figures matches the number of decimal places as the measurement with the least number of decimal places. In other words, if the measured number contains three decimal places, then the answer should be rounded so that it contains three significant figures.

56. SI Unit System
In chemistry we use the international system of units. This is a modern version of the metric system.
Unfortunately this system of units is not widely used in everyday life in the USA.
Being able to use conversion factors and formulas to transform measurements between systems of units is extremely important.
This procedure is called unit analysis

57. The Fundamental SI Units
SI means system international, or international system of units
Physical Quantity
Name of Unit
Abbreviation
Mass
kilogram kg
Length
meter m
Time
second s
Temperature
kelvin K
Electric current
ampere A
Amount of substance
mole
mol
Copyright © Cengage Learning. All rights reserved

58. Basic Metric Units
Some of the common units for measurements and their abbreviations are shown below.
Measurement
Units
Abbreviation
Mass
grams g
Volume
liters L
Distance
meters m
Time
seconds s
A much more extensive table is given on page 24 of the text.

59. Common Metric Unit Prefixes
In chemistry we are often dealing with very large or very small quantities. To help with this a system of prefix modifiers have been developed. Please Memorize this list. (Note a more extensive list is on page 26 of your text)
Prefix
Abbreviation
Multiplier
mega M
(106)
kilo k
1000 (103)
deci d
0.1 (10-1)
centi c
0.01 (10-2)
milli m
0.001 (10-3)
micro μ
(10-6)

60. Application of Metric Prefixes
Length (m) Mass (g) Time (s)
103 m = km 103 g = kg 103 s = ks
10-2 m = cm 10-2 g = cg 10-2 s = cs
10-3 m = mm 10-3 g = mg 10-3 s = ms
10-6 m = µm 10-6 g = µg s= µs
Note: The memorized number always is in front of the single letter.

61. Unit Conversion Accidents
There have been many serious incidents that have resulted from errors in converting between systems of units.
Air Canada Flight 143
(Google it for more details)

62. Unit Conversion Accidents
$125 million Mars Climate Orbiter.
Lost in Space.
Dp you think there is the potential to make errors in the conversion of units for health care providers?

63. Conversion Problem Steps
Write down the number and unit.
Draw lines; a vertical line after the number an unit and horizontal line below the number and unit.
Insert a fractional fact to cancel out the original unit.
Compare the new unit to the asked for unit
a. If the same, you are done.
b. If not the same, repeat step 3.

64. Step 1. Write down the number and unit.
47.2 mg

65. Step 1. Write down the number and unit.
47.2 mg
Step 2. Draw lines
47.2 mg

66. Step 1. Write down the number and unit.
47.2 mg
Step 2. Draw lines
47.2 mg
Step 3. Insert fractional fact crossing out original unit

67. Step 1. Write down the number and unit.
47.2 mg
Step 2. Draw lines
47.2 mg
Step 3. Insert fractional fact crossing out original unit
47.2 mg
10-3 g mg

68. Step 1. Write down the number and unit.
47.2 mg
Step 2. Draw lines
47.2 mg
Step 3. Insert fractional fact crossing out original unit
47.2 mg
10-3 g mg
Step 4. Compare new unit to the asked for unit.

69. Step 1. Write down the number and unit.
47.2 mg
Step 2. Draw lines
47.2 mg
Step 3. Insert fractional fact crossing out original unit
47.2 mg
10-3 g mg
Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

70. Step 1. Write down the number and unit.
47.2 mg
Step 2. Draw lines
47.2 mg
Step 3. Insert fractional fact crossing out original unit
47.2 mg
10-3 g
= g mg
Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

71. Step 1. Write down the number and unit.
702 cL
Step 2. Draw lines
702 cL
Step 3. Insert fractional fact crossing out original unit
702 cL
10-2 L cL
Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

72. Step 1. Write down the number and unit.
Not a match repeat step #3
Step 1. Write down the number and unit.
702 cL
Step 2. Draw lines
702 cL
Step 3. Insert fractional fact crossing out original unit
702 cL
10-2 L cL
Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

73. It’s a match, done
Step 1. Write down the number and unit.
702 cL
Step 2. Draw lines
702 cL
Step 3. Insert fractional fact crossing out original unit μL
702 cL
10-2 L cL
10-6 L
Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

74. It’s a match, done
Step 1. Write down the number and unit.
702 cL
Step 2. Draw lines
702 cL
Step 3. Insert fractional fact crossing out original unit
702 cL
10-2 L μL
= 7.02 x 106 μL cL
10-6 L
Step 4. Compare new unit to the asked for unit. A. If the same you are done b. If not the same repeat step 3.

75. DENSITY What is heavier 5 pounds of lead or 5 pounds of feathers?
What takes up more space, 5 pounds of lead or 5 pounds of feathers?

76. DENSITY
What is heavier 5 pounds of lead or 5 pounds of feathers? Both the same. This is an old riddle to confuse density with weight
What takes up more space, 5 pounds of lead or 5 pounds of feathers?

77. DENSITY
What is heavier 5 pounds of lead or 5 pounds of feathers? Both the same. This is an old riddle to confuse density with weight
What takes up more space, 5 pounds of lead or 5 pounds of feathers? Feathers, since they are less dense.

78. g/mL, g/cm3, (for solids and liquids), or g/L for gases
DENSITY UNITS
g/mL, g/cm3, (for solids and liquids), or g/L for gases
Grams is a measure of mass using a laboratory balance, while mL, cm3 and liters are volume measurements.
Cm3 is found be measuring the length, width, and height of a regularly shaped object and multiplying these measurements together. The volume measurement of mL is found from a graduated cylinder shown on the next slide. Archimedes principle is used to determine the volume on an irregularly shaped object, also on the next slide.

79. Archimedes Principle
Archimedes was the scientist for king Hiero II of Sicily. King Hiero ordered a new crown and provided the goldsmith with several pounds of gold. When the crown was finished, King Hiero suspected the goldsmith of substituting some of the gold with silver and keeping the rest of the gold for himself. The King requested that Archimedes determine if his crown was pure gold or not. Since Archimedes knew the density of gold, the trick was to determine the density of the crown, if the same than the crown was pure gold. The challenge was how to find the volume of the royal crown, which is illustrated on the next slide.

80. Archimedes Principle
We can determine the volume of irregularly shaped objects by displacement.
How can we determine the volume of a gas?
Gases fill whatever container they are placed in. So it’s the volume of the container !

81. DENSITY PROBLEM SOLVING STRATEGY
Use the four step unit analysis. Organize the measurements to give density units.
Sample Problems
Calculate the density of a 4.07 g sample of rock that displaces 1.22 mL of water.
Calculate the density of a 4.22 g sample of wood that measures 2.0 cm by 1.33 cm by 3.56 cm.
Mercury has a density of 13.6 g/mL. Find the mass of 125 mL of mercury.
Water has a density of 1.00 g/mL. Find the volume, in liters, of a 3.22 kg sample of water.
What does an object do in water with
A density greater than water?
A density less than water?
A density equal to water?

82. English/Metric Conversions
Definitions
2.54 cm = in
946 ml = qt
454 g = lb
cm3 = mL
Please Remember
Definitions are not measurements and do not contain significant figures

83. Sample English/Metric Conversion Problems
Convert 708 pounds to kilograms.
Convert 50.0 liters to gallons.
Convert the density of water to pounds per gallon.
How many cubic meters are contained in 33 liters?
The density of aluminum is 2.70 g/mL. Find the thickness of aluminum foil that measures 2.0 cm by 5.66 cm that has a mass of 1.23 g.

84. The End