1. Chapter 3 Scientific Measurement 3.1 Using and Expressing Measurements
3.2 Units of Measurement
3.3 Solving Conversion Problems
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2. How do you write numbers in scientific notation?
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3. A measurement is a quantity that has both a number and a unit.
Scientific Notation
A measurement is a quantity that has both a number and a unit.
Your height (66 inches), your age (15 years), and your body temperature (37°C) are examples of measurements.
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4. Scientific Notation
In chemistry, you will often encounter very large or very small numbers.
A single gram of hydrogen, for example, contains approximately 602,000,000,000,000,000,000,000 hydrogen atoms.
You can work more easily with very large or very small numbers by writing them in scientific notation.
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5. Scientific Notation
In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power.
For example, the number 602,000,000,000,000,000,000,000 can be written in scientific notation as 6.02 x 1023.
The coefficient in this number is The power of 10, or exponent, is 23.
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6. Scientific Notation
In scientific notation, the coefficient is always a number greater than or equal to one and less than ten. The exponent is an integer.
A positive exponent indicates how many times the coefficient must be multiplied by 10.
A negative exponent indicates how many times the coefficient must be divided by 10.
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7. Scientific Notation
When writing numbers greater than ten in scientific notation, the exponent is positive and equals the number of places that the original decimal point has been moved to the left.
6,300,000. = 6.3 x 106
94,700. = 9.47 x 104
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8. Scientific Notation
Numbers less than one have a negative exponent when written in scientific notation. The value of the exponent equals the number of places the decimal has been moved to the right.
= 8 x 10–6
= 7.36 x 10–3
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9. Multiplication and Division
Scientific Notation
Multiplication and Division
To multiply numbers written in scientific notation, multiply the coefficients and add the exponents.
(3 x 104) x (2 x 102) = (3 x 2) x = 6 x 106
(2.1 x 103) x (4.0 x 10–7) = (2.1 x 4.0) x 103+(–7) = 8.4 x 10–4
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10. ( ) Multiplication and Division
Scientific Notation
Multiplication and Division
To divide numbers written in scientific notation, divide the coefficients and subtract the exponent in the denominator from the exponent in the numerator.
3.0 x ( ) =
x 105–2 =
0.5 x 103 = 5.0 x 102
6.0 x 102
6.0
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11. Addition and Subtraction
Scientific Notation
Addition and Subtraction
If you want to add or subtract numbers expressed in scientific notation and you are not using a calculator, then the exponents must be the same.
In other words, the decimal points must be aligned before you add or subtract the numbers.
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12. Addition and Subtraction
Scientific Notation
Addition and Subtraction
For example, when adding 5.4 x 103 and 8.0 x 102, first rewrite the second number so that the exponent is a 3. Then add the numbers.
(5.4 x 103) + (8.0 x 102)
= (5.4 x 103) + (0.80 x 103)
= ( ) x 103
= 6.2 x 103
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13. Using Scientific Notation
Sample Problem 3.1
Using Scientific Notation
Solve each problem and express the answer in scientific notation.
a. (8.0 x 10–2) x (7.0 x 10–5)
b. (7.1 x 10–2) + (5 x 10–3)
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14. Analyze Identify the relevant concepts.
Sample Problem 3.1
Analyze Identify the relevant concepts.
To multiply numbers in scientific notation, multiply the coefficients and add the exponents. To add numbers in scientific notation, the exponents must match. If they do not, then adjust the notation of one of the numbers. 1
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15. Solve Apply the concepts to this problem.
Sample Problem 3.1
Solve Apply the concepts to this problem.
Multiply the coefficients and add the exponents. a. 2
(8.0 x 10–2) x (7.0 x 10–5)
= (8.0 x 7.0) x 10–2 + (–5)
= 56 x 10–7
= 5.6 x 10–6
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16. Solve Apply the concepts to this problem.
Sample Problem 3.1
Solve Apply the concepts to this problem.
Rewrite one of the numbers so that the exponents match. Then add the coefficients. b. 2
(7.1 x 10–2) + (5 x 10–3)
= (7.1 x 10–2) + (0.5 x 10–2)
= ( ) x 10–2
= 7.6 x 10–2
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17. The mass of one molecule of water written in scientific notation is 2
The mass of one molecule of water written in scientific notation is 2.99 x 10–23 g. What is the mass in standard notation?
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18. The mass of one molecule of water written in scientific notation is 2
The mass of one molecule of water written in scientific notation is 2.99 x 10–23 g. What is the mass in standard notation?
The mass of one molecule of water in standard notation is gram.
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19. Accuracy, Precision, and Error
How do you evaluate accuracy and precision?
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20. Accuracy, Precision, and Error
Accuracy and Precision
In chemistry, the meanings of accuracy and precision are quite different.
Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured.
Precision is a measure of how close a series of measurements are to one another, irrespective of the actual value.
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21. Accuracy, Precision, and Error
Accuracy and Precision
To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.
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22. Accuracy, Precision, and Error
Accuracy and Precision
Darts on a dartboard illustrate the difference between accuracy and precision.
Good Accuracy, Good Precision
Poor Accuracy, Good Precision
Poor Accuracy, Poor Precision
The closeness of a dart to the bull’s-eye corresponds to the degree of accuracy. The closeness of several darts to one another corresponds to the degree of precision.
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23. Accuracy, Precision, and Error
Determining Error
Suppose you use a thermometer to measure the boiling point of pure water at standard pressure.
The thermometer reads 99.1°C.
You probably know that the true or accepted value of the boiling point of pure water at these conditions is actually
100.0°C.
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24. Accuracy, Precision, and Error
Determining Error
There is a difference between the accepted value, which is the correct value for the measurement based on reliable references, and the experimental value, the value measured in the lab.
The difference between the experimental value and the accepted value is called the error.
Error = experimental value – accepted value
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25. Accuracy, Precision, and Error
Determining Error
Error can be positive or negative, depending on whether the experimental value is greater than or less than the accepted value.
For the boiling-point measurement, the error is 99.1°C – 100°C, or –0.9°C.
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26. Are precise measurements always accurate?
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27. Are precise measurements always accurate?
No, measurements are precise if they are easily reproducible, but not accurate if they do not reflect the accepted value.
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28. Chapter 3 Scientific Measurement 3.2 Units of Measurement
3.1 Using and Expressing
Measurements
3.2 Units of Measurement
3.3 Solving Conversion Problems
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29. Using SI Units What makes metric units easy to use? Using SI Units
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30. Using SI Units
The standards of measurement used in science are those of the metric system.
All metric units are based on multiples of 10. As a result, you can convert between units easily.
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31. The metric system was originally established in France in 1795.
Using SI Units
The metric system was originally established in France in 1795.
The International System of Units (abbreviated SI after the French name, Le Système International d’Unités) is a revised version of the metric system.
The SI was adopted by international agreement in 1960.
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32. There are seven SI base units.
Using SI Units
There are seven SI base units.
From these base units, all other SI units of measurement can be derived.
Derived units are used for measurements such as volume, density, and pressure.
SI Base Units
Quantity
SI base unit
Symbol
Length
meter m
Mass
kilogram kg
Temperature
kelvin K
Time
second s
Amount of substance
mole
mol
Luminous intensity
candela cd
Electric current
ampere A
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33. In SI, the basic unit of length, or linear measure, is the meter (m).
Using SI Units
Units of Length
In SI, the basic unit of length, or linear measure, is the meter (m).
All measurements of length can be expressed in meters.
For very large and very small lengths, however, it may be more convenient to use a unit of length that has a prefix.
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34. Commonly Used Metric Prefixes
Using SI Units
Units of Length
The table below lists the prefixes in common use.
Commonly Used Metric Prefixes
Prefix
Symbol
Meaning
Factor
mega M
1 million times larger than the unit it precedes
106
kilo k
1000 times larger than the unit it precedes
103
deci d
10 times smaller than the unit it precedes
10-1
centi c
100 times smaller than the unit it precedes
10-2
milli m
1000 times smaller than the unit it precedes
10-3
micro μ
1 million times smaller than the unit it precedes
10-6
nano n
1 billion times smaller than the unit it precedes
10-9
pico p
1 trillion times smaller than the unit it precedes
10-12
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35. A hyphen (-) measures about 1 mm.
Using SI Units
Units of Length
For example, the prefix milli- means 1/1000 (one-thousandth), so a millimeter (mm) is 1/1000 of a meter, or m.
A hyphen (-) measures about 1 mm.
For large distances, it is most appropriate to express measurements in kilometers (km).
The prefix kilo- means 1000, so 1 km equals 1000 m.
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36. The space occupied by any sample of matter is called its volume.
Using SI Units
Units of Volume
The space occupied by any sample of matter is called its volume.
You calculate the volume of any cubic or rectangular solid by multiplying its length by its width by its height.
The unit for volume is thus derived from the units of length.
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37. This volume is a cubic meter (m3).
Using SI Units
Units of Volume
The SI unit of volume is the amount of space occupied by a cube that is 1 m along each edge.
This volume is a cubic meter (m3).
A more convenient unit of volume for everyday use is the liter, a non-SI unit.
A liter (L) is the volume of a cube that is 10 centimeters (10 cm) along each edge (10 cm x 10 cm x 10 cm = 1000 cm3 = 1 L).
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38. Using SI Units
Units of Volume
A smaller, non-SI unit of volume is the milliliter (mL); 1 mL is 1/1000 of a liter.
Thus, there are 1000 mL in 1 L.
Because 1 L is defined as 1000 cm3, 1 mL and 1 cm3 are the same volume.
The units milliliter and cubic centimeter are thus used interchangeably.
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39. Using SI Units
Units of Volume
There are many devices for measuring liquid volumes, including graduated cylinders, pipets, burets, volumetric flasks, and syringes.
The volume of substances will change with temperature, so accurate volume-measuring devices are calibrated at a given temperature—usually 20 degrees Celsius (20°C), which is about normal room temperature.
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40. Units of Volume Using SI Units
The relationships among common metric units of volume are shown in the table below.
Metric Units of Volume
Unit
Symbol
Relationship
Example
Liter L
base unit
quart of milk ≈ 1 L
Milliliter mL
103 mL = 1 L
20 drops of water ≈ 1 mL
Cubic centimeter
cm3
1 cm3 = 1 mL
cube of sugar ≈ 1 cm3
Microliter μL
103 μL = 1 L
crystal of table salt ≈ 1 μL
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41. Units of Mass Using SI Units
The mass of an object is measured in comparison to a standard mass of 1 kilogram (kg), which is the basic SI unit of mass.
A kilogram was originally defined as the mass of 1 L of liquid water at 4°C.
A cube of water at 4°C measuring 10 cm on each edge would have a volume of 1 L and a mass of 1000 grams (g), or 1 kg.
A gram (g) is 1/1000 of a kilogram; the mass of 1 cm3 of water at 4°C is 1 g.
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42. Units of Mass Using SI Units
The relationships among units of mass are shown in the table below.
Metric Units of Mass
Unit
Symbol
Relationship
Example
Kilogram (base unit) kg
1 kg = 103 g
small textbook ≈ 1 kg
Gram g
1 g = 10-3 kg
dollar bill ≈ 1 g
Milligram mg
103 mg = 1 g
ten grains of salt ≈ 1 mg
Microgram μg
106 μg = 1 g
particle of baking powder ≈ 1 μg
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43. Weight is a force that measures the pull on a given mass by gravity.
Using SI Units
Units of Mass
Weight is a force that measures the pull on a given mass by gravity.
Weight, a measure of force, is different from mass, which is a measure of the quantity of matter.
The weight of an object can change with its location.
An astronaut in orbit is weightless, but not massless.
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44. What is the SI unit of volume?
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45. What is the SI unit of volume?
The SI unit of volume is the cubic meter (m3). A liter (L) can be converted to the SI unit of volume, because a liter is defined as 1000 cm3.
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46. Temperature Scales What temperature units do scientists commonly use?
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47. Temperature is a measure of how hot or cold an object is.
Temperature Scales
Temperature is a measure of how hot or cold an object is.
An object’s temperature determines the direction of heat transfer.
When two objects at different temperatures are in contact, heat moves from the object at the higher temperature to the object at the lower temperature.
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48. A very important exception is water.
Temperature Scales
Almost all substances expand with an increase in temperature and contract as the temperature decreases.
A very important exception is water.
These properties are the basis for the common bulb thermometer.
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49. Temperature Scales
The liquid in a thermometer expands and contracts more than the volume of the glass, producing changes in the column height of liquid.
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50. Temperature Scales
Scientists commonly use two equivalent units of temperature, the degree Celsius and the kelvin.
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51. Temperature Scales
The Celsius scale sets the freezing point of water at 0°C and the boiling point of water at 100°C.
The distance between these two fixed points is divided into 100 equal intervals, or degrees Celsius (°C).
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52. This scale is named for Lord Kelvin (1824–1907), a Scottish physicist.
Temperature Scales
Another temperature scale used in the physical sciences is the Kelvin, or absolute, scale.
This scale is named for Lord Kelvin (1824–1907), a Scottish physicist.
On the Kelvin scale, the freezing point of water is kelvins (K), and the boiling point is (K).
Note that with the Kelvin scale, the degree sign is not used.
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53. The figure below compares the Celsius and Kelvin scales.
Temperature Scales
The figure below compares the Celsius and Kelvin scales.
The zero point on the Kelvin scale, 0 K, or absolute zero, is equal to –273.15°C.
Celsius
Kelvin
100
divisions
100°C
Boiling point
of water K
0°C
Freezing point K
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54. You simply add or subtract 273, as shown in the following equations.
Temperature Scales
Because one degree on the Celsius scale is equivalent to one kelvin on the Kelvin scale, converting from one temperature to another is easy.
You simply add or subtract 273, as shown in the following equations.
K = °C + 273
°C = K – 273
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55. Converting Between Temperature Scales
Sample Problem 3.7
Converting Between Temperature Scales
Normal human body temperature is 37°C. What is this temperature in kelvins?
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56. Is the change of one degree on the Celsius scale equivalent to one kelvin on the Kelvin scale?
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57. Is the change of one degree on the Celsius scale equivalent to one kelvin on the Kelvin scale?
Yes, a change of one degree on the Celsius scale is equivalent to a change of one kelvin on the Kelvin scale.
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58. Density What determines the density of a substance? Density
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59. This relationship is called density.
The relationship between an object’s mass and its volume tells you whether it will float or sink.
This relationship is called density.
Density is the ratio of the mass of an object to its volume.
mass
volume
Density =
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60. The SI unit of density is kilograms per cubic meter (kg/m3).
When mass is measured in grams, and volume in cubic centimeters, density has units of grams per cubic centimeter (g/cm3).
The SI unit of density is kilograms per cubic meter (kg/m3).
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61. Density
This figure compares the density of four substances: lithium, water, aluminum, and lead.
Increasing density (mass per unit volume)
10 g
0.53 g/cm3
19 cm3
10 cm3
3.7 cm3
0.88 cm3
1.0 g/cm3
2.7 g/cm3
0.88 g/cm3
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62. Density
Density is an intensive property that depends only on the composition of a substance, not the size of the sample.
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63. Because of differences in density, liquids separate into layers.
As shown at right, corn oil floats on top of water because it is less dense.
Corn syrup sinks below water because it is more dense.
Corn oil
Water
Corn syrup
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64. Densities of Some Common Materials
Interpret Data
Densities of Some Common Materials
Solids and Liquids
Gases
Material
Density at 20°C (g/cm3)
Density at 20°C (g/L)
Gold
19.3
Chlorine
2.95
Mercury
13.6
Carbon dioxide
1.83
Lead
11.3
Argon
1.66
Aluminum
2.70
Oxygen
1.33
Table sugar
1.59
Air
1.20
Corn syrup
1.35–1.38
Nitrogen
1.17
Water (4°C)
1.000
Neon
0.84
Corn oil
0.922
Ammonia
0.718
Ice (0°C)
0.917
Methane
0.665
Ethanol
0.789
Helium
0.166
Gasoline
0.66–0.69
Hydrogen
0.084
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65. Water is an important exception.
Density
What happens to the density of a substance as its temperature increases?
The volume of most substances increases as the temperature increases, while the mass remains the same.
Since density is the ratio of an object’s mass to its volume, the density of a substance generally decreases as its temperature increases.
Water is an important exception.
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66. Sample Problem 3.8
Calculating Density
A copper penny has a mass of 3.1 g and a volume of 0.35 cm3. What is the density of copper?
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67. Can you assume that something with a low weight will float in water?
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68. Can you assume that something with a low weight will float in water?
No, it is the relationship between an object’s mass and its volume, its density, that tells you whether it will float or sink.
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69. Key Concepts
All metric units are based on multiples of 10. As a result, you can convert between units easily.
Scientists commonly use two equivalent units of temperature, the degree Celsius and the kelvin.
Density is an intensive property that depends only on the composition of a substance.
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70. K = °C + 273 °C = K – 273 Key Equations mass Density = volume
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71. Chapter 3 Scientific Measurement 3.3 Solving Conversion Problems
3.1 Using and Expressing
Measurements
3.2 Units of Measurement
3.3 Solving Conversion
Problems
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72. Conversion Factors
Conversion Factors
What happens when a measurement is multiplied by a conversion factor?
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73. 1 dollar = 4 quarters = 10 dimes = 20 nickels = 100 pennies
Conversion Factors
If you think about any number of everyday situations, you will realize that a quantity can usually be expressed in several different ways.
For example:
1 dollar = 4 quarters = 10 dimes = 20 nickels = 100 pennies
These are all expressions, or measurements, of the same amount of money.
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74. The same thing is true of scientific quantities.
Conversion Factors
The same thing is true of scientific quantities.
For example:
1 meter = 10 decimeters = 100 centimeters = 1000 millimeters
These are different ways to express the same length.
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75. Conversion Factors
Whenever two measurements are equivalent, a ratio of the two measurements will equal 1, or unity.
For example, you can divide both sides of the equation 1 m = 100 cm by 1 m or by 100 cm.
1 m =
100 cm 1 or
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76. A conversion factor is a ratio of equivalent measurements.
Conversion Factors
The ratios 100 cm/1 m and 1 m/100 cm are examples of conversion factors.
A conversion factor is a ratio of equivalent measurements.
conversion factors
1 m =
100 cm 1 or
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77. Conversion Factors
The measurement in the numerator (on the top) is equivalent to the measurement in the denominator (on the bottom).
The conversion factors shown below are read “one hundred centimeters per meter” and “one meter per hundred centimeters.”
conversion factors
1 m =
100 cm 1 or
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78. Conversion Factors
Conversion factors are useful in solving problems in which a given measurement must be expressed in some other unit of measure.
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79. Conversion Factors
When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same.
For example, even though the numbers in the measurements 1 g and 10 dg (decigrams) differ, both measurements represent the same mass.
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80. Conversion Factors
In addition, conversion factors within a system of measurement are defined quantities or exact quantities.
Therefore, they have an unlimited number of significant figures and do not affect the rounding of a calculated answer.
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81. The relationship between grams and kilograms is 1000 g = 1 kg.
Conversion Factors
Here are some additional examples of pairs of conversion factors written from equivalent measurements.
The relationship between grams and kilograms is 1000 g = 1 kg.
The conversion factors are
1000 g
1 kg
and
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82. The possible conversion factors are
The relationship between nanometers and meters is given by the equation 109 nm = 1 m.
The possible conversion factors are
109 nm
1 m
and
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83. The relationship 1 L = 106μL yields the following conversion factors:
Common volumetric units used in chemistry include the liter and the microliter.
The relationship 1 L = 106μL yields the following conversion factors:
1 L
106 μL
and
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84. What is the relationship between the two components of a conversion factor?
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85. What is the relationship between the two components of a conversion factor?
The two components of a conversion factor are equivalent measurements with different units. They are two ways of expressing the same quantity.
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86. Dimensional Analysis
Dimensional Analysis
What kinds of problems can you solve using dimensional analysis?
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87. Dimensional Analysis
Many problems in chemistry are conveniently solved using dimensional analysis, rather than algebra.
Dimensional analysis is a way to analyze and solve problems using the units, or dimensions, of the measurements.
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88. Using Dimensional Analysis
Sample Problem 3.9
Using Dimensional Analysis
How many seconds are in a workday that lasts exactly eight hours?
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89. Using Dimensional Analysis
Sample Problem 3.10
Using Dimensional Analysis
The directions for an experiment ask each student to measure 1.84 g of copper (Cu) wire. The only copper wire available is a spool with a mass of 50.0 g. How many students can do the experiment before the copper runs out?
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90. Simple Unit Conversions
Dimensional Analysis
Simple Unit Conversions
In chemistry, as in everyday life, you often need to express a measurement in a unit different from the one given or measured initially.
Dimensional analysis is a powerful tool for solving conversion problems in which a measurement with one unit is changed to an equivalent measurement with another unit.
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91. Using Density as a Conversion Factor
Sample Problem 3.12
Using Density as a Conversion Factor
What is the volume of a pure silver coin that has a mass of 14 g? The density of silver (Ag) is 10.5 g/cm3.
Density can be used to write two conversion factors. To figure out which one you need, consider the units of your given quantity and the units needed in your answer.
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92. Dimensional Analysis
Multistep Problems
Many complex tasks in your life are best handled by breaking them down into smaller, manageable parts.
Similarly, many complex word problems are more easily solved by breaking the solution down into steps.
When converting between units, it is often necessary to use more than one conversion factor.
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93. What kind of problems can you solve using dimensional analysis?
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94. What kind of problems can you solve using dimensional analysis?
Problems that require the conversion of a measurement from one unit to another can be solved using dimensional analysis.
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95. Key Concepts
When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same.
Dimensional analysis is a powerful tool for solving conversion problems in which a measurement with one unit is changed to an equivalent measurement with another unit.
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96. BIG IDEA
Dimensional analysis is a problem-solving method that involves analyzing the units of the given measurement and the unknown to plan a solution.
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