ChapterChapter Two TwoMeasurements in
Chemistry
Fundamentals of General, Organic, and Biological Chemistry
5th Edition
James E. MayhughJames E. MayhughOklahoma City UniversityOklahoma City University2007 Prentice Hall, Inc.2007 Prentice Hall, Inc.
OutlineOutline► 2.1 Physical Quantities► 2.2 Measuring Mass► 2.3 Measuring Length and Volume► 2.4 Measurement and Significant Figures► 2.5 Scientific Notation► 2.6 Rounding Off Numbers► 2.7 Converting a Quantity from One Unit to Another► 2.8 Problem Solving: Estimating Answers► 2.9 Measuring Temperature► 2.10 Energy and Heat► 2.11 Density► 2.12 Specific Gravity
Chapter Two 2
2.1 Physical Quantities2.1 Physical Quantities
Physical properties such as height, volume, and temperature that can be measured are called physical quantities. Both a number and a unit of defined size is required to describe physical quantity.
Chapter Two 3
► A number without a unit is meaningless.► To avoid confusion, scientists have agreed on a standard set
of units.► Scientists use SI or the closely related metric units.
Prentice Hall © 2007 Chapter Two 4
► Scientists work with both very large and very small numbers.
► Prefixes are applied to units to make saying and writing measurements much easier.
► The prefix pico (p) means “a trillionth of.” ► The radius of a lithium atom is 0.000000000152
meter (m). Try to say it.► The radius of a lithium atom is 152 picometers
(pm). Try to say it.
Chapter Two 5
Frequently used prefixes are shown below.Frequently used prefixes are shown below.
Chapter Two 6
2.2 Measuring Mass2.2 Measuring Mass
► Mass is a measure of the amount of matter in an object. Mass does not depend on location.
► Weight is a measure of the gravitational force acting on an object. Weight depends on location.
► A scale responds to weight.► At the same location, two objects with identical
masses have identical weights.► The mass of an object can be determined by
comparing the weight of the object to the weight of a reference standard of known mass.
Chapter Two 7
a) The single-pan balance with sliding counterweights. (b) A modern electronic balance.
Chapter Two 8
Relationships between metric units of mass and the mass units commonly used in the United States are shown below.
Chapter Two 9
2.3 Measuring Length and Volume2.3 Measuring Length and Volume
► The meter (m) is the standard measure of length or distance in both the SI and the metric system.
► Volume is the amount of space occupied by an object. A volume can be described as a length3.
► The SI unit for volume is the cubic meter (m3).
Chapter Two 10
Relationships between metric units of length and volume and the length and volume units commonly used in the United States are shown below and on the next slide.
Chapter Two 11
Chapter Two12
A m3 is the volume of a cube 1 m or 10 dm on edge. Each m3 contains (10 dm)3 = 1000 dm3 or liters. Each liter or dm3 = (10cm)3 =1000 cm3 or milliliters. Thus, there are 1000 mL in a liter and 1000 L in a m3.
The metric system is based on factors of 10 and is much easier to use than common U.S. units. Does anyone know how many teaspoons are in a gallon?
Chapter Two 13
14
Exact NumbersExact Numbers
An exact number is obtained when you count objects or use a defined relationship.
Counting objects are always exact2 soccer balls4 pizzas
Exact relationships, predefined values, not measured1 foot = 12 inches1 meter = 100 cm
For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches.
15
Learning Check Learning Check
A. Exact numbers are obtained by 1. using a measuring tool
2. counting3. definition
B. Measured numbers are obtained by 1. using a measuring tool
2. counting3. definition
16
SolutionSolution
A. Exact numbers are obtained by 2. counting
3. definition
B. Measured numbers are obtained by 1. using a measuring tool
17
Learning CheckLearning Check
Classify each of the following as an exact or ameasured number.
1 yard = 3 feet
The diameter of a red blood cell is 6 x 10-4 cm.
There are 6 hats on the shelf.
Gold melts at 1064°C.
18
Classify each of the following as an exact (1) or ameasured(2) number. This is a defined relationship.A measuring tool is used to determine length.The number of hats is obtained by counting.A measuring tool is required.
SolutionSolution
2.4 Measurement and Significant 2.4 Measurement and Significant FiguresFigures
► Every experimental measurement has a degree of uncertainty.
► The volume, V, at right is certain in the 10’s place, 10mL<V<20mL
► The 1’s digit is also certain, 17mL<V<18mL
► A best guess is needed for the tenths place.
Chapter Two 19
20
What is the Length?
1 2 3 4 cm
►We can see the markings between 1.6-1.7cm►We can’t see the markings between the .6-.7►We must guess between .6 & .7►We record 1.67 cm as our measurement►The last digit an 7 was our guess...stop there
Learning CheckLearning Check
What is the length of the wooden stick?1) 4.5 cm 2) 4.54 cm 3) 4.547 cm
22
8.00 cm or 3 (2.2/8)?
► To indicate the precision of a measurement, the value recorded should use all the digits known with certainty, plus one additional estimated digit that usually is considered uncertain by plus or minus 1.
► No further insignificant digits should be recorded.
► The total number of digits used to express such a measurement is called the number of significant figures.
► All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate.
Chapter Two 23
Chapter Two 24
Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.
► When reading a measured value, all nonzero digits should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not.
► RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, 94.072 g has five significant figures.
► RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four.
Chapter Two 25
► RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m.
► RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point.
Chapter Two 26
Practice Rule #1 Zeros
45.8736
.000239
.00023900
48000.
48000
3.982106
1.00040
6
3
5
5
2
4
6
•All digits count
•Leading 0’s don’t
•Trailing 0’s do
•0’s count in decimal form
•0’s don’t count w/o decimal
•All digits count
•0’s between digits count as well as trailing in decimal form
2.5 Scientific Notation2.5 Scientific Notation
► Scientific notation is a convenient way to write a very small or a very large number.
► Numbers are written as a product of a number between 1 and 10, times the number 10 raised to power.
► 215 is written in scientific notation as: 215 = 2.15 x 100 = 2.15 x (10 x 10) = 2.15 x 102
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Chapter Two 29
Two examples of converting standard notation to scientific notation are shown below.
Chapter Two 30
Two examples of converting scientific notation back to Two examples of converting scientific notation back to standard notation are shown below. standard notation are shown below.
► Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point.
► The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures.
► Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 108 indicates 2 and writing it as 1.500 x 108 indicates 4.
► Scientific notation can make doing arithmetic easier. Rules for doing arithmetic with numbers written in scientific notation are reviewed in Appendix A.
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2.6 Rounding Off Numbers2.6 Rounding Off Numbers
► Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified.
► How do you decide how many digits to keep?► Simple rules exist to tell you how.
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► Once you decide how many digits to retain, the rules for rounding off numbers are straightforward:
► RULE 1. If the first digit you remove is 4 or less, drop it and all following digits. 2.4271 becomes 2.4 when rounded off to two significant figures because the first dropped digit (a 2) is 4 or less.
► RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 is 4.6 when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater.
► If a calculation has several steps, it is best to round off at the end.
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Practice Rule #2 RoundingPractice Rule #2 Rounding
Make the following into a 3 Sig Fig numberMake the following into a 3 Sig Fig number
1.5587
.0037421
1367
128,522
1.6683 106
1.56
.00374
1370
129,000
1.67 106
Your Final number must be of the same value as the number you started with,129,000 and not 129
Examples of RoundingExamples of RoundingFor example you want a 4 Sig Fig number
4965.03
780,582
1999.5
0 is dropped, it is <5
8 is dropped, it is >5; Note you must include the 0’s
5 is dropped it is = 5; note you need a 4 Sig Fig
4965
780,600
2000.
RULE 1. RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.
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►RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers.
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Multiplication and divisionMultiplication and division
32.27 1.54 = 49.6958
3.68 .07925 = 46.4353312
1.750 .0342000 = 0.05985
3.2650106 4.858 = 1.586137 107
6.0221023 1.66110-24 = 1.000000
49.7
46.4
.05985
1.586 107
1.000
Addition/SubtractionAddition/Subtraction
25.5 32.72 320 +34.270 0.0049‑ + 12.5 59.770 32.7151 332.5
59.8 32.72 330
__ ___ __
Addition and SubtractionAddition and Subtraction
.56 + .153 = .713
82000 + 5.32 = 82005.32
10.0 - 9.8742 = .12580
10 – 9.8742 = .12580
.71
82000
.1
0
Look for the last important digit
Mixed Order of OperationMixed Order of Operation
8.52 + 4.1586 18.73 + 153.2 =
(8.52 + 4.1586) (18.73 + 153.2) =
239.6
2180.
= 8.52 + 77.89 + 153.2 = 239.61 =
= 12.68 171.9 = 2179.692 =
TryTry
i
i
n
xx
Find the standard deviation for the following numbers: 7.691 g, 7.23 g, 7.892 g
2
12
1
n
xx
s ii
TryTry
i
i
n
xx
7.691 g 7.23 g 7.892 g22.813 g
22.81 g = 7.603 g 3
7.691 g, 7.23 g, 7.892 g
TryTry
= 7.603 g
7.691 g – 7.603 g = .088 g 7.23 g – 7.603 g = -.37 g 7.892 g – 7.603 g = .289 g
.01 g
2
12
1
n
xx
s ii
TryTry
2
12
1
n
xx
s ii
.25
.01 = .01 g
007.
2
01.2/12
s
2.7 Problem Solving: Converting a 2.7 Problem Solving: Converting a Quantity from One Unit to AnotherQuantity from One Unit to Another
► Factor-Label Method: A quantity in one unit is converted to an equivalent quantity in a different unit by using a conversion factor that expresses the relationship between units.
Chapter Two 46
(Starting quantity) x (Conversion factor) = Equivalent quantity
Chapter Two 47
Writing 1 km = 0.6214 mi as a fraction restates it in the form of a conversion factor. This and all other conversion factors are numerically equal to 1.
The numerator is equal to the denominator. Multiplying by a conversion factor is equivalent to multiplying by 1 and so causes no change in value.
When solving a problem, the idea is to set up anequation so that all unwanted units cancel, leaving only the desired units.
Chapter Two 49
2.8 Problem Solving: Estimating 2.8 Problem Solving: Estimating AnswersAnswers
► STEP 1: Identify the information given.► STEP 2: Identify the information needed to answer.► STEP 3: Find the relationship(s) between the known
information and unknown answer, and plan a series of steps, including conversion factors, for getting from one to the other.
► STEP 4: Solve the problem.► BALLPARK CHECK: Make a rough estimate to be
sure the value and the units of your calculated answer are reasonable.
Chapter Two 51
54
How many minutes are in 1.4 days?Initial unit: 1.4 days
Unit plan: days hr min
Set up problem: 1.4 days x 24 hr x 60 min = 2.0 x 103 min
1 day 1 hr
Example: Problem SolvingExample: Problem Solving
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An adult human has 4650 mL of blood. How many gallons of blood is that?Unit plan: mL qt gallon
Equalities: 1 quart = 946 mL 1 gallon = 4 quarts
Learning Check Learning Check
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Unit plan: mL qt gallon
Setup: 4650 mL x 1 qt x 1 gal = 1.23 gal
946 mL 4 qt 3 SF 3 SF exact 3 SF
SolutionSolution
►150 pounds (American) is how many stones (British)?
1 pound = 265 dram1 Gram = 1.71 dram 1.13 pennyweights = 1 Gram1.21 scruples = 1 pennyweight17.2 scruples = 1 stone
57
?
►The following relationships are British liquid units.1 hogshead = 7 firkin18 pottle = 1 firkin140 pottle = 1 puncheon504 pottle = 1 tun
How many hogsheads in 12.5 tuns?
58
pottle
firkin
18
1hogsheads0.50
►A 60 watt light bulb is the how many horsepower?3.41 Btu/hr = 1 watt4.20 calorie/minute = 1 Btu/hr.0514 Foot-pound-force/second = 1 calorie/minute1.825×10-3 horsepower = 1 Foot-pound-force/second
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Learning CheckLearning Check
►If your pace on a treadmill is 65 meters per minute, how many minutes will it take for you to walk a distance of 7500 feet?
►The dosage ordered is 485 mg of Erythromycin four times a day (q.i.d). If the oral suspension contains 200 mg Erythromycin/5 mL, how many mL will be given in a day?
►The dimensions of a box are 12 inch by 11 inch by 5.5 inch. Calculate the volume of the box in cm3. There is 2.54 cm per inch.
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2.9 Measuring Temperature2.9 Measuring Temperature
► Temperature is commonly reported either in degrees Fahrenheit (oF) or degrees Celsius (oC).
► The SI unit of temperature is the Kelvin (K).► 1 Kelvin, no degree, is the same size as 1 oC.► 0 K is the lowest possible temperature, 0 oC =
273.15 K is the normal freezing point of water. To convert, adjust for the zero offset.
► Temperature in K = Temperature in oC + 273.15► Temperature in oC = Temperature in K - 273.15
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Freezing point of H2O Boiling point of H2O
32oF 212oF0oC 100oC
212oF - 32oF = 180oF covers the same range of temperature as 100oC - 0oC = 100oC covers. Therefore, a Celsius degree is exactly 180/100 = 1.8 times as large as a Fahrenheit degree. The zeros on the two scales are separated by 32oF.
Chapter Two 62
Fahrenheit, Celsius, and Kelvin temperature scales.Chapter Two 63
► Converting between Fahrenheit and Celsius scales is similar to converting between different units of length or volume, but is a little more complex. The different size of the degree and the zero offset must both be accounted for.
► ooF = (1.8 x F = (1.8 x ooC) + 32C) + 32► ooC = (C = (ooF – 32)/1.8F – 32)/1.8
Chapter Two 64
2.10 Energy and Heat2.10 Energy and Heat
► Energy: The capacity to do work or supply heat.► Energy is measured in SI units by the Joule (J); the
calorie is another unit often used to measure energy.► One calorie (cal) is the amount of heat necessary to
raise the temperature of 1 g of water by 1°C.► A kilocalorie (kcal) = 1000 cal. A Calorie, with a capital
C, used by nutritionists, equals 1000 cal.► An important energy conversion factor is:
1 cal = 4.184 J1 cal = 4.184 J
Chapter Two 65
► Not all substances have their temperatures raised to the same extent when equal amounts of heat energy are added.
► One calorie raises the temperature of 1 g of water by 1°C but raises the temperature of 1 g of iron by 10°C.
► The amount of heat needed to raise the temperature of 1 g of a substance by 1°C is called the specific heat of the substance.
► Specific heat is measured in units of cal/gC
Chapter Two 66
Chapter Two 67
► Knowing the mass and specific heat of a substance makes it possible to calculate how much heat must be added or removed to accomplish a given temperature change.
► (Heat Change) = (Mass) x (Specific Heat) x (Temperature Change)
► Using the symbols Δ for change, H for heat, m for mass, C for specific heat, and T for temperature, a more compact form is:
ΔH = m×C×Δ T
Learning CheckLearning Check
1. How much energy is required to change the temperature of 15.0 g Fe from 18.5 C to 56.8 C? The specific heat of iron is 0.451 J/g·K.
2. (2.67) copper has specific heat of .092 cal/(g °C). ∙When 52.7 cal of heat is added to a piece of copper, the temperature increases from 22.4 °C to 38.6 °C. What is the mass of the piece of copper?
3. If 34.8 J is required to change the temperature of 10.0 g of mercury by 25 K, what is the specific heat of mercury?
68
Learning CheckLearning Check
1. How much energy is required to change the temperature of 15.0 g Fe from 18.5 C to 56.8 C? The specific heat of iron is 0.451 J/g·K.
69
ΔH = 15.0 g×0.451 J×(56.8-18.5)C g·K
ΔH = m×C×Δ T
ΔH = 259 J
Learning CheckLearning Check
2. (2.67) copper has specific heat of .092 cal/(g °C). ∙When 52.7 cal of heat is added to a piece of copper, the temperature increases from 22.4 °C to 38.6 °C. What is the mass of the piece of copper?
70
ΔH = m×C×Δ T
52.7 cal = m×.092 cal× (38.6-22.4)C gC
Learning CheckLearning Check
3. If 34.8 J is required to change the temperature of 10.0 g of mercury by 25 K, what is the specific heat of mercury?
71
ΔH = m×C×Δ T
34.8 J = 10.0 g×C×25 K
2.11 2.11 DensityDensity
Density relates the mass of an object to its volume. Density is usually expressed in units of grams per cubic centimeter (g/cm3) for solids, and grams per milliliter (g/mL) for liquids.
Chapter Two 72
Density = Mass (g)
Volume (mL or cm3)
►Which is heavier, a ton of feathers or a ton of bricks?
►Which is larger?
►If two objects have the same mass, the one with the higher density will be smaller.
Chapter Two 73
Mercury has a density of 13.6 g/mL. How many milliliters of mercury weigh 475 grams?
1. 0.000155 mL2. 0.0286 mL3. 34.9 mL4. 6460 mL
Learning CheckLearning Check
►(2.75) What is the density of lithium metal ( in g/cm3) if a cylindrical wire with a diameter of 2.40 mm and a length of 15.0 cm has a mass of .3624 g; vcyln=r2l.
►The density of acetic acid is 1.05 g/mL. What is the volume of 275 g of acetic acid?
►A cube of iron has a mass of 15.37 g. If each side of the cube has dimensions of 1.25 cm, what is the density of iron?
75
Learning CheckLearning Check
►(2.75) What is the density of lithium metal ( in g/cm3) if a cylindrical wire with a diameter of 2.40 mm and a length of 15.0 cm has a mass of .3624 g; vcyln=r2l.
76
Density = Mass (g)
Volume (mL or cm3)
Learning CheckLearning Check
►The density of acetic acid is 1.05 g/mL. What is the volume of 275 g of acetic acid?
77
Density = Mass (g)
Volume (mL or cm3)
Learning CheckLearning Check
►A cube of iron has a mass of 15.37 g. If each side of the cube has dimensions of 1.25 cm, what is the density of iron?
78
Density = Mass (g)
Volume (mL or cm3)
2. 12 Specific Gravity2. 12 Specific Gravity
Specific gravity (sp gr): density of a substance divided by the density of water at the same temperature. Specific gravity is unitless. The density of water is so close to 1 g/mL that the specific gravity of a substance at normal temperature is numerically equal to the density.
Chapter Two 79
Density of substance (g/ml)Density of substance (g/ml)
Density of water at the same temperature (g/ml)Density of water at the same temperature (g/ml)Specific gravity =Specific gravity =
The specific gravity of a liquid can be measured using an instrument called a hydrometer, which consists of a weighted bulb on the end of a calibrated glass tube. The depth to which the hydrometer sinks when placed in a fluid indicates the fluid’s specific gravity.
Chapter Two 80
►Galileo’s Galileo’s ThermometerThermometer
As temperature As temperature changes so do changes so do the density's of the density's of the solutions in the solutions in the floating the floating bulbs.bulbs.
81
Chapter SummaryChapter Summary► Physical quantities require a number and a unit.► Preferred units are either SI units or metric units.► Mass, the amount of matter an object contains, is
measured in kilograms (kg) or grams (g). ► Length is measured in meters (m). Volume is
measured in cubic meters in the SI system and in liters (L) or milliliters (mL) in the metric system.
► Temperature is measured in Kelvin (K) in the SI system and in degrees Celsius (°C) in the metric system.
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Chapter Summary Cont.Chapter Summary Cont.► The exactness of a measurement is indicated by
using the correct number of significant figures.► Significant figures in a number are all known with
certainty except for the final estimated digit.► Small and large quantities are usually written in
scientific notation as the product of a number between 1 and 10, times a power of 10.
► A measurement in one unit can be converted to another unit by multiplying by a conversion factor that expresses the exact relationship between the units.
Chapter Two 83
Chapter Summary Cont.Chapter Summary Cont.
► Problems are solved by the factor-label method.► Units can be multiplied and divided like numbers.► Temperature measures how hot or cold an object is.► Specific heat is the amount of heat necessary to
raise the temperature of 1 g of a substance by 1°C. ► Density relates mass to volume in units of g/mL for
a liquid or g/cm3 for a solid.► Specific gravity is density of a substance divided by
the density of water at the same temperature.
Chapter Two 84
End of Chapter 2End of Chapter 2
Chapter Two 85
