CHAPTER 2: THE CHEMISTS’ TOOLBOX
CHAPTER 2: THE CHEMISTS’ TOOLBOX
Problems: 7-14, 15-29 (odd only), 31-34, 37-41 (odd only), 44, 48
measurement: a number with attached units
To measure, one uses instruments = tools such as a ruler, balance, etc.
All instruments have one thing in common: UNCERTAINTY!
→ INSTRUMENTS CAN NEVER GIVE EXACT MEASUREMENTS!
EXPRESSING UNCERTAINTY IN MEASURED QUANTITIES
APPENDIX 1: SIGNIFICANT FIGURES (also called “Sig Figs” or “Significant Digits”)
When you a record a measurement, you know all the numbers with certainty except the last
number, which you must estimate. All the digits are significant because removing any digit
changes the measurement's uncertainty.
Example: Record the length of each wood sample for each ruler with units of centimeters (cm).
A B C
Increment of the smallest
markings on ruler
# of sig figs
Thus, a measurement taken with an instrument is always recorded to one more
decimal place than the smallest markings on the instrument.
CHEM110: Chapter 2 page 1
Guidelines for Determining Number of Sig Figs (if the measurement is given):
Count the number of digits in a measurement from left to right:
1. When a decimal point is present:
– For measurements ≥1, count all the digits (even zeros).
– 60.2 cm has 3 sig figs, 5.0 m has 2 sig figs, 150.00 g has 5 s.f.
– For measurements less than 1, start with the first nonzero digit and count all digits
(even zeros) after it.
– 0.011 mL, 0.0050 g, and 0.00022 kg each have 2 sig figs
2. When there is no decimal point:
– Count all non-zero digits and zeros between non-zero digits
– 125 g has 3 sig figs and 107 mL has 3 sig figs
– Placeholder zeros may or may not be significant
– 1000 g may have 1, 2, 3 or 4 sig figs
Example: How many significant digits do the following measurements have?
a. 105 mm _____ b. 90.40 cm _____ c. 100.00 m ____ d. 0.0050 g _____
2.3 SCIENTIFIC NOTATION
Some numbers are very large or very small → difficult to express.
Avogadro’s number = 602,000,000,000,000,000,000,000
an electron’s mass = 0.000 000 000 000 000 000 000 000 000 91 kg
To handle such numbers, we use a system called scientific notation. Regardless of their
magnitude, all numbers can be expressed in the form
N × 10n
where N =digit term= a number between 1 and 10, so there can only be
one number to the left of the decimal point: #.####
n = an exponent = a positive or a negative integer (whole #).
To express a number in scientific notation:
– Count the number of places you must move the decimal point to get N between 1 and 10.
Moving decimal point to the right (if # < 1) → negative exponent.
Moving decimal point to the left (if # > 1) → positive exponent.
CHEM110: Chapter 2 page 2
Example: Express the following numbers in scientific notation (to 3 sig figs):
555,000 → __________________
0.000888 → __________________
602,000,000,000,000,000,000,000 → ___________________________
Also, in some cases the number of sig figs in a measurement may be unclear:
For example, Ordinary form Scientific Notation
Express 100.0 g to 3 sig figs: ___________ → ______________
Express 100.0 g to 2 sig figs: ___________ → ______________
Express 100.0 g to 1 sig fig: ___________ → ______________
Thus, some measurements—usually those expressing large amounts—must be expressed
in scientific notation to accurately convey the number of sig figs.
ROUNDING OFF NONSIGNIFICANT DIGITS
How do we eliminate nonsignificant digits?
• If first nonsignificant digit < 5, just drop ALL nonsignificant digits
• If first nonsignificant digit ≥ 5, raise the last sig digit by 1 then
drop ALL nonsignificant digits
last significant digit
For example, express 72.58643 with 3 sig figs:
72.58643 to → _______________
3 sig figs
first nonsignificant digit
Example: Express each of the following with the number of sig figs indicated:
a. 376.276 to 3 sig figs _______________________
b. 500.072 to 4 sig figs _______________________
c. 0.00654321 to 3 sig figs _______________________
d. 1,234,567 to 5 sig figs _______________________
e. 2,975 to 2 sig figs _______________________
Be sure to express measurements in scientific notation when necessary to make it clear how
many sig figs there are in the measurement.
CHEM110: Chapter 2 page 3
When adding and subtracting measurements, your final value is limited by the measurement
with the largest uncertainty—i.e. the number with the fewest decimal places.
Ex 1: 106.61 + 0.25 + 0.195 = 107.055 → 107.055 of sig figs → ______________
to correct #
Ex 2: 725.50 – 103 = 622.50 → 622.50 of sig figs → ______________
to correct #
When multiplying or dividing measurements, the final value is limited by the measurement with
the least number of significant figures.
Ex 1: 106.61 × 0.25 × 0.195 = 5.1972375 → 5.1972375 of sig figs → ____________
to correct #
Ex 2: 106.61 × 91.5 = 9754.815 → 9754.815 of sig figs → _____________
to correct #
MULTIPLYING/DIVIDING WITH EXPONENTIAL NUMBERS:
When multiplying or dividing measurements with exponents, use the digit term (N in “N ×10n”)
to determine number of sig figs.
Ex. 1: (6.02×1023)(4.155×109) = 2.50131×1033
How do you calculate this using your scientific calculator?
Step 1. Enter “6.02×1023” by pressing:
6.02 then EE or EXP (which corresponds to “×10”) then 23
→ Your calculator should look similar to: 6.02 x1023
Step 2. Multiply by pressing: ×
Step 3. Enter “4.155×109” by pressing:
4.155 then EE or EXP (which corresponds to “×10”) then 9
→ Your calculator should now read 4.155 x109
Step 4. Get the answer by pressing: =
→ Your calculator should now read 2.50131 x1033
The answer with the correct # of sig figs = ___________________
CHEM110: Chapter 2 page 4
Be sure you can do exponential calculations with your calculator. Most of the
calculations we do in chemistry involve very large and very small numbers with
to correct # of sig figs
Ex. 2: (3.75×1015) (8.6×104) = 3.225×1020 → ___________________
1.90 × 1015
Ex. 3: 8
= 760000 of sig figs → ___________________
to correct #
2.500 × 10
SIGNIFICANT DIGITS AND EXACT NUMBERS
Although measurements can never be exact, we can count an exact number of items. For
example, we can count exactly how many students are present in a classroom, how many
M&Ms are in a bowl, how many apples in a barrel.
2.5 CONVERTING BETWEEN UNITS
Unit equation: Simple statement of two equivalent values
Conversion factor = unit factor = equivalents:
− Ratio of two equivalent quantities
1 dollar 10 dimes
Unit equation: 1 dollar = 10 dimes Unit factor: or
10 dimes 1 dollar
Unit factors are exact if we can count the number of units equal to another or if both
quantities are in the same system of measurement—i.e., both in the metric system
(e.g. cm and meters) or in the English system (inches and feet).
For example, the following unit factors and unit equation are exact:
365.25 days 1 day 12 in. 1m
and 1 yard ≡ 3 feet
1 year 24 hours 1 foot 100 cm
Exact equivalents have an infinite number of sig figs
→ never limit number of sig figs!
Note: When the relationship between two units or items is exact, the “≡” (meaning
“equals exactly”) is used instead of the basic “=” sign.
Equivalents based on measurements or relating measurements from two different systems are
inexact or approximate because they contain uncertainty, such as
1.61 km 65 mi 3.00 × 10 m
1 mile hour s
Approximate equivalents do limit the sig figs for the final answer.
CHEM110: Chapter 2 page 5
THE DIMENSIONAL ANALYSIS (or UNIT-CONVERSION) METHOD
1. Write the units for the answer.
2. Determine what information to start with.
3. Arrange all other unit factors—showing them as fractions—with the correct units
in the numerator and the denominator, so all units cancel except for the units
needed for the final answer.
4. Check for correct units and number of sig figs in the final answer.
Example 1: If a marathon is 26.2 miles, then a marathon is how many yards?
(1 mile≡5280 feet, 1 yard≡3 feet)
Example 2: You and a friend decide to drive to Portland, which is about 175 miles from
Seattle. If you average 99 kilometers per hour with no stops, how many hours
does it take to get there? (1 mile = 1.609 km)
Example 3: The speed of light is about 2.998×108 meters per second.
Express this speed in miles per hour. (1 mile=1.609 km, 1000 m≡1 km)
CHEM110: Chapter 2 page 6
THE METRIC SYSTEM
metric system: A unified decimal system of measurement with a basic unit for each type of
quantity basic unit symbol
length meter m
mass gram g
volume liter L
time second s
− Multiples or fractions of a basic unit are expressed as a prefix
→ Each prefix = power of 10
→ The prefix increases or decreases the base unit by a power of 10.
Prefix Symbol Multiple/Fraction
giga G 1,000,000,000 = 109
mega M 1,000,000 = 106
kilo k 1000
deci d 0.1 ≡
centi c 0.01 ≡
milli m 0.001 ≡
micro µ (Greek “mu”) 0.000 001 ≡
nano n 0.000 001 ≡
KNOW the metric units above (included in Table 2 on page 36 in Tro textbook)!
METRIC CONVERSION FACTORS
Ex. 1 Complete the following unit equations:
a. 1 dollar ≡ __________ cents → 1 m ≡ __________ cm
b. 1 dollar ≡ __________ dimes → 1 m ≡ __________ dm
Note: To help remember the number of centimeters or decimeters in a meter, just think of
the number of cents or dimes in a dollar!
CHEM110: Chapter 2 page 7
Ex. 2 Complete the following unit equations:
a. 1 kg ≡ ________ g c. 1 L ≡ ________ mL e. 1 Gwatts ≡ _____ watts
b. 1 g ≡ ________ cg d. 1 s ≡ ________ ns f. 1 g ≡ _____ µg
Ex. 3: Complete the following unit equations then write two unit factors for each equation:
a. 1 km ≡ __________ m b. 1 g ≡ ___________ mg
Ex. 1 Convert 175 ms into units of seconds.
Ex. 2 Convert 0.120 kilograms into milligrams.
Ex. 3 Convert 3.00×108 m/s into kilometers per hour.
Ex. 4 Convert 3.50×107 cm to units of kilometers.
CHEM110: Chapter 2 page 8
English system: Our general system of measurement.
Scientific measurements are exclusively metric. However, most Americans are more familiar
with inches, pounds, quarts, and other English units.
→ A method of conversion between the two systems is necessary.
These conversions will be given to you on quizzes and exams.
Quantity English unit Metric unit English–Metric conversion
length 1 inch (in.) 1 cm 1 in. ≡ 2.54 cm (exact)
mass 1 pound (lb) 1g 1 lb = 453.6 g (approximate)
volume 1 quart (qt) 1 mL 1 qt = 946 mL (approximate)
Ex. 1 What is the mass in kilograms of a person weighing 155 lbs?
Ex. 2 A 2.0-L bottle can hold how many cups of liquid? (1 qt. ≡ 2 pints, 1 pint ≡ 2 cups)
Ex. 3 A light-year (about 5.88×1012 miles) is the distance light travels in one year.
Calculate the speed of light in meters per second. (1 mile=1.609 km)
CHEM110: Chapter 2 page 9
Volume is determined in three principal ways:
1. Volume of any liquid can be measured directly using calibrated glassware
(graduated cylinder, pipets, burets, etc.)
2. Volume of a solid with a regular shape (rectangular, cylindrical, uniformly spherical or
cubic, etc.) can be determined by calculation.
3. Volume of an irregular solid is found indirectly by the amount of liquid it displaces.
This technique is called volume by displacement.
VOLUME BY CALCULATION
The volume of a rectangular solid can be calculated as follows:
volume = length × width × thickness
An easy method for solving for volume, length, width, or thickness:
length width thickness
Cover the value you are solving for → equation you need!
Ex. 1 What is the volume of a gold bar that is 5.25 cm long, 3.50 cm wide, and 2.75 cm thick?
Ex. 2 A rectangular bar of gold with a volume of 35.5 cm3 is 7.50 cm long and 3.50 cm wide.
How thick is the bar?
CHEM110: Chapter 2 page 10
VOLUME BY DISPLACEMENT
a. Fill a graduated cylinder halfway with water, and record the initial volume.
b. Carefully place the object into the graduated cylinder so as not to splash or lose water.
c. Record the final volume.
d. Volume of object = final volume – initial volume
Ex. 1 What is the volume of the Green jade sample shown below?
Ex. 2 A chunk of gold metal is placed in a graduated cylinder containing 24.6 mL of water.
The volume of water now reads 33.6 mL. Determine the volume of the gold chunk.
1.11 DENSITY: The amount of mass in a unit volume of matter
density = or d= generally in units of g/cm3 or g/mL
For water: 1.00 g of water occupies a volume of 1.00 cm3
m 1.00 g
d= = = 1 . 00 g /cm 3
V 1.00 cm 3
CHEM110: Chapter 2 page 11
Density also expresses the concentration of mass
– i.e., the more concentrated the mass in an object
→ the heavier the object → the higher its density
Sink or Float
Note how some objects float on water (e.g. a cork), but others sink
(e.g. a penny). That's because objects that have a higher density than
a liquid will sink in the liquid, but those with a lower density than the
liquid will float. Since water's density is about 1.00 g/cm3, cork's
density must be less than 1.00 g/cm3, and a penny's density must be
greater than 1.00 g/cm3.
Ex.: Consider the figure at the right and the following
solids and liquids and their densities:
ice (d=0.917 g/cm3) honey (d=1.50 g/cm3)
iron cube (7.87 g/cm3) hexane (d=0.65 g/cm3)
rubber cube (d=1.19 g/cm3)
Identify L1, L2, S1, and S2 by filling in the blanks below:
L1= _______________ and L2= _______________
S1= _______________, S2= _______________, and S3= _______________
Applying Density as a Unit Factor
Given the density for any matter, you can always write two unit factors. For example, the
density of ice is 0.917 g/cm3.
Two unit factors would be: or
Ex. 1 Give 2 unit factors for each of the following:
a. density of lead = 11.3 g/cm3 b. density of chloroform = 1.48 g/mL
CHEM110: Chapter 2 page 12
Ex. 2 Aluminum has a density of 2.70 g/cm3. What is the volume (in mL) of a piece of
aluminum with a mass of 0.125 kg?
Ex. 3 Ethanol is used in alcoholic beverages and has a density of 0.789 g/mL. What is the
mass of ethanol that has a volume of 1.50 L?
Ex. 4 A chunk of silver metal weighing 168 g is placed in a graduated cylinder with 21.0 mL of
water. The volume of water now reads 37.0 mL. Calculate the density of silver.
Ex. 5 A 25.0 g rectangular piece of lead measures 2.000 cm by 1.500 cm by 0.735 cm.
If the density of liquid mercury is 13.5 g/mL, will the piece of lead sink or float when
placed in a glass container with a sample of mercury.
CHEM110: Chapter 2 page 13
Percent: Ratio of parts per 100 parts → 10% is , 25% is , etc.
To calculate percent, divide one quantity by the total of all quantities in sample:
Percentage = × 100%
Ex. 1 In a chemistry class with 25 women and 20 men, what percentage of the class is
female? What percentage is male? (Express your answers to 3 sig figs.)
Writing out Percentage as Unit Factors
Ex. 1: Water is 88.8% oxygen by mass. Write two unit factors using this info.
Ex. 2: A 1968 penny was cast from a mixture of 95.0% copper and 5.0% zinc by mass. Write
four unit factors using this information.
CHEM110: Chapter 2 page 14
Percentage Practice Problems
Ex. 1 An antacid sample was analyzed and found to be 10.0% aspirin by mass. What mass of
aspirin is present in 3.50 g tablet of antacid?
Ex. 2 Water is 88.8% oxygen and 11.2% hydrogen by mass. How many grams of hydrogen
are present in 250.0 g of water?
Ex. 3 Steel is an alloy of iron mixed with other elements like carbon and chromium. If a
sample of high carbon steel contains is 1.35% carbon by mass, what mass of the steel
(in kg) contains 50.0 g of carbon?
Ex. 4 Air is 20.9% oxygen by volume. What volume (in L) of air contains 275 mL of oxygen?
CHEM110: Chapter 2 page 15
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