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CHAPTER 2: MEASUREMENT AND CALCULATIONS
Active Learning Problems: 1-4, 6, 8-21
End-of-Chapter Problems: 1-97, 99-101, 103-107, 109-110, 112-119, 121-137, 141-153
measurement: a number with attached units
When scientists collect data, they record the measurements as accurately as possible, and they
report the measurements taken to reflect the accuracy and precision of the instruments they used
to collect that data.
Consider the following plot of global land-ocean temperatures based on measurements taken
from meteorological stations and ship and satellite temperature (SST) measurements:
“This graph illustrates the change in global surface temperature relative to 1951-1980 average
temperatures. The 10 warmest years in the 134-year record all have occurred since 1998, with
2010 and 2005 ranking as the warmest years on record (Source: NASA/GISS). This research is
broadly consistent with similar constructions prepared by the Climatic Research Unit and
the National Atmospheric and Oceanic Administration.”
Source: http://climate.nasa.gov/vital-signs/global-temperature/
Ex. 1: Between about 1950 and 1980, what was the general range in variation in the average
global surface temperature?
Ex. 2: Between about 1980 and 2010, what was the general range in variation in the average
global surface temperature?
CHEM 139: Zumdahl Chapter 2 page 1 of 17 2.5 SIGNIFICANT FIGURES (or SIG FIGS): Writing Numbers to Reflect Precision
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!
When a measurement is recorded, all the given numbers are known with certainty (given the
markings on the instrument), except the last number is estimated.
→ The digits are significant because removing them changes the measurement's uncertainty.
– Thus, when measurements are recorded,
– they are recorded to one more decimal place than the markings for analog instruments;
– they are recorded exactly as displayed on electronic (digital) instruments.
LENGTH
– generally reported in meters, centimeters, millimeters, kilometers, inches, feet, miles
– Know the following English-English conversions: 1 foot ≡ 12 inches
1 yard ≡ 3 feet
Example: Using Rulers A, B, and C below, indicate the measurement to the line indicated for
each ruler. Assume these are centimeter rulers, so show the each measurement
has units of cm. Circle the estimated digit for each measurement.
Ruler A
0
1
2
3
4
5
0
1
2
3
4
5
Ruler B
Ruler C
4.1
A
4.2
4.3
4.4
B
C
Measurement
Increment of the smallest
markings on ruler
# of sig figs
Thus, for analog instruments (e.g. a ruler) a measurement is always recorded with one
more digit than the smallest markings on the instrument used, and measurements with
more sig figs are usually more accurate.
CHEM 139: Zumdahl Chapter 2 page 2 of 17 2.4 UNCERTAINTY IN MEASUREMENT
Guidelines for Sig Figs (if measurement is given):
Count the number of digits in a measurement from left to right:
1. When there is a decimal point:
– For measurements greater than 1, count all the digits (even zeros).
– 62.4 cm has 3 sig figs, 5.0 m has 2 sig figs, 186.100 g has 6 s.f.
– For measurements less than 1, start with the first nonzero digit and count all digits
(even zeros) after it.
– 0.011 mL 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
– e.g. 125 g has 3 sig figs, 1007 mL has 4 sig figs
– Placeholder zeros may or may not be significant
– e.g. 1000 may have 1, 2, 3 or 4 sig figs
Example: Indicate the number of significant digits for the following:
a. 165.3 g
_____
c. 90.40 m _____
e. 0.19600 g
_____
b. 105 cm
_____
d. 100.00 L _____
f. 0.0050 cm
_____
2.3 MEASUREMENTS OF LENGTH, VOLUME, AND MASS
VOLUME: Amount of space occupied by a solid, gas, or liquid.
– generally in units of liters (L), milliliters (mL), or cubic centimeters (cm3)
– Know the following:
1 L ≡ 1 dm3
1 mL ≡ 1 cm3
(These are both exact!)
Note: When the relationship between two units or items is exact, we use the “≡” to mean
“equals exactly” rather than the traditional “=” sign.
– also know the following equivalents in the English system
1 gallon ≡ 4 quarts
MASS:
1 quart ≡ 2 pints
1 pint ≡ 2 cups
a measure of the amount of matter an object possesses
– measured with a balance and NOT AFFECTED by gravity
– usually reported in grams or kilograms
WEIGHT: a measure of the force of gravity
– usually reported in pounds (abbreviated lbs)
mass ≠ weight = mass × acceleration due to gravity
CHEM 139: Zumdahl Chapter 2 page 3 of 17 Mass is not affected by gravity!
2.1 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.
Example: Express the following numbers in scientific notation (to 3 sig figs):
Height of Mt. Rainier: 14,400 ft.
→
Diameter of a human hair: 0.0110 cm
__________________
→
__________________
Avogadro’s Number: 602,000,000,000,000,000,000,000 → _____________________
CHEM 139: Zumdahl Chapter 2 page 4 of 17 Some measurements may be rounded to a number of sig figs requiring scientific notation.
For example,
Express 100.0 g to 3 sig figs: ___________
→ ______________
Express 100.0 g to 2 sig figs: ___________
→ ______________
Express 100.0 g to 1 sig fig:
→ ______________
___________
UNBIASED ROUNDING (or ROUND-TO-EVEN METHOD)
How do we eliminate nonsignificant digits?
•
If first nonsignificant digit < 5, just drop the nonsignificant digits
•
If first nonsignificant digit ≥ 6, raise the last sig digit by 1 and drop nonsignificant digits
•
If first nonsignificant digit =5 and
– nonzero digits follow 5, raise the last sig digit by 1 and drop nonsignificant digits
to 3 s.f.
– e.g. 3.14501 ⎯⎯
⎯⎯→ 3.15 (since nonsig figs are “501” in 3.14501)
– no digits or only zeros follow the 5, leave it alone or raise the last sig digit to get an
even number and drop nonsignificant zeros
to 3 s.f.
– e.g. 3.145 or 3.145000 ⎯⎯
⎯⎯→ 3.14 (to get last sig fig to be an even number)
to 3 s.f.
– e.g. 3.175 or 3.175000 ⎯⎯⎯⎯→ 3.18 (to get last sig fig to be an even number)
Express each of the following with the number of sig figs indicated:
to 3 sig figs
a. 648.75
⎯⎯⎯⎯⎯
⎯→
b. 23.6500
⎯⎯ ⎯ ⎯ ⎯
⎯→
c. 64.55
⎯⎯ ⎯ ⎯ ⎯
⎯→
d. 0.00123456
⎯⎯ ⎯ ⎯ ⎯
⎯→
e. 1,234,567
⎯⎯⎯⎯⎯
⎯→
f. 1975
⎯⎯ ⎯ ⎯ ⎯
⎯→
to 3 sig figs
to 3 sig figs
to 3 sig figs
to 5 sig figs
to 2 sig figs
_______________________
_______________________
_______________________
_______________________
_______________________
_______________________
Express measurements in scientific notation whenever the number of sig figs is unclear!
CHEM 139: Zumdahl Chapter 2 page 5 of 17 SIGNIFICANT FIGURES IN CALCULATIONS
ADDING/SUBTRACTING MEASUREMENTS
When adding or subtracting measurements, the final value is limited by the measurement with
the largest uncertainty—i.e. the measurement with the fewest number of decimal places.
MULTIPLYING/DIVIDING MEASUREMENTS
When multiplying or dividing measurements, the final value is limited by the measurement
with the least number of significant figures.
Ex. 1:
7.4333 g + 8.25 g + 10.781 g = _________________________
Ex. 2:
13.50 cm × 7.95 cm × 4.00 cm = _________________________
Ex. 3:
9.75 mL − 7.35 mL = _________________________
Ex. 4:
101.755 g
= _________________________
25.75 cm × 10.25 cm × 8.50 cm
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: (6.022×1023)(4.50×109) = 2.7099×1033
How do you calculate this using your scientific calculator?
Step 1.
Enter “6.02×1023” by pressing:
6.022 then
EE or EXP (which corresponds to “×10”) then 23
6.022
→ Your calculator should look similar to:
Step 2.
Multiply by pressing: ×
Step 3.
Enter “4.50× 109” by pressing:
4.50 then
23
EE or EXP (which corresponds to “×10”) then 9
4.50
→ Your calculator should look similar to:
Step 4.
x10
x10
9
Get the answer by pressing: =
→ Your calculator should now read
2.7099
x10
33
The answer with the correct # of sig figs = ___________________
CHEM 139: Zumdahl Chapter 2 page 6 of 17 Be sure you can do exponential calculations with your calculator. Many calculations we
do in chemistry involve numbers in scientific notation.
to correct # of sig figs
Ex. 2: (3.25×1012) (8.6×104) = 2.795 ×1017 ⎯⎯
⎯⎯⎯⎯⎯→ _________________
Ex. 3:
3.75 × 1015
8.605 × 10
4
to correct # of sig figs
= 4.357931435×1010 ⎯⎯
⎯⎯⎯⎯⎯→ __________________
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. We say that exact numbers of objects have an
infinite number of significant figures.
UNIT EQUATIONS AND UNIT FACTORS
Unit equation: Simple statement of two equivalent values
Conversion factor = unit factor = equivalents:
- Ratio of two equivalent quantities
Unit equation
Unit factor
1 dollar
10 dimes
1 dollar ≡ 10 dimes
or
10 dimes
1 dollar
Note: The “≡” is used to mean “exactly equal to” in place of the usual equal sign, “=”.
Unit factors are exact if we can count the number of units equal to another.
For example, the following unit factors and unit equations are exact:
7 days
1 week
24 hours
1 day
1 gallon
4 quarts
100 cm
1m
and
1 yard ≡ 3 feet
Exact equivalents have an infinite number of sig figs
→ never limit the number of sig figs in calculations!
Other equivalents are inexact or approximate because they are measurements or
approximate relationships, such as
1.61 km
1 mile
55 miles
1 hour
454 g
lb
Approximate equivalents do limit the sig figs for the final answer.
CHEM 139: Zumdahl Chapter 2 page 7 of 17 2.6 PROBLEMS SOLVING AND DIMENSIONAL ANALYSIS
1. Write the units for the answer.
2. Determine what information to start with.
3. Arrange all unit factors (showing them as fractions with units), so all of the units cancel
except those needed for the final answer.
4. Check for the correct units and the correct number of sig figs in the final answer.
Ex. 1: The Hope Diamond is a large, blue diamond that weighs 0.020 lb. How many carats is the
Hope Diamond? (1 lb. = 453.6 g and 1 carat = 0.200 g)
Ex. 2 The distance from the Earth to the Sun is about 93 million miles. If light travels at a speed
of 2.998×108 m/s, how many minutes does it take for light from the Sun to reach the
Earth? (1 mile = 1.609 km and
1 km≡1000 m)
2.2 UNITS
Show “Powers of Ten” video.
International System or SI Units (from French "le Système International d’Unités")
– standard units for scientific measurement
Metric system: A decimal system of measurement with a basic unit for each type of
measurement
quantity
basic unit (symbol)
quantity
SI unit (symbol)
length
meter (m)
length
meter (m)
mass
gram (g)
mass
kilogram (kg)
volume
liter (L)
time
second (s)
time
second (s)
temperature
Kelvin (K)
CHEM 139: Zumdahl Chapter 2 page 8 of 17 Metric Prefixes
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
tera
T
1,000,000,000,000 ≡ 1012
giga
G
1,000,000,000 ≡ 109
mega
M
1,000,000 ≡ 106
kilo
k
1000 ≡ 103
deci
d
0.1 ≡
centi
c
0.01 ≡
milli
m
0.001 ≡
micro
µ (Greek “mu”)
10–6
nano
n
10–9
pico
p
10–12
femto
f
10–15
1
10
1
≡ 10-1
100
1
≡ 10-2
1000
≡ 10-3
Metric Conversion Factors
Example 1: Complete the following unit equations:
a. 1 kg ≡ ________ g
d. 1 L ≡ ________ mL
g. 1 s ≡ _______ fs
b. 1 m ≡ ________ nm
e. 1 g ≡ ________ µg
h. 1 m ≡ _______ pm
c. 1 cm ≡ ________ m
f.
1 megaton ≡ ________ tons
Writing Unit Factors: Write two unit factors using the unit equations for examples a, b, and c.
CHEM 139: Zumdahl Chapter 2 page 9 of 17 Metric-Metric Conversions: Solve the following using dimensional analysis.
Ex. 1 The Space Needle is 184.41 m tall.
a. Convert this to units of kilometers.
b. Convert this to units of centimeters.
Ex. 2 Human hair can vary from about 17-181 µm (micrometers, also called “microns”) in
diameter. What is this range in millimeters?
Metric-English Conversions
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.
→ Conversions between the two systems are often 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?
CHEM 139: Zumdahl Chapter 2 page 10 of 17 Ex. 2 A 2.0-L bottle can hold how many cups of liquid? (1 qt. ≡ 2 pints, 1 pint ≡ 2 cups)
Ex. 3: The speed of sound in dry air is about 343 meters per second. Express this speed in miles
per hour. (1 mile=1.609 km)
Ex. 4: If a car averages 6.72 L per 100 kilometers, what is its fuel efficiency in miles per gallon?
(1 mile=1.609 km)
2.7 TEMPERATURE CONVERSIONS: AN APPROACH TO PROBLEM SOLVING
Temperature: A measure of the average energy of a single particle in a system.
The instrument for measuring temperature is a thermometer.
Temperature is generally measured with these units:
References
Fahrenheit scale (°F)
English system
Celsius scale (°C)
Metric system
freezing point for water
32°F
0°C
boiling point for water
212°F
100°C
Nice summer day in Seattle
77°F
25°C
CHEM 139: Zumdahl Chapter 2 page 11 of 17 Conversion between Fahrenheit and Celsius scales:
°C =
(°F - 32)
1.8
°F = (°C ×1.8) + 32
Kelvin Temperature Scale
– There is a third scale for measuring temperature: the Kelvin scale.
– The unit for temperature in the Kelvin scale is Kelvin (K, NOT °K!).
– The Kelvin scale assigns a value of zero kelvins (0 K) to the lowest possible
temperature, which we call absolute zero and corresponds to –273.15°C.
– The term absolute zero is used to indicate the theoretical lowest temperature.
Conversion between °C and K:
K = ˚C + 273.15
˚C = K – 273.15
Ex. 1 Liquid nitrogen boils at 77 K, so anything placed in a container with liquid nitrogen can be
cooled to extremely low temperatures. For example, a rose can be cooled in liquid
nitrogen until the water in it crystallizes, so the rose can shatter just like glass.
What is the equivalent temperature in ˚C and in ˚F?
DETERMINING VOLUME
– Volume is determined in three principal ways:
1. The volume of any liquid can be measured directly using calibrated glassware in the
laboratory (e.g. graduated cylinder, pipets, burets, etc.)
2. The volume of a solid with a regular shape (rectangular, cylindrical, uniformly spherical
or cubic, etc.) can be determined by calculation.
– e.g. volume of rectangular solid = length × width × thickness
4
volume of a sphere = πr3
3
3. Volume of solid with an irregular shape can be found indirectly by the amount of liquid it
displaces. This technique is called volume by displacement.
CHEM 139: Zumdahl Chapter 2 page 12 of 17 VOLUME BY DISPLACEMENT
a. Fill a graduated cylinder halfway with water, and record the initial volume.
b. Carefully place the object in the graduated cylinder so as not to splash or lose any water.
c. Record the final volume.
d. Volume of object = final volume – initial volume
Example: What is the volume of the piece of green jade in the figure below?
2.8 DENSITY: The amount of mass in a unit volume of matter
The Density Concept: The amount of mass in a unit volume of matter
density =
mass
volume
or
d=
m
V
generally in units of g/cm3 or g/mL
For water, 1.00 g of water occupies a volume of 1.00 cm3: d =
m
V
=
1.00 g
3
1.00 cm
= 1.00 g/cm3
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:
CHEM 139: Zumdahl Chapter 2 0.917g
cm3
or
cm3
0.917g
page 13 of 17 Example: 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
Solve the following problems:
Ex. 1
The density of pure silver is 10.5 g/cm3. A silvery piece of metal weighing 55.195 g is
placed in a graduated cylinder containing 3.50 mL of water. The water level rises until
the volume of water is 9.25 mL. Calculate the density of the metal, and determine if it is
pure silver.
Ex. 2
In the opening sequence of “Raiders of the Lost Ark,” Indiana Jones steals a gold statue
by replacing it with a bag of sand. If the statue has a volume of about 1.5 L and gold has
a density of 19.3 g/cm3, how much does the statue weigh in pounds?
Ex. 3
The average density of the Earth is 5,515 kg/m3. What is its density in grams per cubic
centimeter?
CHEM 139: Zumdahl Chapter 2 page 14 of 17 Ex. 4:
Given 25.0 g of each of the following, rank the objects in terms of increasing volume:
a. lead (Pb) cube (d=11.4 g/cm3)
b. gold (Au) cube (d=19.3 g/cm3)
c. ice cube (d=0.917 g/cm3)
d. aluminum (Al) cube (d=2.70 g/cm3)
______________ < ______________ < ______________ < ______________
smallest volume
largest volume
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. 1: Consider the figure at the right and the following
solids and liquids and their densities:
ice (d=0.917 g/cm3)
iron cube (7.87 g/cm3)
rubber cube (d=1.19 g/cm3)
honey (d=1.50 g/cm3)
hexane (d=0.65 g/cm3)
Identify L1, L2, S1, and S2 by filling in the blanks below:
L1= ___________________ and L2= ___________________
S1= ___________________ S2= ___________________ and S3= ___________________
PERCENTAGES
Percent: Ratio of parts per 100 parts → 10% is
10
25
, 25% is
, etc.
100
100
To calculate percent, divide one quantity by the total of all quantities in sample:
Percentage =
CHEM 139: Zumdahl Chapter 2 one part
×100%
total sample
page 15 of 17 Example: Carbon dioxide consists of carbon and oxygen atoms. A 0.500 g sample of
carbon dioxide was analyzed and found to contain 0.136 g of carbon. Calculate the
percentage by mass of carbon and oxygen in carbon dioxide.
Writing out Percentage as Unit Factors
Example:
Pennies cast between 1963 and 1982 are a mixture of 95.0% copper and 5.0%
zinc by mass. Write four unit factors using this information.
Percentage Practice Problems
Ex. 1 Steel is an alloy of iron mixed with elements like carbon and chromium. If high carbon
steel is 1.35% carbon by mass, what mass (in kg) of steel contains 25.0 g of carbon?
Ex. 2 What mass (in kg) of copper is present in 100.0 lbs. of pennies cast in the 1970’s?
CHEM 139: Zumdahl Chapter 2 page 16 of 17 Ex. 3 A 1.35 g cube made of African teak wood has an edge length of 1.114 cm. Will the cube of
teak wood sink or float in a sample of ethanol which has a density of 0.789 g/mL?
Ex. 4 Pennies cast between 1963 and 1982 are a mixture of 95.0% copper and 5.0% zinc by
composition (i.e., by number of atoms). Copper’s density is 8.96 g/cm3 while zinc’s
density is 7.14 g/cm3. What is the density of a 1968 copper penny?
Ex. 5 Pennies cast between 1963 and 1982 are a mixture of 95.0% copper and 5.0% zinc by
mass. Copper’s density is 8.96 g/cm3 while zinc’s density is 7.14 g/cm3.
a. Assuming 100.0 g of pennies, indicate the mass of copper and zinc present below.
In 100.0 g of pennies, __________ g of copper and __________ g of zinc are present.
b. Calculate the volume occupied by the copper in the pennies and the volume occupied
by the zinc. Afterwards, use the masses and volumes to calculate the density of a 1968
penny.
CHEM 139: Zumdahl Chapter 2 page 17 of 17