Ch. 2 “Scientific Measurement
& Problem Solving”
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Types of Observations and
Measurements
• We make QUALITATIVE observations
of reactions — Describes using words
Ex. Odor, color, texture, and physical
state.
• We also make QUANTITATIVE
observations, using numbersmeasurements
• Ex. 25.3 mL, 4.239 g
Standards of Measurement
When we measure, we use a measuring tool to
compare some dimension of an object to a
standard.
For example, at one time the
standard for length was the
king’s foot. What are some
problems with this standard?
Our measurements must be both
Accurate & precise!
Accuracy –
how close a measurement
comes to the true value of what is
measured
Precision –
is concerned with the
reproducibility of the measurement
Can you hit the bull's-eye?
Three targets
with three
arrows each to
shoot.
How do
they
compare?
Both
accurate
and precise
Precise
but not
accurate
Neither
accurate
nor precise
Stating a Measurement
In every measurement there is a
Number followed by a
 Unit from a measuring device
The number should also be as precise as the
measurement!
SI Measurement
• Le Systeme International d’Unites : SI Metrics
• System of measurement agreed on all over the
world in 1960
• Contains 7 base units
• units are defined in terms of standards of
measurement that are objects or natural
occurrence that are of constant value or are
easily reproducible
• We still use some non-SI units!
Le Système international d'unités
• SI units — based on the
metric system
• The only countries that have not
officially adopted SI are Liberia (in
western Africa) and Myanmar
(a.k.a. Burma, in SE Asia), but
now these are reportedly using
metric regularly
• Among countries with non-metric
usage, the U.S. is the only country
significantly holding out. The U.S.
officially adopted SI in 1966.
Information from U.S. Metric
Association
The 7 Base Units of SI
S.I. (the ones you’re responsible for
knowing!)
Selected S.I. base (or standard) units
Mass
kg
Length
m
Time
sec
Temperature
K
mole mol
Amount of Substance
Derived Units
made by combining Base Units!
• Volume
• Density
length cubed
m3
cm3
mass/volume g/mL kg/L g/ cm3
g/L
• Speed
length /time
• Area
length squared
mi/hr m/s
m2
cm2
km/hr
We use prefixes to expand on the base units!
S.I. prefixes you must memorize!
Prefix
Abbreviation
Value
kilo
k
103
deci
d
10-1
centi
c
10-2
milli
m
10-3
micro
m
10-6
nano
n
10-9
Metric System
• These prefixes are based on powers of 10.
• From each prefix every “step” is either:
• 10 times larger
or
• 10 times smaller
• For example
• Centimeters are 10 times larger than
millimeters
• 1 centimeter = 10 millimeters
kilo
hecto
deca
Base Units
meter
gram
liter
deci
centi
milli
Metric System
• An easy way to move within the metric
system is by moving the decimal point one
place for each “step” desired
Example: change meters to centimeters
1.00 meter = 10.0 decimeters = 100. centimeters
kilo
hecto
deca
meter
liter
gram
deci
centi
milli
mass – measure of the quantity of matter
SI unit of mass is the kilogram (kg)
1 kg = 1000 g = 1 x 103 g
Not to be confused with weight – mass + gravity
force that gravity exerts on an object
Mass does not vary from place to place!
A 1 kg bar has a
A 1 kg bar will weigh
mass of 1 kg
1 kg on earth
on earth and
0.1 kg on moon
on the moon
Volume – Amount of space occupied by matter
SI derived unit for volume is cubic meter (m3)
m3 = m x m x m
We often use the Liter (L) when
working with liquid volumes!
1 L = 1000 mL = 1000 cm3 = 1 dm3
1 mL = 1 cm3
1 dm3 = 1 L
Temperature Scales
• Fahrenheit
• Celsius
• Kelvin
Anders Celsius
1701-1744
Lord Kelvin
(William Thomson)
1824-1907
TEMPERATURE SCALES
In Chemistry, the terms heat and
temperature are often used to describe
specific properties of a sample.
HEAT is the most common form of energy
in nature and is directly related to the
motion of particles of matter.
The faster the motion of particles in a
sample the greater its heat content.
TEMPERATURE is associated only with
the intensity of heat and is not affected
by the size of the sample.
A forest fire and a lit match may both
be at the same temperature, but there
is a large difference in the amount of
heat each possess.
Heat always spontaneously flows from a
hotter system (higher temp.) to a colder
system (lower temp.).
Temperature Scales
Boiling point
of water
Freezing point
of water
Fahrenheit
Celsius
Kelvin
212 ˚F
100 ˚C
373 K
180˚F
100˚C
32 ˚F
0 ˚C
Notice that 1 Kelvin = 1 degree Celsius
100 K
273 K
Temperature
Scientists do not know of any limit on how
high a temperature may be.
The temperature at the center of the sun
is about 15,000,000 °C.
However, nothing can have a temperature
lower than –273°C. This temperature is
called absolute zero.
It forms the basis of the Kelvin scale.
Because the Kelvin scale begins at
absolute zero, 0 K equals –273°C, and
273 K equals 0 °C.
Calculations Using
Temperature
• Many chemistry equations require temp’s to be
in Kelvin
• K = ˚C + 273
• Body temp = 37 ˚C + 273 = 310 K
˚C = K - 273
• Liquid nitrogen = 273 -77 K = -196 ˚C
DENSITY –
an important and useful physical property
(Derived Unit)
ratio of mass per unit of volume
Density 
mass (g)
volume (cm3)
platinum
mercury
Mercury
Platinum
Aluminum
13.6 g/cm3
21.5 g/cm3
2.7 g/cm3
DENSITY
• Density is an
INTENSIVE
property of matter.
• Since it is a ratio
of mass to volume
-does NOT
depend on
quantity of matter.
The density of 1 g of gold =
The density of 5 kg of gold!
Styrofoam
Brick
Get out those calculators!
Problem A piece of copper has a
mass of 57.54 g. It is 9.36 cm long,
7.23 cm wide, and 0.095 cm thick.
Calculate density (g/cm3).
Pure copper metal
Copper ore
SOLUTION
1. Make sure dimensions are in common
units. (all are in cm’s)
2. Calculate volume in cubic centimeters.
L x W x H = volume
(9.36 cm)(7.23 cm)(0.095 cm) = 6.4 cm3
3. Calculate the density.
57.54 g
3
=
9.0
g
/
cm
6.4 cm3
Learning Check
Which diagram represents the liquid
layers in the cylinder?
(K) Karo syrup (1.4 g/mL), (V) vegetable
oil (0.91 g/mL,) (W) water (1.0 g/mL)
1)
2)
3)
V
W
K
W
K
V
K
V
W
Solution
(K) Karo syrup (1.4 g/mL), (V)
vegetable oil (0.91 g/mL,) (W) water
WATER!!
(1.0 g/mL)
1)
V
W
K
Denser materials ‘sink’ in
less dense materials!
Most solids sink in their liquid form.
Can you think of an exception to this?!
Finding Volume of an IrregularSolid by
Water Displacement
A solid displaces a matching volume of
water when the solid is placed in water.
Volume of solid
is 8 mL
33 mL
25 mL
Calculator Time!
What is the density (g/cm3) of 48 g of a
metal if the metal raises the level of water in
a graduated cylinder from 25 mL to 33 mL?
a) 0.2 g/ cm3 b) 6.0 g/cm3 c) 252 g/cm3
33 mL
25 mL
Percent Error
• Percent Error:
• Measures the inaccuracy of experimental
data
• Can have + or – value
• Accepted value : correct value based on reliable
references
• Experimental value: value you measured in the lab
accepted  experiment al
100%
accepted
Scientific Notation
The number of atoms in 12 g of carbon:
602,200,000,000,000,000,000,000
6.022 x 1023
The mass of a single carbon atom in grams:
0.0000000000000000000000199
1.99 x 10-23
N x 10n
N is a number
between 1 and 10
(1 non-zero digit to left of dec. pt.)
n is a positive or
negative integer
To change standard form to
scientific notation…
• Place the decimal point so that there is
one non-zero digit to the left of the
decimal point.
• Count the number of decimal places
the decimal point has “moved” from
the original number. This will be the
exponent on the 10.
• If the original number was less than 1,
then the exponent is negative. If the
original number was greater than 1,
then the exponent is positive.
Examples
• Given: 289,800,000
• Use: 2.898 (moved 8 places)
• Answer: 2.898 x 108
• Given: 0.000567
• Use: 5.67 (moved 4 places)
• Answer: 5.67 x 10-4
To change scientific notation
to standard form…
• Simply move the decimal point to
the right for positive exponent 10.
• Move the decimal point to the left
for negative exponent 10.
(Use zeros to fill in places.)
Example
• Given: 5.093 x 106
• Answer: 5,093,000 (moved 6
places to the right)
• Given: 1.976 x 10-4
• Answer: 0.0001976 (moved 4
places to the left)
Learning Check
• Express these numbers in
Scientific Notation:
1)
2)
3)
4)
5)
405789
0.003872
3000000000
2
0.478260
Scientific Notation
568.762
0.00000772
move decimal left
move decimal right
n>0
n<0
568.762 = 5.68762 x 102
0.00000772 = 7.72 x 10-6
Addition or Subtraction
1. Write each quantity with
the same exponent n
2. Combine N1 and N2
3. The exponent, n, remains
the same
4.31 x 104 + 3.9 x 103 =
4.31 x 104 + 0.39 x 104 =
4.70 x 104
Scientific Notation
Calculations
Multiplication
-5) x (7.0 x 103) =
(4.0
x
10
1. Multiply N1 and N2
(4.0 x 7.0) x (10-5+3) =
2. Add exponents n1 and n2
-2 =
28
x
10
3. Put in proper format, if necessary
2.8 x 10-1
Division
1. Divide N1 and N2
2. Subtract exponents n1 and n2
3. Put in proper format, if necessary
8.5 x 104 ÷ 5.0 x 109 =
(8.5 ÷ 5.0) x 104-9 =
1.7 x 10-5
Significant Figures
The numbers reported in a
measurement are limited by the
measuring tool
Significant Figures in a
measurement include all certain
digits plus one estimated digit
•7.50 cm
• 19.5 mL
Significant Figures
• All certain digits plus one
estimated digit (used when
recording measurements)
Known + Estimated Digits
In 2.85 cm…
• Known digits 2 and 8 are 100% certain
(there are lines on the ruler for these!)
•The third digit, 5, is estimated (uncertain)
• In the reported length, all three digits (2.76
cm) are significant including the estimated
one
Figure 5.5: Measuring a pin.
There are not really lines on the
scale here – just estimates!
Reading a Meterstick
. l2. . . . I . . . . I3 . . . .I . . . . I4. .
First digit (known) = 2
Second digit (known)
cm
2.?? cm
= 0.8
2.8? cm
Third digit (estimated) between 0.03- 0.05
Length reported
=
2.83 cm
or
2.84 cm
or
2.85 cm
Learning Check
. l8. . . . I . . . . I9. . . . I . . . . I10. .
cm
What is the length of the line?
1) 9.3 cm
2) 9.40 cm
3) 9.30 cm
How does your answer compare with your
neighbor’s answer?
Rules for Counting Significant Figures
RULE 1. All non-zero digits in a measured
number are significant.
Number of Significant Figures?
38.15 cm
5.6 mL
65.6 kg
122.55 m
4
2
3
5
Sandwiched Zeros
RULE 2. Zeros between nonzero numbers are
significant.
Number of Significant Figures?
50.8 mm
3
2001 min
4
.702 mg
3
400005 m
6
Leading Zeros (in front)
RULE 3. Leading zeros in decimal
numbers are NOT significant.
Number of Significant Figures?
0.008 mm
1
0.0156 g
3
0.0042 cm
2
0.0002602 mL
4
Trailing Zeros (at end)
RULE 4. Trailing zeros in numbers
without decimals are NOT significant.
They are only serving as place holders.
Number of Significant Figures?
25,000 m
2
200 L
1
48,600 mg
3
5
25,005,000 kg
Trailing Zeros, cont.
RULE 5. Trailing zeros in numbers with
decimals ARE significant.
Number of Significant Figures?
35,000.0 m
6
700. s
3
48.600 L
5
25,005.000 g
8
How many significant figures are in
each of the following measurements?
24 mL
2 significant figures
3001 g
4 significant figures
0.0320 m3
3 significant figures
6.4 x 104 molecules
2 significant figures
560 kg
2 significant figures
Significant Numbers in Calculations
A calculated answer cannot be more
precise than the measuring tool.
A calculated answer must match the
least precise measurement.
Significant figures are needed for final
answers from
1) adding or subtracting
2) multiplying or dividing
Rounding
• Need to use rounding to write a calculation
involving measurements correctly.
• Calculator gives you lots of insignificant
numbers so you must round to the correct
decimal place
• When rounding, look at the digit
after the one you can keep
• Greater than or equal to 5, round
up
• Less than 5, keep the same
Examples
Round each of the following measurements
so they have 3 sig figs:






761.50
14.334
10.44
10789
8024.50
203.514
762
14.3
10.4
10800
8020
204
Series of operations: keep all non-significant
digits during the intermediate calculations,
and round to the correct number of SF only
when reporting an answer.
Ex: (4.5 + 3.50001) x 2.00 =
(8.00001) x 2.00 = 16.0002 → 16
Adding and Subtracting
The answer has the same number of
decimal places as the measurement with
the fewest decimal places.
25.2
one decimal place (to right of decimal pt.)
+ 1.34 two decimal places (to right of decimal pt.)
26.54
Answer: 26.5 (one decimal place)
Using Sig Figs in Calculations
• Adding/Subtracting:
• end with the least number of decimal places
Using Sig Figs in Calculations
• Adding/Subtracting:
• end with the least number of decimal places
Significant Figures
Addition or Subtraction (con’t,)
89.332
+1.1
90.432
3.70
-2.9133
0.7867
one significant figure after decimal point
round off to 90.4
two significant figures after decimal point
round off to 0.79
Learning Check
In each calculation, round the answer to
the correct number of significant
figures.
A. 235.05 + 19.6 + 2.1 =
1) 256.75 2) 256.8
3) 257
B.
58.925 - 18.2 =
1) 40.725 2) 40.73
3) 40.7
Multiplying and Dividing
Round (or add zeros) to the calculated
answer until you have the same number of
significant figures as the measurement with
the fewest significant figures.
(Sometimes you’ll need to put the answer
into Sci. Notation to get correct # of sig
figs!)
Using Sig Figs in Calculations
• Multiplying/Dividing:
• end with the least number of sig figs
(Counting sig figs from left)
Using Sig Figs in Calculations
• Multiplying/Dividing:
• end with the least number of sig figs
Significant Figures
Multiplication or Division
The number of significant figures in the result is set by the original
number that has the smallest number of significant figures
4.51 x 3.6666 = 16.536366 = 16.5
3 sig figs
round to
3 sig figs
6.8 ÷ 112.04 = 0.0606926 = 0.061
2 sig figs
round to
2 sig figs
Learning Check
A. 2.19 X 4.2 =
1) 9
2) 9.2
B.
C.
3) 9.198
4.311 ÷ 0.07 =
1) 61.58
2) 62
3) 60
2.54 X 0.0028
=
0.0105 X 0.060
1) 11.3
2) 11
3) 0.041
For exact numbers (e.g. 4 beakers) and those
used in conversion factors (e.g. 1 inch = 2.54
cm), there is no uncertainty in their
measurement. Therefore, IGNORE exact
numbers when finalizing your answer with the
correct number of significant figures.
(Numbers from definitions or numbers of objects are considered
to have an infinite number of significant figures)
The average of three measured lengths, 6.64, 6.68 and 6.70 is:
6.64 + 6.68 + 6.70
= 6.67333 = 6.67
3
Because 3 is an exact number
=7
Chemistry In Action
On 9/23/99, $125,000,000 Mars Climate Orbiter entered Mar’s
atmosphere 100 km lower than planned and was destroyed by
heat.
1 lb = 1 N
1 lb = 4.45 N
“This is going to be the
cautionary tale that will be
embedded into introduction
to the metric system in
elementary school, high
school, and college science
courses till the end of time.”
Conversion Factors
• Ratio that comes from a statement of
equality between 2 different units
• every conversion factor is equal to 1
Example:
statement of equality
conversion factor
4quarters  1dollar
4 quarters
1dollar
1 =
4quarters
1 dollar
Conversion Factors (con’t.)
Fractions in which the numerator and
denominator are EQUAL quantities
expressed in different units
Example:
1 in. = 2.54 cm
Factors: 1 in.
2.54 cm
and
2.54 cm
1 in.
Learning Check
Write conversion factors that relate each of
the following pairs of units:
1. Liters and mL
1 L.
and
1000 mL
1000 mL
1L
2. Hours and minutes
1 hr.
and
60 mins.
60 mins.
1 hr
3. Meters and kilometers
1000 m
1 km
and
1 km__
1000 m
Conversion Factors
• can be multiplied by other numbers
without changing the value of the
number
(since you are just multiplying by 1)
4quarters
3dollars 
 12quarters
1dollar
Dimensional Analysis
Method of Solving Problems
1. Start with the given
2. Determine what unit label is needed on the answer
3. Add conversion factor(s) & cancel units until you are left
with the desired unit label!
How many mL are in 1.63 L?
1 L = 1000 mL
1000 mL
1.63 L x
= 1630 mL
1L
2
1L
L
1.63 L x
= 0.001630
1000 mL
mL
1.9
Sample Problem
• You have $7.25 in your pocket in
quarters. How many quarters do
you have?
7.25 dollars X
4 quarters
1 dollar
= 29 quarters
Learning Check
How many seconds are in 1.4 days?
Unit plan: days
1.4 days x
hr
min
seconds
Solution
Unit plan: days
hr
min
seconds
1.4 day x 24 hr x 60 min x 60 sec
1 day
1 hr
1 min
= 1.2 x 105 sec
Example
Convert 5.2 cm to mm
• Known: 100 cm = 1 m
1000 mm = 1 m
• MUST use m as an intermediate
1m
1000mm
5.2cm 

 52mm
100cm
1m
Example
Convert 0.020 kg to mg
• Known: 1 kg = 1000 g
1000 mg = 1 g
• Must use g as an intermediate
1000 g 1000mg
0.020kg 

 20,000mg
1kg
1g
Advanced Conversions
• A more difficult type of conversion deals
w/units that are fractions themselves
• Be sure convert one unit at a time; don’t try
to do both at once
• Setup your work the exact same way
When unit labels are fractions (or ratios),
unzip them!
11.3 g/mL can be written as 11.3 g
1 mL
OR
1 mL
11.3 g
Ex. Convert 11.3 g/mL to g/L
11.3 g 1000 mL
1 mL
1L
= 1.13 x 104 g/L
PROBLEM: Mercury (Hg) has a density of
13.6 g/cm3. What is the mass of 95 cm3 of
Hg in grams?
Solve the problem using
DIMENSIONAL
ANALYSIS.
13.6 g
3
3 g
95 cm •
=
1.3
x
10
cm3
The speed of sound in air is about 343 m/s.
What is this speed in miles per hour?
What is the given? What do you have to convert?
meters to miles
1 mi = 1609 m
343
seconds to hours
1 min = 60 s
m
1 mi
60 s
s x 1609 m x 1 min
1 hour = 60 min
60 min
mi
x
= 767
hour
1 hour
1.9
Advanced Conversions
• Another difficult type of conversion deals
with squared or cubed units
• Be sure to square or cube the conversion
factor you are using to cancel all the units
• If you tend to forget to square or cube the
number in the conversion factor, try
rewriting the conversion factor instead of
just using the exponent
Square and Cubic units
• Use the conversion factors you already
know, but when you square or cube the
unit, don’t forget to cube the number also!
• Best way: Square or cube the ENTIRE
conversion factor
• Example: Convert 4.3 cm3 to mm3
4.3 cm3 10 mm
(
1 cm
3
)
=
4.3 cm3 103 mm3
13 cm3
= 4300 mm3
Example
• Convert:
2000 cm3 to m3
• No intermediate
needed
Known:
100 cm = 1 m
cm3 = cm x cm x cm
m3 = m x m x m
 1m   1m   1m 
3
2000cm  cm  cm 


  0.002m
 100cm   100cm   100cm 
3
OR
 1m 
3
2000cm  

0
.
002
m

 100cm 
3