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Instructional Improvement Program
Significant Figures, Measurement, and
Calculations in Chemistry
Carl Hoeger, Ph.D.
University of California, San Diego
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Part 2: Calculations in Science:
Dimensional Analysis
Carl Hoeger, Ph.D.
University of California, San Diego
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Dimensional Analysis
•  Also known as the Factor-Label Method of problem
solving.
•  A quantity in one unit is converted into an equivalent
quantity in a different unit.
•  A way to analyze and solve problems by using the units (or
dimensions) of the measurement; based on the (correct)
assumption that if the units of your answer are right,
chances are good that your answer is as well.
•  You will use the units to solve the problem, doing the
actual math at the end.
•  Requires the use of conversion factors: equivalence
statements that allow us to convert units of one type to
another.
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Conversion Factors
•  Derived from equivalence statements: an equality relating
one quantity to another. Write in equation form if not
already.
One foot is 12 inches; there are 15,125 bolts in a Saturn Vue; a mole
of carbon atoms weighs 12.01 grams
1 ft = 12 in
15,125 screws = 1 Saturn Vue
1 mol C = 12.01 g
•  Most useful when expressed as an equivalence “ratio”.
1 ft
12 in
or
12 in
1 ft
;
1 mol C
12.01 g
or
12.01 g
1 mol C
•  When “1” by itself appears in an equivalence statement or
ratio, that “1” is an exact number.
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Equivalence Ratios
•  Let’s examine how an equivalence statement is converted
into an equivalence ratio:
1 atm = 101.325 kPa
1 atm
101.325 kPa
=
101.325 kPa
101.325 kPa
=1
or
1 atm = 101.325 kPa
1 atm
1 atm
=
101.325 kPa
1 atm
=1
SO, Conversion factors are just a creative way to express “1”!
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Conversion Factors (cont)
For unit interconversions in the same measurement system (i.e.
metric to metric or English to English) conversion factors are
defined quantities and therefore have unlimited significant
figures.
2 nm = 2 x 10–7 cm; 16 oz = 1 lb
For unit interconversions between different systems (i.e. metric to
English) cf’s are measured values and DO have sig fig limits!
–  EXCEPTION: 1 in = 2.54 cm; this is an EXACT
conversion (only one!)
•  Equivalence statements always have the following relationship:
big # of a small unit = small # of a big unit
1000 mm = 1 m; 6.022 x 1023 atoms = 1 mol
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Conversion Factors (cont)
•  The units in a conversion factor can be treated the same way
you normally treat numbers: they can be squared, rooted,
canceled with identical units, etc.
⎛ 2.54 cm ⎞
⎜⎝ 1 in ⎟⎠
Note that ONLY
the units cancel!
15 in
3
3
3
=
×
(2.54) cm
3
(1) in
16.4 cm
1 in
3
3
3
=
16.4 cm
1 in
3
3
Remember to
change your
numerical portion
accordingly!
3
= 246 cm
3
•  Thus, conversion factors allow us to convert one
measurement in a given set of units into another.
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Conversion Factors (cont)
•  Conversion factors can also be derived from physical properties,
chemical measurements, or constants:
d Hg = 13.55 g ⋅ cm ; heat of vaporization of Hg = 59.23 kJ ⋅ mol
–3
h = 6.626 × 10
–34
–1
J ⋅s
•  Conversion factors derived from physical properties or chemical
measurements are considered measured quantities and therefore have
significant figures limits. It is therefore important to use most precise
value you can find when using them and apply sig fig rules as needed:
d Hg = 13.55 g ⋅ cm (4sf ) vs. d Hg = 13.6 g ⋅ cm (3sf )
–3
–3
•  Constants also have significant figure limits, but most constants have
values with such high precision that it is rarely necessary to invoke sig
fig rules:
–34
–34
h = 6.6260755 × 10 J ⋅ s, commonly used as just 6.626 × 10 J ⋅ s
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Conversion Factors and Dimensional Analysis:
How To
•  Begin by creating a conversion path.
•  What do we have; what do we need; what do we know?
•  Determine what equivalence ratios (conversion factors) are
needed.
•  Put conversion string together, adjust and cancel units as
called for.
•  Put in actual numerical values.
•  Do the math CAREFULLY!
•  Does the answer make sense?
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DA Example #1
•  A ruler is 12.0 inches long. How long is it in meters? (1 inch =
2.54 cm)
•  Set up a conversion path, putting needed equivalence statements
in it:
?
in
1 in = 2.54 cm
m
100 cm= 1 m
cm
•  Put conversion string together:
12 in ×
2.54 cm
1 in
×
1m
= 0.3048 m = 0.30 m (2sf )
100 cm
Answer has 2 sf; all
conversions here are exact
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DA Example #2
Jules Verne wrote a book 20,000 leagues under the sea. How far is
this in feet?
Much more difficult; need a lot of ‘uncommon conversions’:
1 league = 3 nautical miles; 1 nautical mile = 10 cable lengths;
1 cable length =100 fathoms; 6 ft = 1 fathom
Set up a conversion path, putting needed equivalence ratios in it:
league
?
feet
1:3
6:1
nautical mile
1:10
fathom
cable length
100:1
Put conversion string together:
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20000 leagues
×
3 nautical mi
1 league
×
100 fathoms
10 cable lengths
×
1 cable lengths
1 nautical mi
×
6 ft
1 fathom
= 3.6 × 108 ft = 4 × 108 ft (1sf )
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DA Example #3
A more common problem seen in chemistry is illustrated here:
A 0.032 molar solution of HCl (hydrochloric acid) has a density of 1.17; how
many mL do you need to measure out to ensure that the volume you have
contains 7.0 g of HCl?
Here you are actually faced with TWO problems: a value with no given units
(1.17) and too much information (0.032 molar)!
Approach: what do you need? Volume of HCl that contains 7.0 g of HCl
what do you know? Density of solution = 1.17 g/mL (!); molar = ??
Set up a conversion path, starting this time with what you need:
g HCl
1.17 g/mL
mL HCl
•  Put conversion string together:
7.0 g HCl needed ×
1 mL HCl soln
1.17 g HCl
= 5.98 mL = 6.0 mL (2sf )
SigFigs limited by starting amt
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Multistep Problems: More Complex DA
• 
Usually involves complex units: values naturally expressed as a ratio,
where units may be “understood”:
Gas mileage; speed; density; molar mass
STRATEGY:
–  Solve problems by breaking the solution into steps.
–  Convert complex units, using dimensional analysis.
Many complex tasks in daily life are handled by breaking them down into
manageable parts; Consider steps in cleaning a car:
I. 
II. 
III. 
IV. 
vacuum the inside
wash the exterior
dry the exterior
apply a coat of wax
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Multistep Problem Example
You are driving your sports car to Tahoe at an average speed of
68 mph. After 250. minutes you decide to stop for lunch. How
many kilometers have you driven in this time and how much money
have you spent on gas to drive this distance? Your car gets 32 mpg
and gas costs 82 cents per liter.
There may be more than one way to approach problems like these.
Here is one solution path for this problem:
1.  Determine time in hours spent driving;
2.  Determine miles driven;
3.  Determine kilometers driven;
4.  Determine money spent.
Unusual Conversions needed:
1 mi = 5280 ft; 1 in = 2.54 cm; 1 gal = 3.785 L
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Multistep Solution-1
1. Convert minutes to hours:
1 h
250. min ×
= 4.17 h (3 sf )
60 min
2. Determine miles driven:
4.17 h ×
68 mi
= 284 mi (3 sf ; only 2 sf allowed)
1 h
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Multistep Solution-2
3. Convert miles to kilometers:
284 mi ×
5280 ft
×
12 in
1 mi
×
2.54 cm
1 ft
×
1 in
1 m
×
100 cm
1 km
= 457 km = 460 km (2 sf )
1000 m
4. Determine money spent:
284 mi ×
1 gal
32 mi
×
3.785 L
1 gal
×
$0.82
= $27.62 = $28 (2 sf )
1 L
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Summary
•  Errors occur during scientific measurements, regardless of
how careful one is!
•  Significant Figures allow us to convey the precision of our
measurements to our audience.
•  Units are as important as the numerical values are.
•  Equivalence ratios (conversion factors) allow us to relate a
value in one unit set to another.
•  Dimensional Analysis is a way of using one or more
equivalence ratio(s) to solve problems.
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Questions? Comments?
This presentation has been brought to you through the
generous support of the University of California, San
Diego’s Instructional Improvement Program.
If you have questions, comments, or wish to use these
presentations in your University’s courses, please feel free
to send an email to Dr. Carl Hoeger at
[email protected]
This is episode “Dimensional Analysis”; if you have
specific comments or questions regarding this episode
please note this in your email
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