The Factor Label Method and Conversion Factors

```Measurement in Chemistry
Factor-Label Method
The Factor-Label Method
At the conclusion of our time
together, you should be able to:
1. Recognize a problem that can be solved with
the factor label method
2. Transform a statement of equality into a
conversion factor
3. Use the appropriate conversion factor in the
correct way so that the labels cancel and the
correct conversion is found
The Factor label Method




A way to solve math problems in chemistry
Used to convert
km to miles, m to km, mol to g, g to mol, etc.
To use this we need:
 1) desired quantity
 2) given quantity
 3) conversion factors
Conversion factors are valid relationships or
equalities expressed as a fraction and equal to
one!
Equalities
State the same measurement in two different units
length
10.0 in.
25.4 cm
Conversion Factors
Fractions in which the numerator and denominator
are EQUAL quantities expressed in different units
but always equal to one. You can always
multiply any equation by this equality and not
change the quantity, just the units.
Example:
Factors:
1 in. = 2.54 cm
1in.
2.54 cm
and
2.54 cm
1 in.
For example: 1 km = 0.6 miles
the conversion factor is
1 km
0.6 miles
or
0.6 miles
1 km
Write conversion factors for 1 foot = 12 inches
What conversion factors can you think of that involve
meters?
Conversion Factors
Conversion factors for 1 ft = 12 in
1 foot
12 inches
or
12 inches
1 foot
There are almost an infinite number of
conversion factors that include meters:
1000 m
1m
1m
,
,
1 km
100 cm 1000 mm
1m
1m
0.9144 yards
,
,
3.28 feet
39.37 inches
1m
The Steps to Follow
Now we are ready to solve problems using the factor
label method. The steps involved are:
1.
2.
3.
4.
5.
Write the desired quantity and =
Write down the given quantity and put it over 1
Determine what conversion factors you will use to
turn the given label into the needed label.
Multiply the given quantity by the appropriate
conversion factors to eliminate units you don’t
want and leave the units you do want
Complete the math
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km
First write down the
desired quantity
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0 mi
Next, equate desired
quantity to the given
quantity
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0 mi
Now we have to
choose a conversion
factor
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0 mi
1 km
0.621 mi
0.621 mi
1 km
What conversion
factors are possible?
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0 mi
1 km
0.621 mi
0.621 mi
1 km
Pick the one that will
allow you to cancel
out miles
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0 mi
1 km
0.621 mi
0.621 mi
1 km
Pick the one that will
allow you to cancel
out miles
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0 mi
1 km
0.621 mi
0.621 mi
1 km
Multiply given
quantity by chosen
conversion factor
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0 mi
x 1 km
0.621 mi
Multiply given
quantity by chosen
conversion factor
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0 mi x 1 km
0.621 mi
Cross out common
factors
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0
x 1 km
0.621
Cross out common
factors
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0
x 1 km
0.621
Are the units now
correct?
Yes – km on both sides!
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0 x 1 km
0.621
= 75.68438003 km
Now finish the math.
The Steps to Follow
Now we are ready to solve problems using the factor
label method. The steps involved are:
1.
2.
3.
4.
Complete the math with no rounding
Make certain the sig figs are correct by rounding to
the correct number of sig figs at the very end
Don’t forget the order of operations when you
complete the math
Conversion factors do not determine sig. figs.!
Factor label Example
How many kilometers are in 47.0 miles?
(note: 1 km = 0.621 miles)
# km = 47.0
x 1 km
0.621
= 75.7 km
75.7 km
Summary
The previous problem was not that hard
In other words, you probably could have done it
faster using a different method
However, for harder problems the factor label
method is easiest
More Examples
1. You want to convert 100.00 U.S. dollars to
Canadian dollars. If the exchange rate is
1 Can\$ = 0.65 US\$, how much will it cost?
# Can\$ = 100.00 US\$x 1 Can\$ = 153.85 Can\$
0.65 US\$
The Factor-Label Method
Let’s see if you can:
1. Recognize a problem that can be solved with
the factor label method
2. Transform a statement of equality into a
conversion factor
3. Use the appropriate conversion factor in the
correct way so that the labels cancel and the
correct conversion is found
Learning Check
Write conversion factors that relate each of the
following pairs of units:
1. Liters and mL
1 Liter = 1000 mL
2. hours and minutes
1 hour = 60 minutes
3. meters and kilometers
1000 meters = 1 kilometer
How many minutes are in 2.5 hours?
Conversion factor
2.5 hr x
60 min
1 hr
= 150 min
By using dimensional analysis/factor-label method,
the UNITS ensure that you have the conversion
right side up, and the UNITS are calculated as well
as the numbers!
Learning Check

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
Measurement in Chemistry
Factor-Label Method
Part 2
The Factor-Label Method
At the conclusion of our time
together, you should be able to:
1. Recognize a problem that can be solved by
moving the decimal point.
2. Use the appropriate conversion factor in the
correct way so that the labels cancel and the
correct conversion is found with two
changes of labels or labels that are squared
or cubed.
Dealing with Two Units
Convert 55.00 km/h to m/s
55.00 km x 1000 m x 1 h___ =
h
1 km
3600 s
15.28 m/s
A patient requires injection of 0.012 g of a pain killer
available in a 15 mg/mL solution. How many
When you see a number with two units like 15 mg/mL, it
can be used as a conversion factor. What it really says is
that 1 ml of the solution contains 15 mg of the drug.
? mL = 0.012 g of drug
0.012 g drug 
mg drug  mL soln
3 mg drug
10
1 mL soln
? mL = 0.012 g of drug
1 g drug
15 mg drug
(
= 0.80 mL soln
)(
)
Dealing with Two Units, Your Turn
meters per minute, how many
seconds will it take for you to walk a
distance of 8450 feet?
1 meter = 3.28 feet
# s = 8450 ft
2380 seconds
x1m
3.28 ft
x 1 min
65 m
x 60 s
1 min
What about Square and Cubic units?
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

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
Learning Check

A Nalgene water
bottle holds 1000 cm3
of dihydrogen
monoxide (DHMO).
How many cubic
decimeters is that?
Solution
1000 cm3
(
)
1 dm
10 cm
3
= 1 dm3
So, a dm3 is the same as a Liter!
A cm3 is the same as a milliliter.
Converting Metric to Metric
A rattlesnake is 2.44 m long. How long is
the snake in cm?
a) 2440 cm
b) 244 cm
c) 24.4 cm
Solution
A rattlesnake is 2.44 m long. How long is
the snake in cm?
b) 244 cm
2.44 m x 100 cm
1m
= 244 cm
Converting Units of Length Made Easy

0.5 kilometer (km) = 500 meters (m)

2.5 meter (m) = 250 centimeters (cm)

1 centimeter (cm) = 10 millimeter (mm)

1 nanometer (nm) = 1.0 x 10-9 meter
O—H distance =
9.4 x 10-11 m
9.4 x 10-9 cm
0.094 nm
An Easier Way
A rattlesnake is 2.44 m long. How long is
the snake in cm?
G _ _ M _ _ k h da _ d c m _ _ μ _ _ n
1. A move from 1 meter to
centimeters is two places right
2. Move the decimal place of the
number two places right
3. 244 cm
Another Example: How many millimeters
are there in 4.5 cm?
G _ _ M _ _ k h da _ d c m _ _ μ _ _ n
1. A move from cm to mm is one
place right
2. Move the decimal place of the
number one place right
3. 45 mm
Another Example: How many kilometers
are there in 4.5 cm?
G _ _ M _ _ k h da _ d c m _ _ μ _ _ n
1. A move from cm to km is five
places left
2. Move the decimal place of the
number five places left
3. 0.000 045 km
The Factor-Label Method
Let’s see if you can:
1. Recognize a problem that can be solved by
moving the decimal point.
2. Use the appropriate conversion factor in the
correct way so that the labels cancel and the
correct conversion is found with two
changes of labels or labels that are squared
or cubed.
Learning Check: 2 kilometers is the same
as how many millimeters
G _ _ M _ _ k h da _ d c m _ _ μ _ _ n
1. A move from km to mm is six
places right
2. Move the decimal place of the
number six places right
3. 2 000 000 mm, 2 x 106 mm
Metric Conversions #1: Write 550 mm as
meters.
G _ _ M _ _ k h da _ d c m _ _ μ _ _ n
1. A move from mm to m is 3
places left
2. Move the decimal place of the
number 3 places left
3. 0.55 m
Learning Check
A person’s blood contains 185 mg of cholesterol per
deciliter of blood. How many grams of cholesterol
are there in 1 liter of this blood?
A. 0.0185 g
B. 0.185 g
C. 1.85 g
D. 18.5 g
E. 1850 g
English and Metric Conversions
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
If you know ONE conversion for each type of
measurement, you can convert anything!
You must use these conversions:
 Mass:
454 grams = 1 pound
 Length:
2.54 cm =
1 inch
 Volume:
0.946 L =
1 quart
Learning Check
An adult human has 4.65 L of blood. How many
gallons of blood is that?
Unit plan: L
Equalities:
qt
gallon
1 quart = 0.946 L
1 gallon = 4 quarts
Your Setup: gal = 4.65 L x 1 quart x 1 gallon
1
0.946 L 4 quarts
= 1.23 gallons
Exit Quiz
There are 12 inches in a foot, 0.394 inches in a
centimeter, and 3 feet in a yard. How many
centimeters are in 1.000 yard?
# cm = 1 yd x 3 ft x 12 in x 1 cm
= 91.37 cm
1 yd 1 ft
0.394 in
Exit Quiz #6 on WS
Change 9.4 miles to km (1 mile = 1.6 km)
# km = 9.4 mi
x 1.6 km
1 mi
= 15 km
Exit Quiz
With a U.S. dollar you can buy 1.1 Euros, 130
Yen, or 25 Rubles. How many Yen can