OpenStax-CNX module: m42293
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Dimensional Analysis
∗
Steve Davis
This work is produced by OpenStax-CNX and licensed under the
†
Creative Commons Attribution License 3.0
Abstract
Units and unit conversions
plain arabic
Chem 105 On-Line Notes
Dimensional Analysis
Chemistry, along with other sciences and engineering, makes use of many dierent units. Some of the
common ones include mass (ton, lb, oz, grains, grams), length (yard, feet, in, meters), and energy (Joule,
erg, kcal, eV). Since there are so many dierent units used to describe the same property, it is necessary to
be able to convert between the various units. To do this, one uses a conversion factor.
1 Unit Conversions
Consider the relationship between feet and inches.
1 feet = 12 inches
(1)
We can divide by either the right or left side of the equation giving
12 inches
1 feet
=1=
12 inches
1 feet
(2)
1
Both quotients are equal to 1. If the units are ignored, the quotients do not numerically equal 1 but 12
or 12. However, with the inclusions of the units, both the numerators and denominators describe the exact
same length, so the quotients are equal to 1 when the units are included. Since the two quotients are equal
to 1, multiplying or dividing by the quotients is the same as multiplying or dividing by 1; it does not change
the equation, only the relative numerical values within the various units.
Now, we can use these quotients to convert from inches to feet or from feet to inches.
12 inches
= 60 inches
1 feet
1 feet
30
5
30 inches
=
feet = feet = 2.5 feet
12 inches
12
2
5 feet
(3)
(4)
If there is confusion regarding which quotient to use in the conversion, just make sure the units work out.
In the conversion of 5 feet to inches represented in equation (1), the unit feet is in both the numerator
∗ Version
1.1: Jan 19, 2012 10:51 am -0600
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and denominator of the expression, so they cancel. The units (words) behave just like numbers in products
and quotients and can be multiplied and divided. As an example, consider the area of a rectangle which is
length×width. Let the length=5 inches and the width=10 inches.
Area = l × w = 5 inches × 10 inches = 50 inches2
(5)
2
So, inches × inches = inches .
For a second example, consider the relationship between mass and moles. For elemental carbon, C,
12.00 g C = 1 mol C
(6)
This equation means that 6.02 × 1023 carbons atoms weighs 12 grams and that 12 grams of carbon contain
6.02 × 1023 atoms. We can form the quotients
1 mol C
12 g C
=
=1
1 mol C
12 g C
(7)
We can use these quotients to convert between weight (in grams) to moles and vice verse.
1 mol C
60 g C
= 5 mol C
12 g C
12 g C
= 96 g C
8 mol C
1 mol C
(8)
(9)
We can string conversion along as products; consider converting the number of carbons atoms to a weight.
Let the number of carbons atoms be 20 × 1018 .
20 × 10
18
atoms C
1 mol C
6.02 × 1023 atoms C
12.00 g C
1 mol C
= 4 × 10−4 g C
(10)
Here we used two conversions:
1 mol C = 6.02 × 1023 atoms C
and
12 g C = 1 mol C
(11)
Now consider the following example: gold bars held by central banks weigh 12.4 kg; the current price of
gold is $1778 per oz. How much is a gold bar worth.
12.4 kg Au
2.2 lb
1 kg
16 oz
1 lb
1778 dollars
1 oz Au
= 780, 000 dollars
(12)
Let's look at the units: kg cancels, lb cancels, oz cancels, and Au cancels. This leaves dollars as the nal
unit. Three conversions were used in this calculation for convenience; the rst two conversions were from
memory so there was no need to spend time looking them up. If desired, a conversion for the price of gold
per kg could have been found and the rst two conversions would have been unnecessary, but sometimes it
is easier or quicker to use a product of conversions from memory rather than trying to nd one in a book or
on the internet that could be used singly.
Dierent units can be multiplied or divided. Consider the equation for kinetic energy:
1
mv 2
(13)
2
Let the units of mass be in kg and velocity be in (ms). If the product mv 2 is performed, the units are
kg m2 s2 . This quotient of units occurs frequently in science and is given the name of Joules and the symbol
KE =
J.
J≡
http://cnx.org/content/m42293/1.1/
kg m2
s2
(14)
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However, for sums and dierences, each term must be in the same units. For example, 12 in can be added
to 36 in to give 48 in: 12 in + 36 in = 48 in. It makes no sense, though, to add 12 in to 3 feet: 12 in + 3 feet =
??? 15 in + feet, haha.
2 Temperature Conversions
Two common temperature scales in use are the Fahrenheit (o F ) and centigrade (o C) ones. The centigrade
temperature scale is based upon the freezing and boiling points of water: the fp is given the value 0 o C , and
the bp is given the value 100 o F . The Fahrenheit scale is based on three points: the freezing point of a brine
mixture (ammonium chloride, ice, and water) is given the value zero, the dierence in the fp of water and
normal body temperature is set at 64 degrees. This makes the fp of water 32 o F and the bp 212 o F .
Consider the conversion between the two temperature scales. First, we recognize that there are 100
centigrade temperature units between the fp and bp of water while there are 212-32=180 Fahrenheit units.
Setting up an algebraic statement gives
100 o C = 180 o F
(15)
180 o F
100 o C
=
=1
180 o F
100 o C
(16)
Forming a conversion factor gives
In order to convert one scale to another, we need to take into account that the zero points are dierent.
Therefore, we have to remove the oset of the Fahrenheit scale so that the zero point matches that of the
centigrade scale. Consider the conversion from o F to o C , and let the Fahrenheit temperature be that of the
bp of water, 212 o F .
o
o
(212 F − 32 F )
100 o C
180 o F
= 100 o C
(17)
The rst term corrects the Fahrenheit scale for the zero point oset. Since the Fahrenheit scale starts at 32
F instead of zero o F , 32 o F is subtracted to get to zero o F . Then the conversion factor can be used.
Consider converting 100 o C to o F .
o
100 o C
180 o F
100 o C
+ 32 o F = 212 o F
(18)
Since the centigrade scale already starts at zero, we do not need to oset it so we just multiply by the
conversion factor. Then we correct the Fahrenheit scale by adding 32 o F to provide it's 32 o F oset.
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