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Unit Conversions/Dimensional Analysis Part 1
Dr. MJ Patterson
Frequently in chemistry you will be asked to convert a number from one set of units to another. For instance, you might be given a length in feet and asked to convert it to yards. Many of these conversions can be accomplished by a technique known by two different names, unit conversion factors and dimensional analysis. Let's work out an example less formally first, and then we can apply to formalism to see how it works. (Word to the wise: this formalism may seem pointless for these first examples. However, these examples are designed to help you get comfortable with the process. We will quickly move to more complicated problems where you simply must be able to apply this formalism.)
Example 1: Convert 6 feet to yards.
Solution 1: First we need a conversion factor. We know that there are 3 feet in a yard. If we divide 6 feet by 3 feet per yard, we can conclude that there are 2 yards.
Example 2: Convert 2 yards to feet.
Solution 2: This is the exact opposite of the first problem. We take the 2 yards and multiply by 3 feet per yard to reach a value of 6 feet.
Formalism: We'll approach the first example using the formalism of dimensional analysis.
First of all, write out the conversion factor as an equation.
3 feet = 1 yard
Next, write the number that you want to convert as a fraction simply by placing it over 1 (Remember that any number divided by 1 is simply equal to the original number, just the same as multiplying by 1.).
(6 feet) 
(1) 
Next, we want to multiply this fraction by our conversion factor written as a fraction. But, which side of the equation do we put in the top, and which in the bottom? The simplest way to think about this is to think about the bottom. In the conversion factor, put in the bottom the side with the same units as the top of the step we already wrote. That way, the units will simply cancel! In this case, we want to write the fraction so that 3 feet is in the bottom, and 1 yard is in the top.
(6 feet)  (1 yard) 
(1)  (3 feet) 
Now we just have to perform the math. We want to multiply everything in the top, and divide by everything in the bottom. Remember the units. Feet in the top cancel with feet in the bottom, leaving only yards as the units. As for the numbers, they reduce to 6 divided by 3 = 2.
(6 feet)  (1 yard)  = 2 yards 
(1)  (3 feet) 

Note that we're treating the units as if they were also numbers to be multiplied and divided. If you remember that every quantity has two parts, a number and a unit, your life in chemistry will be much simpler.
To summarize the formalism of dimensional analysis:
 Write the quantity to be converted as a fraction divided by 1.
 Write the conversion factor as a fraction so that the bottom units cancel with the units on the top of the original number in the previous step.
(Note for the mathematically inclined (skip to the examples if desired!): The reason that this technique is sometimes called unit conversion factors is because each conversion factor expressed as a fraction is actually a form of 1 or unity!. To prove this, take the conversion factor written as an equation, and divide both sides by 1 yard.
3 feet  = 1 yard  = 1 
1 yard  1 yard 

Since this fraction is equal to unity or 1, we can multiply it times any other quantity without changing the value of the quantity. Remember that one times any number equals the original number. Note that the way we express the quantity is changing (from 6 feet to 2 yards) but the intrinsic value of the quantity is unchanged.)
Examples (for everyone!):
Useful conversion factors:
1 yd = 3 ft
1 in = 2.54 cm
1 kg = 2.2 lbs
Example 3: Convert 2 yards to feet.
Solution 3:
(2 yds)  (3 ft)  = 6 ft 
(1)  (1 yd) 

Example 4: Convert 25 centimeters to inches.
Solution 4:
(25 cm)  (1 in)  = 9.8 in 
(1)  (2.54 cm) 

Example 5: Convert 125 pounds to kilograms.
Solution 5:
(125 lbs)  (1 kg)  = 56.8 kg 
(1)  (2.2 lbs) 
