Examples of Conversion Factors
In chemistry, we aren't likely to encounter cookie problems often. We will, instead, use conversion factors to change between English and metric units, to quantify moles or atoms, or to use bigger or smaller units within a unit system. Most often, we'll be converting between like units, so we'll convert from one unit of time to another unit of time, or from one unit of mass to another unit of mass, for example. Or maybe, we'll just quantify a number of atoms and put it into moles. It is also possible to convert between dimensions with a known conversion factor, such as 1 kilogram of pure water equals 1 liter.
You might already know some conversion factors without having to look them up, like 12 inches per foot, 60 seconds per minute, 60 minutes per hour, and 24 hours per day. If you will be doing a lot of molar mass calculations, for instance, you might want to memorize the definition of a mole, 6.022 x 10^23. A mole is just a number, the way a dozen means 12. It is easy to look up common conversion factors though, too, in a table like this.
Setting Up Unit Conversions
We have described conversion factors as fractions equaling 1. When we use conversion factors to convert units, we will multiply our original measurement by the conversion factor to get the same measurements expressed in new units. We need to be careful in setting up our multiplication problem correctly so that we cancel out our old units and introduce the new units in the numerator of the conversion factor fraction.
Using the table above, we see that 1 kilometer is equal to approximately 0.62137 miles. If we want to convert 5.0 miles to kilometers, we multiply our original measurement (5.0 miles) by the conversion factor. If we multiplied it by (0.62137 miles/1 kilometer), which is, indeed, a fraction equaling 1, we would end up with final units of miles squared divided by kilometers. Those units are not what the problem called for and don't make sense as a unit of length. So it's important that we look ahead to the final units we are looking for. In this case, those units are kilometers.
To correctly solve this unit conversion problem, we multiply 5.0 miles by 1 kilometer / 0.62137 miles. The miles units in both the numerator and denominator cancel out, and we're left with just units of kilometers.
Practice Problems
Now let's practice a few unit conversion problems using conversion factors.
Practice Problem #1
Convert 45.0 inches to centimeters, given that 1 inch equals 2.54 centimeters.
To solve this problem, we start with 45.0 inches and multiply it by our conversion factor. We'll need to cancel out inches, and we want to introduce units of centimeters, so centimeters will need to go in the numerator of the conversion factor fraction, and inches will go in the denominator. Finally, we multiply the numbers to get our final answer.
Practice Problem #2
Using the conversion factor 1 liter per 1 kg of water, convert 6.0 liters of water to kilograms of water.
To solve this problem, we start with the initial measurement of 6.0 liters and multiply it by our conversion factor. We intend to cancel out units of liters and introduce units of kilograms, so we'll put kilograms in the numerator and liters in the denominator of our conversion factor fraction. Multiplying across, we'll find that the value is the same in this case.
Practice Problem #3
Use two conversion factors (60 seconds per minute and 60 minutes per hour) to convert 18,000 seconds to hours.
Starting with 18,000 seconds, we'll need to use a conversion factor to cancel out the original units of seconds, so we multiply the original time by 1 minute / 60 seconds. Our remaining units at this point are minutes, which are not the final units we are looking for in this problem, so we'll need to use another conversion factor. To cancel out minutes and introduce units of hours, we put hours in the numerator and minutes in the denominator (1 hour / 60 minutes). Multiplying across, we should anticipate a much smaller numerical value than our original value because we are moving to a bigger unit of time.
Lesson Summary
A conversion factor is a ratio expressed as a fraction that equals 1. When we use conversion factors to convert units, we multiply our original measurement by the conversion factor to get the same measurements expressed in new units. We need to be careful to set up our multiplication problem correctly so that we cancel out our old units and introduce the new units in the numerator of the conversion factor fraction. We use conversion factors every day, so it is important to understand this concept.
Definitions to Remember
- Conversion factor: a ratio expressed as a fraction that equals 1
- Original measurement: what the conversion factor needs to be changed to, i.e., measurement, weight, etc.
- Final units: the answer arrived at after the conversion in new units, i.e., feet to meters, kilograms to pounds
Learning Outcome
After concluding this lesson, you might understand what conversion factors are and how to use various formulas to find solutions.
