2.4: Converting Units

 Learning Objective

• Convert from one unit to another unit of the same type.

You may already know how to convert common units that you use in your everyday life. Continue to use that process when it works for you! However, in this course you may encounter unfamiliar problems that are more difficult to figure out. In that case, I recommend using the unit analysis method which will be demonstrated here. 

Consider a simple example: how many feet are there in 4 yards? Many people can answer that there are 12 feet in 4 yards. If there are 3 feet in 1 yard, and there are 4 yards, then there are 4 × 3 = 12 feet in 4 yards.

To solve a problem like this one using unit analysis, we use equalities which are pairs of values that are equal to one another. In this case, 1 yard (yd) equals 3 feet (ft):

\[1\, yd = 3\, ft\nonumber \]

Conversion Factors

In the unit analysis problem solving method we will use equalities as conversion factors by rewriting them as fractions equal to one. For example:

\[1= \dfrac{3\,ft}{1\,yd}\nonumber \] or \[1= \dfrac{1\,yd}{3\,ft}\nonumber \]

This is a strange way to write 1, but it makes sense: 3 ft equal 1 yd, so the quantities in the numerator and denominator are the same quantity, just expressed with different units.

Unit Analysis

To use the unit analysis method, start with the original quantity. This will generally be a measurement with a simple (not compound) unit that is not already part of a relationship or equality. In the problem above our original quantity was 4 yd.

Next, multiply this quantity by a conversion factor that is equal to 1 (so that it does not change the value of the quantity). Remember that conversion factors are fractions made up of two values that are equal. Eventually we want the unit of yards to divide out leaving only a unit of feet for the answer, so we set up the conversion factor so that yards is on the bottom where it will cancel the original unit and feet is on the top. That will become the unit for the answer. 

\[4\,yd\times \dfrac{3\,ft}{1\,yd}\nonumber \]

You'll follow this pattern when solving other unit analysis problems in the future. The unit for the original value will go on the bottom of the conversion factor and the desired unit will go on the top of the conversion factor.

The 4 yd term can be thought of as 4yd/1; that is, it can be thought of as a fraction with 1 in the denominator. We are essentially multiplying fractions. If the same thing appears in the numerator and denominator of a fraction, they cancel. (You may have seen this with variables in algebra where an x that appears on both the top and bottom of a fraction divides out. It is the same with units). In this case, what cancels is the unit yard:

\[4\,yd\times \dfrac{3\,ft}{1\,yd}\nonumber \]

That is all that we can cancel. Now, multiply the fractions to get the final answer. To multiply fractions start with the number at the far left. Multiply by any values on the top of fractions and divide by any values on the bottom of fractions:

\[\dfrac{4\times 3\, ft}{1}= \dfrac{12\,ft}{1}= 12\,ft\nonumber \]

Again, we get an answer of 12 ft, just as we did originally. But in this case, we used a more formal procedure that is applicable to a variety of problems.

How many millimeters are in 14.66 m? To answer this, we need to construct a conversion factor between millimeters and meters and apply it correctly to the original quantity. We start with the definition of a millimeter which is:

\[1000\,mm=1\,m\nonumber \]

Now we construct a conversion factor by dividing one quantity into both sides. Can you figureout which version of the conversion factor is needed for this problem?

\[1=\dfrac{1\,m}{1000\,mm}\nonumber \]

or

\[\dfrac{1000\,mm}{1\,m}=1\nonumber \]

The answer is based on what unit you want to get rid of in your initial quantity. The original unit of our quantity is meters, which we want to convert to millimeters. Because the original unit is assumed to be in the numerator, to get rid of it, we want the meter unit in the denominator; then they will cancel. Therefore, we will use the second conversion factor. Canceling units and performing the mathematics, we get:

\[14.66m\times \dfrac{1000\,mm}{1\,m}= 14660\,mm\nonumber \]

Note how \(m\) cancels, leaving \(mm\), which is the unit of interest.

The ability to construct and apply proper conversion factors is a very powerful mathematical technique in chemistry. You need to master this technique if you are going to be successful in this and future courses.

Example \(\PageIndex{1}\)

1. Convert 35.9 kL to liters.
2. Convert 555 mm to meters.

Solution

1. We will use the fact that 1 kL = 1,000 L. Of the two conversion factors that can be defined, the one that will work is 1000L/ 1kL. Applying this conversion factor, we get:

\[35.9\, kL\times \dfrac{1000\,L}{1\,kL}= 35,900\, L \nonumber \nonumber \]

2. We will use the fact that 1,000 mm = 1 m. Of the two possible conversion factors, the appropriate one has the mm unit in the denominator:

\[\dfrac{1\,m}{1000\,mm} \nonumber \nonumber \]

Applying this conversion factor, we get:

\[555\,mm\times \dfrac{1m}{1000 mm}= 0.555\,m \nonumber \nonumber \]

Exercise \(\PageIndex{1}\)

1. Convert 67.08 μL to liters.
2. Convert 56.8 m to kilometers.

Answer a

6.708 × 10⁻⁵ L

Answer b

5.68 × 10⁻² km

Exercise \(\PageIndex{4}\)

How many milliliters are in 607.8 kL?

Answer

6.078 × 10⁸ mL

 

Chemistry is Everywhere: The Gimli Glider

On July 23, 1983, an Air Canada Boeing 767 jet had to glide to an emergency landing at Gimli Industrial Park Airport in Gimli, Manitoba, because it unexpectedly ran out of fuel during flight. There was no loss of life in the course of the emergency landing, only some minor injuries associated in part with the evacuation of the craft after landing. For the remainder of its operational life (the plane was retired in 2008), the aircraft was nicknamed "the Gimli Glider."

The Gimli Glider is the Boeing 767 that ran out of fuel and glided to safety at Gimli Airport. The aircraft ran out of fuel because of confusion over the units used to express the amount of fuel. Source: Photo courtesy of Will F., (CC BY-SA 2.5; Aero Icarus).

The 767 took off from Montreal on its way to Ottawa, ultimately heading for Edmonton, Canada. About halfway through the flight, all the engines on the plane began to shut down because of a lack of fuel. When the final engine cut off, all electricity (which was generated by the engines) was lost; the plane became, essentially, a powerless glider. Captain Robert Pearson was an experienced glider pilot, although he had never flown a glider the size of a 767. First Officer Maurice Quintal quickly determined that the aircraft would not be able make it to Winnipeg, the next large airport. He suggested his old Royal Air Force base at Gimli Station, one of whose runways was still being used as a community airport. Between the efforts of the pilots and the flight crew, they managed to get the airplane safely on the ground (although with buckled landing gear) and all passengers off safely.

What happened? At the time, Canada was transitioning from the older English system to the metric system. The Boeing 767s were the first aircraft whose gauges were calibrated in the metric system of units (liters and kilograms) rather than the English system of units (gallons and pounds). Thus, when the fuel gauge read 22,300, the gauge meant kilograms, but the ground crew mistakenly fueled the plane with 22,300 pounds of fuel. This ended up being just less than half of the fuel needed to make the trip, causing the engines to quit about halfway to Ottawa. Quick thinking and extraordinary skill saved the lives of 61 passengers and 8 crew members—an incident that would not have occurred if people were watching their units.

Key Takeaways

• Units can be converted to other units using the proper conversion factors.
• Conversion factors are constructed from equalities that relate two different units.
• Conversions can be a single step or multi-step.
• Unit conversion is a powerful mathematical technique in chemistry that must be mastered.
• Exact numbers do not affect the determination of significant figures.