7 Canceling Units
When using mathematical equations in chemistry, the numbers you plug into the equation will also have units. The units follow the same math as the numbers and cancel accordingly. It’s important to include the units with the numbers you put into an equation and follow through with canceling them, because that is how you know whether or not you need to the change units of the given quantities you are plugging into the equation. You won’t be told ahead of time if you need to change units, you will discover it during the unit cancellation process.
Example 8.1
The density of honey is 1.42 g/mL. What is the mass of 1.45 L of honey?
Answer: Solving the definition of density [latex]density=\frac{mass}{volume}[/latex] for mass gives
[latex]mass=density*volume[/latex]
Plugging in the numbers and their units gives
[latex]mass=\frac{1.42g}{mL}*1.45L[/latex]
The answer from multiplying these numbers is not correct because the units would be g·L/mL, which is not a unit of mass. Notice that mL in the denominator does not cancel with L in the numerator. They are both volume, but because they aren’t in the same units, they don’t cancel. You can go back and change one of the quantities so that both have the same units (it doesn’t matter if you put them both in mL or both in L) and redo the calculation, or you can just multiply by the appropriate unit conversion during the calculation
[latex]mass=\frac{1.42g}{mL}*1.45L*\frac{1000mL}{1L}=2060g[/latex]
Now the mL and L cancel, and the only unit that remains is g, which is a unit of mass.
Following through with the cancellation of units will also help you catch any algebra errors that you have made, as shown in the following example.
Example 8.2
The density of honey is 1.42 g/mL. What is the volume of 32.6 g of honey?
Answer with Algebra Error: Suppose that when you solve the definition of density [latex]density=\frac{mass}{volume}[/latex] for volume, you made an algebra error and came up with [latex]volume=\frac{density}{mass}[/latex], which is wrong. You would catch this error when plugging in the numbers if you follow through with the units, since the units in the answer aren’t volume units.
[latex]volume=\frac{1.42g/mL}{32.6g}=0.436\frac{g/mL}{g}[/latex]
Remembering that dividing is the same as multiplying by the reciprocal, the units simplify to
[latex]\frac{g}{mL}*\frac{1}{g}=\frac{1}{mL}[/latex]
Notice that the ending units are 1/mL. This is not volume units, they are 1/volume units, which is not what we want. Taking the time to simplify the units on the answer and noticing that they weren’t right revealed that something was wrong with the calculation.
Correct Answer: Solving density for volume correctly gives
[latex]volume=\frac{mass}{density}=\frac{32.6g}{1.42g/mL}=23.0\frac{g}{g/mL}[/latex]
Remembering that dividing is the same as multiplying by the reciprocal, the units simplify to
[latex]\frac{g}{g/mL}=g*\frac{mL}{g}=mL[/latex]
Since mL is a volume unit, the correct answer is 23.0 mL.
