## Thursday, September 12, 2013

### Density Problems and Conversions Primer

The density formula Density = mass/ volume might be simple but once one starts switching up units, things can get very complicated quite quickly.

How do you know when to change units and what to change them to? Which units are which?

Some examples of mass units are grams (g) and kilograms (kg).
Some examples of volume units are cm3, mL, and L.
Density must have a mass component and a volume component as part of its unit. So things like g/mL, g/cm3 , kg/L. These are all examples of density units.

If a density problem does not ask you for a specific unit type, then you are free to choose which unit you would like to report your answer in. The problem itself will usually indicate when a unit change needs to take place.

Example:

If the density of a gas is 3.12 g/L, what is the mass of 648 cm3 of the gas?
cm^3 and L are different units of volume...therefore, they will not cancel one another.
You need to do a conversion to get them to match (be the same thing) so they will cancel. The "easier" conversion in this case would be to take 648 cm^3 and convert to L. "Fencepost" conversions illustrate this concept nicely, though they're a little hard to describe in words.

Take the 648 cm^3 and draw a line under it. Let the line spill over to the right a bit and put a vertical line down next to it (like a big plus sign with the starting value in the top left quadrant). In the bottom right quadrant, write the value you would like to cancel and the appropriate conversion value. In the top right quadrant, write the equivalent value of the unit you would like to convert to. 1 cm^3 = 1 mL. So, if I put 1 cm^3 in the bottom right quadrant and 1 mL in the top, I have now switched units to mL. Since 1000 mL = 1 L, I can now set up a second set of fencepost conversions. Continue your horizontal line right and draw another vertical hatch mark, creating another "+". In the bottom right quadrant, write 1000 mL (since this is the unit we would like to get rid of). In the top right, write 1 L.

Multiply across the top and divide across the bottom. In other words, in this example, you end up dividing by 1000 to get .648 L.

The density formula was D = m/v, but if I'm solving for mass, I need to use this version of the formula:
mass = density x volume

Now, plug in: mass =  (3.12 g/L) (.648 L) = 2.02 g

The units in the problem drove our conversions. I needed the volumes to match, so I forced cm^3 to become L.

If I had a density problem with a mass in kg and a volume in L, then my end unit would be kg/L. I could keep that unit unless the problem specified that I needed to report the answer in a particular unit.