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## Density via Displacement

If you completed Chapter 3 of Middle School Chemistry, you learned that density is a characteristic property of a substance that relates its mass (m) to the amount of space it takes up, also called the volume (v).  You can also express this relationship mathematically, using the following equation:

D=m/v

This equation tells us that the density of a substance is equal to its mass divided by its volume.

One of the main ways that we commonly encounter density is when trying to predict if a substance will sink or float in water.  Substances that are less dense than water will float, while substances that are more dense than water will sink.

By redshirtjosh on Flickr.

Of course, to calculate density, you need to calculate volume.  And one way you can calculate volume is through the displacement method.

Defining Displacement

The displacement method involves putting an object into water and carefully recording how much the water level rises.  The amount that the water volume rises is equal to the volume of the object.

This method was discovered by the ancient Greek mathematician and scientist Archimedes.  According to ancient texts, Archimedes was inspired to discover the displacement method of calculating volume, and thus, density, by a question posed to him by the King.  The King had given a local goldsmith some gold, so that he might fashion the King a crown out of it.  The King was concerned that the goldsmith was dishonest, and wanted to know if he had mixed some silver in with the gold, keeping the extra gold for himself.  The King asked Archimedes to determine if the crown was pure gold.

After contemplating how he might solve the problem for a great while, Archimedes is reported to have had a moment of inspiration when taking a bath.  As he lowered himself into the bathtub, he noticed that the water level rose.

By Sputnik Beanburger III on Flickr.

From there, he quickly realized that he could submerge the crown in water to determine its volume, and ultimately, its density.  By comparing the density of the crown to the density of a sample of pure gold, he would be able to discern if the crown were composed entirely of pure gold.

Here’s how you can visualize displacement: imagine putting a rock into a cup of water.  When you do so, the rock actually displaces a volume of water equal to its own volume.  You may have learned that you can use this method to find the volume of a substance in Chapter 3, Lesson 2 of Middle School Chemistry.  When the rock pushes the water out of the way, the water level in the cup has to increase, because the water has nowhere to go but up.

This way of measuring density can be pretty handy when you have an irregular object with imperfect edges or a complicated shape, like a key or a tree branch.  This is because in order to find the density of these complicated shapes, you would first need to find the volume.  This is difficult to do when it’s hard to measure the dimensions of the irregularly shaped object.

The limits of volume equations

Imagine, for example, that your rock was a perfect cube.  If so, then finding the volume would be easy.  You could simply measure the length of one side and use the equation:

V=side3

By Joe Shlabotnik on Flickr.

This mathematical equation says that the volume of a cube is equal to one of its sides cubed (how appropriate!).  This equation expresses the same idea:

V=side x side x side

The exponent 3 in the first equation is a mathematical operation that means you multiply the number by itself three times, which is exactly what we’ve shown above.  Let’s suppose we had a cube with an edge of 2cm.  Using equations above, we would calculate the volume of the cube to be:

V = 2cm x 2cm x 2cm = 8cm3

So, once you know the volume of your cube, you would just weigh the cube using a scale to find the mass (m), and then you could find the density using D=m/v.

And this is really great science stuff, but what about when your sample isn’t a cube?  What if it’s a mint condition 1959 Les Paul Standard electric guitar, or a icosahedron?  Although you could very carefully find the volumes of both of those items, it might just be easier to use the displacement method, at least to get a rough idea of the density of the object, assuming that you had a big enough volume of water and that you could accurately measure any increase.

Get Your Archimedes On

To get a good idea for how accurate the displacement method can be, you can try using the displacement method on some objects whose volume you already know.  Then you can compare your volume calculations from displacement to your volume calculations from equations and measurement.

By Waifer X on Flickr.

Were the results similar or far apart?  Does Archimedes method work well?

Adam M. Boyd is a Sr. Education Associate in the office of K–8 science at the American Chemical Society. In his work at ACS, Adam focuses on chemistry, writing, and the web.