How To Average Out Density

​Density​ in physics is a measure of the amount of something that exists within a given physical space (volume). Most of the time, "density" is taken by convention to mean "mass density," but as a concept it simply describes how crowded something is.

The population density of Hong Kong, for example, is extremely high, whereas that of Siberia is extremely low. But in each case, "people" is the subject of analysis.

For substances consisting of a single element in some quantity (for example, a gram of pure gold or silver) or a homogeneous blend of elements (such as a liter of distilled water, which includes hydrogen and oxygen in a known, fixed ratio), it can be assumed that there are no meaningful variations in density within the sample.

That means that if the density of a 60-kg homogeneous object in front of you is 12 kg/L, any selected small portion of the object should have this value for its density.

Density is assigned the Greek letter rho (ρ) and is simply mass ​m​ divided by volume ​V​. The SI units are kg/m³, but g/mL or g/cc (1 mL = 1 cc) are more common units in lab settings. These units were in fact chosen to define the density of water as 1.0 at room temperature.

•​Density of everyday materials:​ Gold, as you might expect, has a very high density (19.3 g/cc). Sodium chloride (table salt) checks in at 2.16 g/cc. 

Depending on the type of substance or substances present, there are a number of ways to approach a density mixture problem.

The simplest is when you are given a set of N objects and asked to determine the average density of the objects in the set. This kind of example would arise in situations in which the elements in the set are of the same basic "type" (e.g., people in England, trees in a given forest in Montana, books in a town library in Tennessee) but can very greatly in the characteristic in question (e.g., weight, age, number of pages).

​EXAMPLE:​ You are given three blocks of unknown composition, which have the following masses and volumes:

•Rock A: 2,250 g, 0.75 L
•Rock B: 900 g, 0.50 L
•Rock C: 1,850 g, 0.50 L

​a) Calculate the average of the densities of the rocks in the set.​

This is done by figuring out the individual densities of each rock, adding these together and dividing by the total number of rocks in the set:

\(\frac{(2,250/0.75) + (900/0.50) + (1,650/ 0.60)}{3} = \frac{(3,000 + 1,800 + 3,700)}{3}=2833\text{ g/L}\)

​b) Calculate the average density of the set of rocks as a whole.​

In this case you just divide the total mass by the total volume:

\(\frac{(2,250 + 900 + 1,850) }{ (0.75 +0.50 + 0.50)}=\frac{5,000 }{ 1.75}= 2857\frac{ g/cc}\)

The numbers differ because the rocks do not contribute in equal ways to these calculations.

​EXAMPLE:​ You are given a 5-L (5,000 cc or mL) chunk of material from another planet and told it consists of three fused pieces of the following elements in the listed proportions by volume:

•Thickium (ρ = 15 g/mL): 15%
•Waterium (ρ = 1 g/mL): 60%
•Thinnium (ρ = 0.5 g/mL): 25%

What is the density of the chunk as a whole?

Here, you first convert the percentages to decimals, and multiply these by the individual densities to get the average density of the mixture:

\((0.15)(15) + (0.60)(1.0) + (0.25)(0.50) = 2.975\text{ g/cc}\)