# Density -- Solids

#### Prerequisite

There’s no way around it.  Matter takes up space and has mass.  How much depends on two key facts.  First, it depends on how big the material’s molecules are.  Second, it depends on how tightly these molecules pack together.

If our matter is uniform, then there is something about the mass that is a property of the material of which the solid is made — not of the particular block of the solid that we have. Our sense of "the mass depends on how big the molecules are and how tightly they pack together" can be summarized in a number — the density. It basically tells how much mass is contained in a small standardized volume (that contains enough molecules that we can treat the matter as smooth).

At room temperature and atmospheric pressure, a solid will have a particular density. This density is defined as the mass per unit volume of the material.  It can be determined by dividing the total mass, $M$, of the object by its total volume, $V$:

$$\rho = M/V$$

So for a rectangular block, we can weigh it to determine its mass (remembering to divide by $g$ if our scale that we use to weight the object gives us a force — pounds or Newtons) and calculate its volume from its dimensions: length ($L$), width ($W$) and height ($H$), so that

$$V = L \times W \times H$$

We should note that this block is at atmospheric pressure. This means that there is 105 Pa of air pushing on each of the surfaces of the block.  At this pressure, the block takes up the volume $V$.

Even for solids, the density of different materials varies by orders of magnitude:

• gold = 1.9 x 104 kg/m3,
• steel = 7.8 x 103 kg/m3,
• Styrofoam = 3.3 x 102 kg/m3

Fluids have a large range of densities as well:

• mercury =1.36 x 104 kg/m3,
• water = 103 kg/m3 — one metric ton per cubic meter.

Air has a density of 1000 times less than that of water — only about 1.2 kg/m3 at standard temperature and pressure — 105 Pa and 20°C.)

The densities of parts of many biological organisms are closer to that of liquid water (1000 kg/m3). However, many organisms do have key components that are solids, designed to provide strength. For vertebrates, these components are the bones. Bones are comprised of collagen fibers with various minerals deposited on or around them. Bones are typically either woven, with the collagen fibers randomly ordered, or lamellar where the collagen fibers have preferred orientations. The mineral deposited is calcium phosphate which has a density around 2000 kg/m3. In hard-shelled invertebrates, the shells are made of calcium carbonate, which has a density of 2700 kg/m3,  almost 3 times the density of water.  Some insects rely on an exoskeleton or cuticle, which is layered sheets composed of fibers in a matrix.  In flying insects, this protective layer is designed to be strong, yet light, with a density of 1200-1300 kg m-3.  Some of the least dense materials are found in plants.  Wood has quite a low density, being only 400-500 kg m-3 -- less than that of water (so it floats).

Karen Carleton and Joe Redish 10/20/11

Article 403