Density

A pound of bricks or a pound of feathers?

It’s an old riddle: which is heavier? Neither. They both weigh a pound. So why does a brick feel “heavier” than feathers? Because for the same size, a brick has far more mass than a pile of feathers. To compare how “heavy” materials are, you have to compare equal volumes, and that’s what density does.

The density密度 of a substance is its mass divided by its volume: the mass of one unit of volume.

\rho = \frac{m}{V}

Here ρ (the Greek letter rho) is density, m is mass, and V is volume.

Common units in chemistry: g/cm³ for solids and liquids. Since 1 cm³ = 1 mL, 1 g/cm³ = 1 g/mL.
SI units: kg/m³. The conversion is 1 g/cm³ = 1000 kg/m³.
The density of water is 1.0 g/cm³: 1 mL of water has a mass of 1 g, and 1 L has a mass of 1 kg.

Substance	Density (g/cm³)	Substance	Density (g/cm³)
Gold	19.3	Ice	0.9
Lead	11.3	Water	1.0
Copper	8.9	Ethanol	0.8
Iron	7.9	Vegetable oil	0.9
Aluminum	2.7	Air	0.00129

Density is a physical property of a substance:

For a given substance (at the same state and temperature), density is fixed and doesn’t depend on the size of the sample. A big block of aluminum and a scrap of foil both have a density of 2.7 g/cm³.
Different substances usually have different densities, so density can identify a substance.
The density of a substance changes with its state and temperature. When water freezes, its mass stays the same but its density drops, so its volume grows. That’s why water pipes can burst in a hard freeze.

Rearranging the formula

If you know any two of the three quantities, you can find the third. The tool card shows how to rearrange the formula:

🔧 Math tool card · Rearranging FormulasSkip it if you already know this

The rule: multiplying (or dividing) both sides of an equation by the same nonzero number keeps it true.

Solving for m: multiply both sides of $\rho = \dfrac{m}{V}$ by V to get $m = \rho V$ .

Solving for V: divide both sides of $m = \rho V$ by ρ to get $V = \dfrac{m}{\rho}$ .

Check the units: for $V = \dfrac{m}{\rho}$ the units are $\dfrac{\text{g}}{\text{g/cm}^3} = \text{cm}^3$ , which is a volume, just as it should be.

Starting from $\rho = \dfrac{m}{V}$ , which expression gives the volume V?

Choose what to find and enter the other two values. The calculator shows how the formula is rearranged and the numbers substituted, and the beaker shows whether the object sinks or floats in water. You can also click a material to fill in its density:

Try it: choose what to find, then enter the other two values

Find

Common materials:

Density ρg/cm³Mass mgVolume Vcm³

Compared with water: objects denser than 1.0 g/cm³ sink, and objects less dense float. Ice is about 0.9 times as dense as water, which is why roughly 90% of an iceberg is hidden underwater.

Example 1: Identify a metal by water displacement

A piece of metal has a mass of 54 g. When it’s lowered into a graduated cylinder holding 30 mL of water, the water level rises to 50 mL. What is the metal’s density, and what metal might it be?

Step-by-step solution

Find the volume. The metal’s volume equals the volume of water it displaces: V = 50 mL − 30 mL = 20 mL = 20 cm³.
Find the density. $\rho = \dfrac{m}{V} = \dfrac{54\,\text{g}}{20\,\text{cm}^3} = 2.7\,\text{g/cm}^3$ .
Compare with the table. 2.7 g/cm³ matches aluminum, so the metal is probably aluminum.
Remember. Read the cylinder with your eye level with the bottom of the meniscus (lesson 1.1.3).

Example 2: Measuring out a mass of liquid

An experiment needs 40 g of ethanol (density 0.8 g/cm³). How many milliliters should you measure out with a graduated cylinder?

Step-by-step solution

Choose the formula. You know m and ρ and want V, so use $V = \dfrac{m}{\rho}$ .
Substitute. $V = \dfrac{40\,\text{g}}{0.8\,\text{g/cm}^3} = 50\,\text{cm}^3$ .
Convert units. 1 cm³ = 1 mL, so measure out 50 mL of ethanol.
Why bother? Liquids are awkward to weigh but easy to measure by volume, so labs often use density to turn a mass into a volume.

Practice: which formula does each problem need? The working appears after you answer.

A rock has a mass of 78 g and a volume of 30 cm³. Find its density.
A bottle holds at most 500 mL of water. What is the mass of the water when it's full?
A piece of iron has a mass of 158 g. Find its volume (density of iron: 7.9 g/cm³).
What is the mass of the oil in a 5 L jug of vegetable oil (density 0.9 g/cm³)?
You need 40 g of ethanol (density 0.8 g/cm³). How many milliliters should you measure?
A metal block has a mass of 89 g and a volume of 10 cm³. Which metal might it be?
A block of aluminum has a volume of 100 cm³. Find its mass (density of aluminum: 2.7 g/cm³).

Common mistakes

Mistake 1: “Density goes up with mass and down with volume.” For a given substance, doubling the mass also doubles the volume, so the ratio stays the same. ρ = m/V is how you calculate density, but the substance itself determines it.

Mistake 2: Converting units the wrong way. 1 g/cm³ = 1000 kg/m³, not 0.001 kg/m³. A quick check: 1 m³ of water has a mass of 1000 kg (one metric ton).

Mistake 3: “A substance’s density never changes.” Density changes with state and temperature. When water freezes, its density drops from 1.0 g/cm³ to about 0.9 g/cm³.

Quick check

An aluminum block with a density of 2.7 g/cm³ is cut in half. What is the density of each half?

An empty bottle has a mass of 200 g. Filled with water, it has a total mass of 700 g. What volume does the bottle hold?

Summary

Density ρ = m/V is the mass of one unit of volume. Common units are g/cm³; 1 g/cm³ = 1 g/mL = 1000 kg/m³. Water’s density is 1.0 g/cm³.
The formula rearranges to m = ρV and V = m/ρ, so any two quantities give you the third. Carry units through to check your work.
Density is a physical property that can identify a substance. It doesn’t depend on sample size, but it does change with state and temperature.
Objects denser than water sink in it; less dense objects float.

On to AP (for later)

In AP Chemistry, density is often the bridge in a calculation: converting a liquid’s volume into mass and then into moles (topics 1.1 and 4.5), converting between the mass and volume of a solution (topic 3.7), and, in topic 3.4, finding the density of a gas from the ideal gas law. AP expects you to show every step with units (dimensional analysis), and checking units as you substitute, as you did today, builds exactly that habit.

Density密度

Learning goals

Before you start

A pound of bricks or a pound of feathers?

Density

Rearranging the formula

Example 1: Identify a metal by water displacement

Example 2: Measuring out a mass of liquid

Common mistakes

Quick check

Summary

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