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Legend has it that Archimedes solved a problem for a king by jumping into a bath. The king suspected that his new gold crown had been secretly mixed with cheaper silver. Archimedes realized he could find the crown’s volume by how much water it displaced, then compare its mass to its volume. Pure gold packs more mass into the same space than a gold-silver mix.
Whether or not the bath story is true, the idea is right: density is a fingerprint for a substance.
The formula
density = mass ÷ volume, often written ρ = m ÷ V
Rearranged:
- mass = density × volume
- volume = mass ÷ density
Units
| Situation | Usual unit |
|---|---|
| Solids and liquids | g/cm³ (= g/mL) |
| Gases | g/L |
| SI unit | kg/m³ |
Useful equivalences: 1 cm³ = 1 mL exactly, and 1 g/cm³ = 1000 kg/m³ = 1000 g/L. Water’s density is very close to 1.00 g/mL at room temperature, which makes it a handy reference point.
Worked example 1: find the density
A metal block has a mass of 157.4 g and measures 2.00 cm × 2.00 cm × 5.00 cm.
- Volume: 2.00 × 2.00 × 5.00 = 20.0 cm³
- Density: 157.4 ÷ 20.0 = 7.87 g/cm³
That matches iron (7.874 g/cm³ on iron’s element page), so the block is probably iron or mild steel.
Worked example 2: find the mass
What is the mass of 250 mL of ethanol (density 0.789 g/mL)?
mass = 0.789 × 250 = 197 g
A 250 mL measuring cylinder of ethanol weighs noticeably less than one of water (250 g), because ethanol is less dense.
Worked example 3: irregular objects
For something with no neat shape, measure volume by displacement, exactly like Archimedes.
A stone is dropped into a measuring cylinder, raising the water level from 50.0 mL to 63.5 mL. The stone’s mass is 35.6 g.
- Volume: 63.5 − 50.0 = 13.5 mL
- Density: 35.6 ÷ 13.5 = 2.64 g/mL — typical of quartz-rich rock like granite.
Why things float
An object floats in a liquid if its average density is lower than the liquid’s. Ice (0.917 g/cm³) floats on water (1.00 g/cm³), which is actually unusual: most substances are denser as solids than as liquids. Water’s odd behaviour comes from hydrogen bonding, which holds ice molecules in an open, spacious lattice.
A steel ship floats because its average density, including all the air inside the hull, is less than water’s.
Density of gases
Gases are around a thousand times less dense than liquids, which is why gas densities are usually given in g/L. Combining density with the ideal gas law gives a neat formula:
ρ = PM ÷ RT
where M is molar mass. At the same temperature and pressure, a gas’s density is proportional to its molar mass. That explains a lot:
- Helium (4 g/mol) is much less dense than air (average about 29 g/mol), so helium balloons rise.
- Carbon dioxide (44 g/mol) is denser than air, so it pools in low places — which is why CO₂ fire extinguishers smother flames, and why CO₂ can collect dangerously in cellars and pits.
Density of CO₂ at 0 °C and 1 atm: ρ = (1.00 × 44.01) ÷ (0.082 06 × 273.15) = 1.96 g/L
The densest and least dense elements
Among elements, densities range enormously. Hydrogen gas at 0 °C and 1 atm is about 0.09 g/L. At the other end, osmium and iridium are both around 22.5 g/cm³ — a single cubic centimetre weighs about as much as three one-euro coins. We go into the rankings in what is the densest element. And you can see densities for every element at once in the density trend map.
Density changes with temperature
Most substances expand when heated, so their density falls. That’s why hot air rises, and why careful density measurements always state the temperature. Water is again the odd one out: it’s densest at about 4 °C, which is why deep lakes stay at 4 °C at the bottom and freeze from the top down, protecting the life below.
Quick answers
Is density the same as weight? No. A tonne of feathers and a tonne of lead weigh the same; lead is far denser, so the tonne of lead takes up much less space.
Does cutting a block in half change its density? No. Density is an intensive property — it doesn’t depend on how much you have.
What’s specific gravity? The ratio of a substance’s density to water’s density. Because water is about 1 g/mL, specific gravity is numerically almost the same as density in g/mL, but it has no units.
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