Density

Divide the mass by the volume: ρ = m/V. Example: an aluminium cube with m = 270 g and V = 100 cm³ has ρ = 2.7 g/cm³, in SI units 2,700 kg/m³. You find the mass with a scale; the volume of regular bodies by measuring, of irregular ones via water displacement in a measuring cylinder, where the rise of the water level equals the body volume. Unit discipline matters: work consistently in grams with cubic centimetres or kilograms with cubic metres, never mixed. For conversion, 1 g/cm³ = 1,000 kg/m³. With the computed density you can then look up in a reference table which substance it probably is.

Because ice, at about 917 kg/m³, is less dense than liquid water at 1,000 kg/m³, a famous anomaly, since almost all substances are denser as solids than as liquids. On freezing, the water molecules arrange into an open lattice with lots of empty space, and the volume grows by about 9%. By the Archimedes principle a body floats if its density is lower than that of the liquid; ice is submerged to about 90% (iceberg effect: most of it lies under water). This anomaly is ecologically crucial: lakes freeze from the top, the ice layer insulates, and the water below stays liquid at 4 °C, letting fish survive winter.

Solved for the mass: m = ρ·V, which lets you estimate weights without a scale. A 200 litre aquarium holds m = 1,000 kg/m³ × 0.2 m³ = 200 kg of water, which is why you should never carry it filled. Solved for the volume: V = m/ρ, which gives the space required: 7.9 kg of steel with ρ = 7,900 kg/m³ fills exactly V = 0.001 m³, one litre. Density is the conversion factor between the worlds of mass and volume; craftsmen and engineers use it constantly (how heavy will the concrete slab be, how much does the bucket of paint weigh?). After every rearrangement check the units: kg/m³ times m³ gives kg, then the formula is right.

With the Archimedes displacement method. First weigh the body (mass m). Then fill a measuring cylinder with water, read the level, submerge the body completely and read again; the difference between the levels is its volume V. From this ρ = m/V follows. Example: a piece of rock weighs 156 g and raises the water level from 50 ml to 110 ml, so V = 60 cm³ and ρ = 156/60 = 2.6 g/cm³, typical of granite. Legend has it that Archimedes exposed an adulterated royal crown this way: gold has 19.3 g/cm³, and admixed silver measurably lowers the density. Requirements: the body must not soak up water and must submerge completely.

Because gas molecules, unlike those in solids and liquids, are far apart and easily compressed or expanded. If you raise the pressure, the particles move closer: same gas, smaller volume, higher density. If you warm the gas at constant pressure, it expands and the density falls, which is why warm air rises, the principle of the hot-air balloon. Quantitatively this sits in the ideal gas law: ρ = p·M/(R·T) with molar mass M. Air at 0 °C and normal pressure has about 1.29 kg/m³, at 20 °C only 1.20 kg/m³. That is why gas density figures always come with the conditions (pressure and temperature), whereas for solids the density is nearly constant.