Lesson 1: Fluids and Density

1.1 What is a fluid?

A fluid is any material that has the ability to flow and has no shape of its own. Fluids include liquids and gases.

Because liquids are practically incompressible, we treat liquid flow as incompressible laminar flow: the density of the moving liquid does not change.

1.2 Density (ρ)

Density is a basic property of every material: it is the mass of a unit volume of the substance.

ρ = m / Vol

• SI unit: kg/m³ (also g/cm³; 1 g/cm³ = 1000 kg/m³).
• Dimensional formula: ML⁻³.

Why does density differ from one material to another?

1. The atoms or molecules have different masses.
2. The spaces between the atoms or molecules (inter-atomic or intermolecular spaces) differ. (In gases the spaces are huge, so gas densities are small.)

Denser materials sink; objects of lower density float on liquids of higher density.

Typical values: water 1000 kg/m³, sea water ≈ 1030, oil ≈ 800–900, mercury 13 600, aluminium 2700, iron ≈ 7800, gold 19 320, air at room temperature ≈ 1.2 kg/m³.

1.3 Relative density

Relative density of a material = (density of the material at a given temperature) ÷ (density of water at 4 °C).

relative density = ρ_material / ρ_water(4 °C)

Since the density of water at 4 °C is 1000 kg/m³ (1 g/cm³),

relative density = mass of a certain volume of the material ÷ mass of the same volume of water at 4 °C.

• It is a ratio of two quantities of the same kind → it has no unit (and no dimensions).
• Its numerical value equals the density in g/cm³.

1.4 Worked examples

Example 1. A liquid of volume 250 cm³ has a mass of 200 g. ρ = 200/250 = 0.8 g/cm³ = 800 kg/m³; relative density = 0.8.

Example 2. Find the mass of 3 litres of mercury (ρ = 13 600 kg/m³). Vol = 3 L = 3 × 10⁻³ m³; m = ρVol = 13 600 × 3 × 10⁻³ = 40.8 kg.

Example 3. An iron block has dimensions 10 cm × 5 cm × 4 cm and mass 1.56 kg. Vol = 200 cm³ = 2 × 10⁻⁴ m³; ρ = 1.56/(2 × 10⁻⁴) = 7800 kg/m³; relative density = 7.8.

Example 4. The relative density of a certain oil is 0.9. Its density is 0.9 × 1000 = 900 kg/m³, and 2 m³ of it has mass 1800 kg.

1.5 Applications of density

Measuring density is an important analytical technique in science and in practical life.

1. Charge state of a car battery. The electrolyte is dilute sulphuric acid. When the battery discharges, sulphuric acid is used up in reacting with the lead plates to form lead sulphate, so the density of the electrolyte decreases. When it is recharged, sulphate ions return to the solution from the plates, so the density increases. Measuring the density of the electrolyte (with a hydrometer) therefore tells us how charged the battery is.
2. Medicine.
   • Blood density normally lies between 1040 and 1060 kg/m³. If it is higher, the concentration of blood cells is higher; if it is lower, the concentration of blood cells is lower, which may indicate anaemia.
   • Urine density is normally about 1020 kg/m³. Some diseases increase the amount of salts excreted, so the density of the urine increases.
3. Gold jewellery (critical thinking). Pure (24-carat) gold has a density of about 19.32 g/cm³; 18-carat gold about 15.45 g/cm³ and 14-carat gold about 13.1 g/cm³. Alloys differ in density because gold is mixed with lighter metals (such as copper or silver), which lowers the average density. By measuring the density of a piece of jewellery (mass ÷ volume) we can estimate its carat value.