Bulk modulus and compressibility

Bulk modulus B = −p/(ΔV/V): hydraulic stress over volume strain. The minus sign records that raising the pressure lowers the volume, so B itself is positive for a body in equilibrium.

B has the unit of pressure, N m⁻² or Pa. Bulk modulus applies to solids, liquids and gases alike.

Compressibility k = 1/B = −(1/Δp)(ΔV/V): the fractional change in volume per unit rise in pressure.

Typical B values (GPa). Solids: nickel 260, steel 160, copper 140, iron 100, aluminium 72, brass 61, glass 37. Liquids: mercury 25, glycerine 4.76, water 2.2, carbon disulphide 1.56, ethanol 0.9. Air at STP: just 1.0 × 10⁻⁴.

Solids have much larger B than liquids, and liquids much larger than gases. Gases are roughly a million times more compressible than solids, and their compressibility changes with pressure and temperature.

The reason is coupling between neighbours: atoms in a solid are tightly bound, molecules in a liquid less so, and molecules in a gas hardly at all.

Indian Ocean, average depth about 3000 m, g taken as 10 m s⁻²: p = hρg = 3 × 10⁷ N m⁻², so ΔV/V = p/B = 3 × 10⁷/2.2 × 10⁹ = 1.36 × 10⁻² or 1.36% for water at the bottom.

Summary of moduli: Y for tensile or compressive stress (shape changes, solids only); G for shear (shape changes, solids only); B for hydraulic stress (volume changes, shape does not; solids, liquids and gases).