Bulk Modulus - Thermodynamic Relation

Thermodynamic Relation

Strictly speaking, the bulk modulus is a thermodynamic quantity, and in order to specify a bulk modulus it is necessary to specify how the temperature varies during compression: constant-temperature (isothermal ), constant-entropy (adiabatic ), and other variations are possible. Such distinctions are especially relevant for gases.

For an ideal gas, the adiabatic bulk modulus is given by

and the isothermal bulk modulus is given by

where

γ is the adiabatic index, sometimes called κ.

P is the pressure.

When the gas is not ideal, these equations give only an approximation of the bulk modulus. In a fluid, the bulk modulus K and the density ρ determine the speed of sound c (pressure waves), according to the Newton-Laplace formula

In solids, and have very similar values. Solids can also sustain transverse waves: for these materials one additional elastic modulus, for example the shear modulus, is needed to determine wave speeds.