Adiabatic compressibility

The adiabatic compressibility (often called the isentropic compressibility) is a response function that measures how the volume of a thermodynamic system responds to changes in pressure when the entropy is held fixed.

It is defined by

with composition fixed (e.g. fixed particle number $N$ for a single-component system).

Physical interpretation

An entropy constraint is appropriate for a reversible process with no heat exchange (an idealized “adiabatic” change, e.g. via an adiabatic wall ). For a small quasistatic isentropic compression,

Because an isentropic compression generally heats the system, the material typically resists compression more strongly than in an isothermal process, so one often finds $\kappa_S < \kappa_T$, where $\kappa_T$ is the isothermal compressibility .

A common physical setting is acoustics: small-amplitude sound waves in fluids are well-approximated as isentropic, and $\kappa_S$ controls the relation between pressure and density changes.

Key relations and properties

• Relation to isothermal compressibility and heat capacities: For a simple compressible system,

  $$\kappa_S = \kappa_T\,\frac{C_V}{C_P},$$

  where $C_V$ and $C_P$ are the constant-volume and constant-pressure heat capacities, respectively.

• Difference between compressibilities: An equivalent identity is

  $$\kappa_T - \kappa_S = \frac{T\,V\,\alpha^2}{C_P},$$

  involving the thermal expansion coefficient $\alpha$.

• Stability sign: In stable single-phase equilibrium, one expects $\kappa_S>0$; violations are linked to thermodynamic stability .

• Connection to sound speed (common form): Writing $\rho$ for the mass density, the isentropic stiffness appears in

  $$\left(\frac{\partial p}{\partial \rho}\right)_S = \frac{1}{\rho\,\kappa_S},$$

  which underlies the standard expression for the adiabatic speed of sound in a fluid.