Isothermal compressibility

The isothermal compressibility is a response function describing how a thermodynamic system in equilibrium changes its volume when the pressure is varied while keeping the temperature fixed (and keeping composition fixed, e.g. fixed particle number $N$).

It is defined by

The minus sign is conventional: for ordinary stable matter, increasing $p$ decreases $V$, so $(\partial V/\partial p)_{T,N}<0$ and $\kappa_T>0$.

Physical interpretation

For a small, quasistatic isothermal change (the system is kept at fixed $T$, e.g. by contact with a thermal reservoir ), the definition implies

Thus $\kappa_T$ is the fractional volume decrease per unit pressure increase under isothermal conditions. Its inverse, often called the (isothermal) bulk modulus, quantifies mechanical stiffness.

Key relations and properties

• Stability sign: In stable single-phase equilibrium, one requires $\kappa_T>0$; a negative value signals mechanical instability and is closely tied to thermodynamic stability .

• Link to other response functions: For a simple compressible system,

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

  relating $\kappa_T$ to the constant-pressure heat capacity $C_P$, the constant-volume heat capacity $C_V$, and the thermal expansion coefficient $\alpha$.

• Comparison with adiabatic compressibility: The adiabatic (isentropic) compressibility satisfies

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

  so typically $\kappa_S \le \kappa_T$ when $C_P \ge C_V$.

• Fluctuation formula (statistical mechanics): In an isothermal–isobaric setting (fixed $T$, $p$, $N$), $\kappa_T$ can be expressed through the variance of volume fluctuations:

  $$\kappa_T = \frac{\langle (\Delta V)^2\rangle}{k_B\,T\,\langle V\rangle},$$

  where $k_B$ is the Boltzmann constant and $\langle\cdot\rangle$ denotes an ensemble expectation .