Positivity of isothermal compressibility from stability

Statement

For a thermodynamically stable equilibrium system at fixed temperature and composition, the isothermal compressibility

satisfies

Equivalently,

Key hypotheses

• Equilibrium description in terms of the Helmholtz free energy $F(T,V,N)$.
• $F$ is twice differentiable in $V$ at fixed $T,N$.
• Mechanical stability (a component of thermodynamic stability ), expressed as convexity of $F$ in $V$ at fixed $T,N$: $\left(\frac{\partial^2 F}{\partial V^2}\right)_{T,N} \ge 0.$

Key conclusions

• Since pressure is given by

  $$P = -\left(\frac{\partial F}{\partial V}\right)_{T,N},$$

  one has

  $$\left(\frac{\partial P}{\partial V}\right)_{T,N} ={} -\left(\frac{\partial^2 F}{\partial V^2}\right)_{T,N} \le 0,$$

  and therefore $\kappa_T \ge 0$.

• At phase coexistence, $\left(\frac{\partial P}{\partial V}\right)_{T,N}$ can approach $0$, leading to very large (formally divergent) $\kappa_T$, consistent with $\kappa_T\ge 0$.

Cross-links to definitions

• isothermal compressibility $\kappa_T$
• Helmholtz free energy
• pressure , temperature
• thermodynamic stability

Proof idea / significance

Differentiate the defining relation $P=-(\partial F/\partial V)_{T,N}$ with respect to $V$ at fixed $T,N$:

Stability gives $(\partial^2 F/\partial V^2)_{T,N}\ge 0$, hence $(\partial P/\partial V)_{T,N}\le 0$. Inverting the derivative yields $\kappa_T \ge 0$.

Significance: $\kappa_T\ge 0$ is the condition that small compressions raise the pressure (restoring force), preventing runaway mechanical collapse or expansion in equilibrium.