Why must heat capacity be positive for a stable phase?

If adding heat lowered a system's temperature, a small fluctuation would grow without bound rather than relax, so a stable equilibrium phase must have positive heat capacity; negative values signal an instability and the breakdown of that phase.

Response Functions and Stability

Response functions measure how a system reacts to changes in temperature, pressure, or field, and thermodynamic stability requires these responses to obey definite sign and convexity conditions.

Definition

Response functions are second derivatives of thermodynamic potentials that quantify how extensive variables change under variations of their conjugate intensive variables, and thermodynamic stability is the requirement that these functions satisfy the sign and convexity conditions implied by the minimization of the potentials.

Scope

This topic covers the thermodynamic response functions -- heat capacities at constant volume and pressure, isothermal and adiabatic compressibilities, and thermal expansivity -- their interrelations, and the stability conditions that follow from the convexity of the thermodynamic potentials. The role of positivity of heat capacity and compressibility in guaranteeing equilibrium against fluctuations is included; the divergence of these quantities at critical points is treated under critical phenomena.

Core questions

How are heat capacities, compressibilities, and expansivity defined as derivatives of potentials?
Why must heat capacity and compressibility be positive in a stable phase?
How do convexity properties of the potentials encode thermodynamic stability?
What relations connect the different response functions to one another?

Key concepts

Heat capacities at constant volume and pressure
Isothermal and adiabatic compressibility
Thermal expansion coefficient
Convexity of thermodynamic potentials
Stability conditions and fluctuations

Key theories

Thermodynamic stability conditions: Equilibrium against fluctuations requires the relevant thermodynamic potential to be a convex or concave function of its natural variables, which translates into positivity of the heat capacity and compressibility.

Clinical relevance

Response functions are directly measured in calorimetry and acoustics, characterize material behavior in engineering and geophysics, and their anomalies signal the approach to phase transitions and instabilities.

History

The stability theory of thermodynamics grew out of Gibbs's analysis of equilibrium and was given its modern convexity-based formulation in twentieth-century treatments, connecting measurable response functions to the curvature of the potentials.

Key figures

J. Willard Gibbs
Herbert Callen

Seminal works

callen1985

Frequently asked questions

Why must heat capacity be positive for a stable phase?: If adding heat lowered a system's temperature, a small fluctuation would grow without bound rather than relax, so a stable equilibrium phase must have positive heat capacity; negative values signal an instability and the breakdown of that phase.