Positivity of $C_V$ from thermodynamic stability

Statement

For a stable equilibrium thermodynamic system, the constant-volume heat capacity satisfies

(at fixed $V$ and fixed composition, e.g. fixed $N$). Equivalently, at fixed $V,N$ the temperature is a nondecreasing function of entropy, and entropy is a nondecreasing function of temperature.

Key hypotheses

A convenient set of sufficient hypotheses is:

• The system is in thermodynamic equilibrium and admits a smooth fundamental relation either as
  • energy representation $U=U(S,V,N)$ (internal energy as a function of entropy ), or
  • entropy representation $S=S(U,V,N)$.
• Thermodynamic stability in the usual sense (see thermodynamic stability ), e.g.
  • $U(S,V,N)$ is convex in $S$ at fixed $V,N$, or equivalently
  • $S(U,V,N)$ is concave in $U$ at fixed $V,N$.
• Temperature $T>0$ in the region of interest (see temperature ).

Key conclusions

• The heat capacity at constant volume is nonnegative:

  $$C_V = \left(\frac{\partial U}{\partial T}\right)_{V,N} = T\left(\frac{\partial S}{\partial T}\right)_{V,N} \ge 0.$$

• If stability is strict (strict convexity/concavity), then $C_V>0$ and $T(S)$ is strictly increasing at fixed $V,N$.

• Vanishing curvature can lead to $C_V$ diverging (a common signature near criticality), consistent with $C_V\ge 0$.

Cross-links to definitions

• $C_V$ (constant-volume heat capacity)
• thermodynamic stability
• internal energy , entropy , temperature

Proof idea / significance

Work at fixed $V,N$. In the energy representation,

Stability (convexity of $U$ in $S$) gives

Assuming local invertibility, this implies $(\partial S/\partial T)_{V,N}\ge 0$, hence

Significance: $C_V\ge 0$ expresses stability against thermal perturbations at fixed volume; negative heat capacity would imply that adding energy lowers $T$, signaling instability within ordinary equilibrium thermodynamics (though exotic nonadditive systems can behave differently).