 Solution

For a spherical homogeneous nucleus, the capillarity approximation givesThe first term penalizes creating an interface; below the melting point the bulk free-energy change per volume $\Delta g_V$ is negative and favors growth. Near $T_m$, neglect the temperature dependence of the latent heat. Since the free-energy difference vanishes at $T_m$,Set the radius derivative to zero:The nonzero stationary point is a maximum, since $\Delta G^{\prime \prime }(r^{\ast })=-8\pi \gamma <0$. Subcritical nuclei lower free energy by shrinking; supercritical nuclei lower it by growing. The positive radius and its inverse-supercooling dependence follow from the negative freezing enthalpy. This derivation assumes an isotropic, size-independent interfacial energy and a bulk-like spherical nucleus; it is not a molecular-scale kinetic theory.