= Solution
Let $g_l-g_s$ be the liquid-minus-solid <free energy> density at the same <temperature>. Near the bulk coexistence <temperature> $T_m$, linearization gives
$$
g_l-g_s\simeq\Delta s_V(T_m-T)
=\frac{\Delta H_V}{T_m}\Delta T.
$$
For a sphere, an interface displacement changes area and volume with $dA/dV=2/r$. Equality of the bulk driving force and the capillary cost at a curved solid–liquid interface therefore gives $g_l-g_s=2\gamma/r^*$. Equivalently, differentiating $4\pi r^2\gamma-(4\pi/3)r^3(g_l-g_s)$ gives the same stationary radius. Combining the expressions yields
$$
\boxed{\Delta T=\frac{2\gamma T_m}{r^*\Delta H_V}.}
$$
This form of the <Gibbs-Thomson effect> assumes isotropic interfacial energy, a spherical interface, bulk-like <latent heat> per volume and a near-coexistence <free energy> expansion. More detailed free-particle models must distinguish solid–vapor, liquid–vapor and solid–liquid energies.
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