At the same undercooling, how does the radius of curvature of the critical nucleus in heterogeneous nucleation compare with the critical nucleus radius in homogeneous nucleation?

Answer: C. It is the same

A heterogeneous nucleus has the shape of a spherical cap (part of a sphere) sitting on the substrate with contact angle θ. Its radius of curvature at the critical state is determined only by the interfacial energy and the free energy change per unit volume, r* = −2γ/ΔGv, the same as for homogeneous nucleation. The difference is that, at the same radius of curvature, the spherical cap has a smaller volume than a sphere, so the number of atoms needed and the energy barrier are reduced by the factor f(θ). It is easy to confuse the lowered barrier with a smaller radius, so be careful. Nor does the presence of a substrate make the critical radius larger.