If copper (which has a melting point of 1085°C) homogeneously nucleates at 849°C, calculate the critical radius given values of –1.77 × 109 J/m3 and 0.200 J/m2, respectively, for the latent heat of fusion and the surface free energy.

The critical radius upon homogenous nucleation is; r_c = -1.25 nm

We want to find critical radius and it is gotten from the formula;

r_c = (2γ × T_f)/(h_f,v × ΔT)

Where;

r_c is critical radius

γ is surface free energy.

T_f is temperature of melting point of element

h_f,v is latent heat of fusion or vaporization

ΔT is change in temperature = T_f - T_h

Where T_h is the temperature at which which the element homogeneously nucleates

We are given;

γ = 0.2 J/m²

T_f = 1085 °C = 1358 K

T_h = 849 °C = 1112 K

latent heat of fusion; h_f = -1.77 × 10^(9) J/m³

Thus;

r_c = (2 × 0.2 × 1358)/(-1.77 × 10^(9) × (1358 - 1112))

r_c = -1.25 × 10^(-9) m

r_c = -1.25 nm

Read more about critical radius in homogenous nucleation at; link

do you have any options?