GATE 2026 MT – Question 57
A volume of liquid is undercooled just below its melting temperature to form a spherical solid nucleus by homogeneous nucleation. The Gibbs free energy of solidification is $\Delta G_v=-0.5\times10^8\ \mathrm{J/m^3}$. The solid–liquid interfacial energy is isotropic with $\gamma=0.1\ \mathrm{J/m^2}$. The critical nucleus size for a stable nucleus is ________ nm. (Answer in integer.)
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Correct answer: 4 to 4 OR 8 to 8
Explanation
The formation energy is $\Delta G(r)=4\pi r^2\gamma+(4\pi/3)r^3\Delta G_v$. Setting $d\Delta G/dr=0$ gives $r^*=-2\gamma/\Delta G_v$.
Thus $r^*=2(0.1)/(0.5\times10^8)=4\times10^{-9}\ \mathrm m=4\ \mathrm{nm}$. The critical diameter is 8 nm. The official key accepts **4 or 8**, since “size” is not defined as radius or diameter; both grading alternatives are retained.