Nucleation Energy Barrier Change Calculation

This solution explains how a change in solid-liquid interfacial energy affects the energy barrier for homogeneous nucleation.

Key Formula

The energy barrier ($G_{max}$) for homogeneous nucleation of a spherical solid nucleus is given by the formula:

Where:

• $\gamma$ is the solid-liquid interfacial energy.
• $\Delta G_v$ is the change in free energy per unit volume upon solidification.

Analysis of Change

The question states that the solid-liquid interfacial energy ($\gamma$) increases by 10%. Let the initial and final values be $\gamma_1$ and $\gamma_2$, respectively. We assume $\Delta G_v$ remains constant.

Initial interfacial energy: $\gamma_1$

New interfacial energy: $\gamma_2 = \gamma_1 + 0.10 \gamma_1 = 1.10 \gamma_1$

Calculating the New Energy Barrier

Let the initial energy barrier be $G_{max,1}$ and the new energy barrier be $G_{max,2}$.

Initial barrier: $G_{max,1} = \frac{16 \pi \gamma_1^3}{3 (\Delta G_v)^2}$

New barrier: $G_{max,2} = \frac{16 \pi \gamma_2^3}{3 (\Delta G_v)^2}$ Substituting $\gamma_2 = 1.10 \gamma_1$: $G_{max,2} = \frac{16 \pi (1.10 \gamma_1)^3}{3 (\Delta G_v)^2}$ $G_{max,2} = (1.10)^3 \times \frac{16 \pi \gamma_1^3}{3 (\Delta G_v)^2}$ $G_{max,2} = (1.10)^3 \times G_{max,1}$

Calculate the cube of 1.10:

Therefore, $G_{max,2} = 1.331 \times G_{max,1}$.

Determining the Percentage Change

The percentage change in the energy barrier is calculated as:

The energy barrier increases by approximately 33.1%.

Conclusion

A 10% increase in solid-liquid interfacial energy results in approximately a 33.1% increase in the energy barrier for homogeneous nucleation, aligning with Option B.