This solution outlines the calculation for the critical radius of a nickel nucleus during solidification, utilizing fundamental principles of nucleation theory.

Critical Radius Formula

The critical radius ($r_c$) for homogeneous nucleation in a liquid is determined by the balance between the reduction in volume free energy and the increase in interfacial energy. The formula is:

$r_c = \frac{2 \sigma T_m}{\Delta H_f \Delta T}$

Where:

• $\sigma$ represents the liquid-solid interfacial energy ($0.5 \text{ J.m}^{-2}$).
• $T_m$ is the equilibrium melting temperature ($1728 \text{ K}$).
• $\Delta H_f$ denotes the enthalpy of fusion ($2.65 \times 10^9 \text{ J.m}^{-3}$).
• $\Delta T$ is the degree of undercooling, calculated as $T_m - T$.

Solidification Parameters

The problem provides the following data for nickel:

• Interfacial energy ($\sigma$): $0.5 \text{ J.m}^{-2}$
• Equilibrium melting temperature ($T_m$): $1728 \text{ K}$
• Enthalpy of fusion ($\Delta H_f$): $2.65 \times 10^9 \text{ J.m}^{-3}$
• Solidification temperature ($T$): $1673 \text{ K}$

Steps to Calculate Critical Radius

Step 1: Calculate the undercooling ($\Delta T$).

The undercooling is the difference between the equilibrium melting temperature and the actual solidification temperature.

$\Delta T = T_m - T = 1728 \text{ K} - 1673 \text{ K} = 55 \text{ K}$

Step 2: Substitute known values into the critical radius formula.

Using the formula $r_c = \frac{2 \sigma T_m}{\Delta H_f \Delta T}$:

$r_c = \frac{2 \times (0.5 \text{ J.m}^{-2}) \times (1728 \text{ K})}{(2.65 \times 10^9 \text{ J.m}^{-3}) \times (55 \text{ K})}$

Step 3: Perform the calculation.

Calculate the numerator:

$2 \times 0.5 \times 1728 = 1728 \text{ J.m}^{-2}\text{.K}$

Calculate the denominator:

$2.65 \times 10^9 \times 55 = 145.75 \times 10^9 = 1.4575 \times 10^{11} \text{ J.m}^{-3}\text{.K}$

Now, divide the numerator by the denominator:

$r_c = \frac{1728 \text{ J.m}^{-2}\text{.K}}{1.4575 \times 10^{11} \text{ J.m}^{-3}\text{.K}} \approx 1.1856 \times 10^{-8} \text{ m}$

Step 4: Convert the radius to nanometers ($nm$) and round.

To convert meters to nanometers, multiply by $10^9$ ($1 \text{ m} = 10^9 \text{ nm}$):

$r_c \approx 1.1856 \times 10^{-8} \text{ m} \times \frac{10^9 \text{ nm}}{1 \text{ m}} \approx 11.856 \text{ nm}$

Rounding to one decimal place, the critical radius is $11.9 \text{ nm}$. This value is consistent with the expected range.