Consider the formation of pure $Cu$ precipitates in a low$-$carbon steel alloyed with $0\mathrm{.}6$

$wt\%Cu.$ The diffusion coefficient for $Cu$ is given by $D_0=4\mathrm{.}2c\frac{m^2}{s}$ and $Q=244k\frac{J}{m}ol.$

The particle$-$matrix interface energy is $0\mathrm{.}5\frac{J}{m^2}.$ Assume perfect fit of the precipitates,

a molar volume of $7c{m}^3$ and a frequency factor, $,$ of ${10}^{13}{s}^{-1}.$ The $Cu$ solubility is

$0\mathrm{.}03$ at $\%$ at $500C$ and $0\mathrm{.}47$ at $\%$ at $700C.$ The molar mass of $Cu$ is $63\mathrm{.}5\frac{g}{m}ol$ and

that of $Fe$ is $56\frac{g}{m}ol.$

$($i$)$ Calculate the dissolution temperature, i$.$e$.$ the temperature where the composition

of the solvus line in the phase diagram is equal to the alloy composition. Further,

determine the dissolution temperature for fine $Cu$ precipitates with a radius of $3nm.$

$($ii$)$ Calculate the homogeneous nucleation rates at $400,500$ and $600C.$

$($iii$)$ Determine the heterogenous nucleation rate at $500C$ assuming the density of

heterogeneous nucleation sites to be ${10}^{18}c{m}^{-3}$ and the activation energy for

nucleation to be determined with that for homogeneous nucleation by adopting a

reduced effective interfacial energy of $0\mathrm{.}25\frac{J}{m^2}?$