Answer the following for a Ag$-$Cu alloy with $7\mathrm{.}5$ wt$\%$Cu that is unidirectionally solidified into a bar under the conditions of no diffusion in the solid and perfect mixing in the liquid.

a$)$ At what temperature does the first solid form?

b$)$ What is the Cu concentration of the first solid?

c$)$ Calculate the composition of the liquid when the ingot is $40\%$ solid.

d$)$ What is the interface temperature at this instant $($when the ingot is $40\%$solid$)?$

e$)$ Plot the variation of copper concentration along the bar at this instant $($when the ingot is $40\%$ solid$).$

f$)$ What fraction of the bar will solidify to a eutectic structure?

g$)$ What is the average composition of the solid that did not go through the eutectic solidification?

h$)$ Plot the variation of copper concentration along the bar.