$($a$)$ Consider the example binary phase diagram of the $Ag-Cu$ phase diagram below and the $Ag-3wt.\%Cu$ alloy. When poured into steel mold, the primary silver rich phase, $(Ag)$ or $-$silver, tends to always grow as a dendrite with an unstable solidification interface, while pure silver solidifies with a planer interface. If the pouring and mold conditions are the same, explain why this might happen. $(6$pts$)$

$($b$)$ During growth of solid $$ from the liquid in the $3wt.\%Cu$ alloy, explain the concept of constitutional undercooling $($or constitutional supercooling$).$ Include in your discussion how: $(1)$ the temperature gradient in the liquid, $(2)$ solute partition and diffusion in the liquid, and $(3)$ growth or solidification rate, control the amount of constitutional undercooling. $(4)$ Explain how constitutional undercooling affects solid$/$liquid interface structure and solidification microstructure. $(12$ points$)$

Explain your answers thoroughly for full credit. Include figures where appropriate to help you explain your answers.

Atomic percent copper