Problem $1.$ The following is a temperature$-$composition phase diagram for the binary system of water $($A$)$ and phenol $($B$)$ at $1$ bar. These liquids are partially miscible below $67C.$ Phenol is denser than water, so the layer with the higher mole fraction of phenol is the bottom layer. Suppose you place $4$mol of $H_{2}O$ and $1$mol of phenol in a beaker at $30C$ and gently stir to allow the layers to equilibrate.

$($a$)$ What are the compositions of the equilibrated top and bottom layers?

$($b$)$ Find the amount of each component in the bottom layer.

$($c$)$ As you gradually stir more phenol into the beaker, maintaining the temperature at $30C,$ what changes occur in the volumes and compositions of the two layers? Assuming that one layer eventually disappears, what additional amount of phenol is needed to cause this to happen?