Define the Process - Liquid-Solid-Liquid?

A small quantity of water of mass m = 50g in a container at temperature T = 273 K is placed inside a vacuum chamber which is expatriate rapidly.  As an effect part of the water freezes and turn into ice and the rest becomes vapor.

a)  What quantity of water primarily transforms into ice? The latent heat of vaporization (water/vapor) q_v = 600 cal/g and the latent heat of fusion (ice/water) q_i = 80 cal/g.

b)  A piece of heated metal alloy of mass M = 325 g and original volume V= 48 cm³  is  placed  inside   the  calorimeter   together  with   the  ice        obtained   as  a result   of the experiment  in  (a). The density of metal at T = 273 K is p0 = 6.8 g/cm³  the thermal capacity is C= 0.12 cal/g K and the coefficient of linear expansion α = 1.1.10^-5 K^-1 How much ice will have melted when equilibrium is reached?