Solved thermodynamics problem - Can you interpret the solution, please?

playmannn

1. The problem statement, all variables and given/known data

"In a closed container there is water in equilibrium with saturated vapor at temperature 100°C (212°F). The ratio of the masses vapor/water is β=0.1. The heat capacity of water is C(o) = 4.2 J/g.K. Find the specific heat capacity of the system, considering that the specific heat of transition is q=2.26 J/g. Identify the specific heat capacity Cv of a thin fog (saturated vapor with water drops, the full mass of which is much lower of the mass of the vapor)."

2. Relevant equations

The solution with the key steps is provided in the attachment!

3. The attempt at a solution

Can you, please, tell me how are the stated equations derived, starting with why is the one for mc like so, as well as the following ones. I'll be so thankful if you do!

Attached Thumbnails