2 kg of ice at $-$20°C is mixed with 5 kg of water at 20°C in an insulating vessel having a negligible heat capacity. Calculate the final mass of water in the vessel. It is given that the specific heat of water and ice are 1 kcal/kg/°C and 0.5 kcal/kg/°C respectively and the latent heat of fusion of ice is 80 kcal/kg. 

Let m kg be the mass of ice melted into water.

 Heat lost by 5 kg of water = 5 kg x 1 kcal/kg/°C x 20°C = 100 kcal. 

Heat gained = m kg x 80 kcal/kg + 2 kg x 0.5 kcal/kg/°C x 20°C

                    = 80 m kcal + 20 kcal. Now,

heat gained = heal lost. Therefore,

       80 m + 20 = 1000
or     m= 1 kg.   Therefore,

 final mass of water = 5 kg + 1 kg = 6 kg.