Physics problem-solving difficult thermodynamics/ latent heat/ heat capacity question

Specific heat capacity of water: 4180 J/kg/K

Specific heat capacity of ice: 2030 J/kg/K

Specific latent heat of fusion of ice: 3.35×10^5 J/kg/K

Specific latent heat of vaporization of water: 2.26×10^6 J/kg/K

A mass of 0.35kg of ice at −15°C is lowered into an insulated beaker containing a certain mass (in kg) of water at 59°C.

What is the minimum mass of water at 59°C needed in the beaker to achieve a final temperature of 0.0°C? What is the maximum mass?

My answer (which is wrong):

There are 3 possible equations, 1 containing LΔm for both ice and water, one containing LΔm for just ice, and one containing LΔm for just water

Equation 1: mcΔT (ice) + LΔm (ice) = mcΔT (water) + LΔm (water) = m(cΔT + L) (water)

Equation 2: mcΔT (ice) + LΔm (ice) = mcΔT (water)

Equation 3: mcΔT (ice) = mcΔT (water) + LΔm (water) = m(cΔT + L) (water)

The minimum mass of water is when the LHS is a minimum and the RHS is a maximum, i.e equation 3:

(0.35) (2030) (15) = m (4180 x 59 + 2.26 x 10^6) --> solving for m, m = 4.25 x 10^-3

The maximum mass of water is when the LHS is a maximum and the RHS is a minimum, i.e equation 2:

(0.35) (2030) (15) + (3.35x10^5) (0.35) = m x (4180) (59) --> solving for m, m = 1.40 kg

Both of my answers are wrong, but I don't understand why.

Thank you!

Reply 1