Question

A 20g cube of ice, initially at temperature of -5°C is put into some water in a Styrofoam cup. The initial temperature of the water is 50°C, and the ice cube absorbs 5kJ of heat before reaching thermal equilibrium with the water. Assume the ice absorbs heat only from the water. What final temperature does the mixture reach?

Answer 1

First, let us calculate, does the ice become water. The heat of fusion (see https://en.wikipedia.org/wiki/Enthalpy_of_fusion) is 333.55 J/g, so the total amount of energy will be

$20\cdot333.55 = 6670\,\mathrm{J}.$ This amount is too much. So not all amount of ice will become water.

Let us determine the energy needed for ice to reach 0°C.

Specific heat capacity of ice is 2100 J/kg, so the amount will be

$0.02\cdot2100\cdot5 = 210\,\mathrm{J},$ so all the ice can reach temperature of 0°C. Next, some amount of ice should be melted, and we can calculate this amount, because we know that ice absorbed additionally 5000 - 210 = 4790 J. This energy will be absorbed to melt the mass of ice equal to$m = \dfrac{4790\,\mathrm{J}}{333.55\,\mathrm{J/g}} \approx 14.4\,\mathrm{g}.$ And 5.6 g will remain as ice. So the mixture should have temperature 0°C.