This question requires five separate calculations: (1) energy needed to warm ice at −24.0ºC to its melting point at 0ºC, (2) energy needed to melt ice at OºC, (3) energy needed to warm liquid water from 0ºC to its boiling point at 100ºC, (4) energy needed to boil water at 100ºC, and (5) energy needed to warm steam from 100ºC to 169ºC. The sum of these five is the answer.
For the two phase changes, the enthalpy of fusion (333.55 kJ/kg) and enthalpy of vaporization (2257 kJ/kg) are needed, and the heat required is q = m • ΔH
For the three warming steps, specific heat capacities are needed for each step: ice is 2.05 J/g•ºC, liquid water is 4.184 J/g•ºC, and steam is 2.080 J/g•ºC, and the heat required is q = m • C • ΔT
Step 1: Energy needed for warming the ice from −24.0ºC to 0ºC. ΔT = 24.0ºC
q = m • C • ΔT = 77.0 g x (2.05 J/g•ºC) x (1 kJ / 1000 J) x (24.0 ºC) = 3.788 kJ
Step 2: Energy needed for melting the ice at 0ºC.
q = m • ΔH = 77.0 g x (1 kg/1000 g) x (333.55 kJ/kg) = 25.68 kJ
Step 3: Energy needed for warming the liquid water from 0ºC to 100ºC. ΔT = 100ºC
q = m • C • ΔT = 77.0 g x (4.184 J/g•ºC) x (1 kJ / 1000 J) x (100 ºC) = 32.22 kJ
Step 4: Energy needed for boiling the water at 100ºC.
q = m • ΔH = 77.0 g x (1 kg/1000 g) x (2257 kJ/kg) = 173.8 kJ
Step 5: Energy needed for warming the steam from 100ºC to 169.0ºC. ΔT = 69.0ºC
q = m • C • ΔT = 77.0 g x (2.080 J/g•ºC) x (1 kJ / 1000 J) x (69.0 ºC) = 11.05 kJ
Summing the energies required for each step yields the answer:
3.788 kJ + 25.68 kJ + 32.32 kJ + 173.8 kJ + 11.05 kJ =
Note on significant figures:
I've assumed the freezing point and boiling point temperatures to be exact.
Also, I kept an extra sig fig in the answer for each step.
Source of constants: