Calorimetry with phase changes

Problem

$200\ \text{g}$ of ice at $-10\ ^\circ\text{C}$ are heated until they become water at $20\ ^\circ\text{C}$. Calculate the total heat absorbed. Data: $c_\text{ice} = 2090\ \text{J/(kg·K)}$, $L_f = 334\ \text{kJ/kg}$, $c_\text{water} = 4186\ \text{J/(kg·K)}$. Indicate which contribution weighs the most.

Solution

Setup. The mass is $m = 0{,}200\ \text{kg}$. The process has three stages: heating the ice up to $0\ ^\circ\text{C}$, melting at $0\ ^\circ\text{C}$, and heating the water up to $20\ ^\circ\text{C}$.

Stage 1 — heating the ice (from $-10$ to $0\ ^\circ\text{C}$, $\Delta T = 10\ \text{K}$):

$Q_1 = m\,c_\text{ice}\cdot 10 = 0{,}2\cdot 2090\cdot 10 = 4180\ \text{J}$

Stage 2 — melting (at $0\ ^\circ\text{C}$, constant temperature):

$Q_2 = m\,L_f = 0{,}2\cdot 334000 = 66800\ \text{J}$

Stage 3 — heating the water (from $0$ to $20\ ^\circ\text{C}$, $\Delta T = 20\ \text{K}$):

$Q_3 = m\,c_\text{water}\cdot 20 = 0{,}2\cdot 4186\cdot 20 = 16744\ \text{J}$

Total.

$Q_\text{tot} = 4180 + 66800 + 16744 = 87724\ \text{J}$

Conclusion. The dominant contribution is melting ($Q_2$), because the latent heat requires a lot of energy to break the bonds of the crystal lattice without a temperature change.

$\ev{Q_\text{tot}\approx 87{,}7\ \text{kJ}}$

Links

Topics: Thermodynamics Concepts: Latent heat and changes of state · Specific heat and heat capacity Skills: Calorimetric equation