Solution

Statement

$5.9.16.$ In the ocean, there is a boat with a $1\text{ kg}$ block of ice at $0^\circ\text{C}$ on board. Determine the maximum work that can be obtained using the ice melting process. The water temperature is $27^\circ\text{C}$.

Solution

The previous author made a fundamental mistake. They multiplied the efficiency ($\eta$) by the heat of melting ($Q_C$), assuming that $Q_C$ is the heat received from the hot reservoir. However, in this case, the ice acts as the cold reservoir (sink), and the ocean is the hot reservoir (source).

The heat engine takes heat $Q_H$ from the ocean, performs useful work $A$, and rejects the remaining heat $Q_C$ to the ice, causing it to melt.
The heat required to melt the ice is:
$$Q_C = \lambda m$$
where $\lambda = 334 \text{ kJ/kg}$, $m = 1 \text{ kg}$, $T_C = 273 \text{ K}$.
The temperature of the ocean (hot reservoir) is $T_H = 300 \text{ K}$.

For a reversible Carnot cycle (which yields maximum work), the following relation holds:
$$\frac{Q_H}{T_H} = \frac{Q_C}{T_C}$$
From this, the heat the engine takes from the ocean is:
$$Q_H = Q_C \frac{T_H}{T_C}$$
The maximum work is the difference between the heats:
$$A_{\max} = Q_H - Q_C = Q_C \left(\frac{T_H}{T_C} - 1\right) = \lambda m \frac{T_H - T_C}{T_C}$$

Substituting the numerical values:
$$A_{\max} = 334000 \cdot \frac{300 - 273}{273} = 334000 \cdot \frac{27}{273} \approx 33033 \text{ J} \approx 33 \text{ kJ}$$

Regarding the heating of the resulting water: the problem explicitly states "using the ice melting process", so the work is calculated only for the phase transition.