What quantity of energy does it take to convert \(0.500 \mathrm{~kg}\) ice at \(-20 .{ }^{\circ} \mathrm{C}\) to steam at \(250 .{ }^{\circ} \mathrm{C}\) ? Specific heat capacities: ice, \(2.03 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C} ;\) liquid, \(4.2 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C} ;\) steam, \(2.0 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C} ; \Delta H_{\text {vep }}=\) 40.7 \(\mathrm{kJ} / \mathrm{mol} ; \Delta H_{\mathrm{fec}}=6.02 \mathrm{~kJ} / \mathrm{mol}\).

Understanding specific heat capacity is critical when examining how substances absorb or release heat during temperature changes. It represents the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). Mathematically, it's expressed as

\( c = \frac{Q}{m\triangle T} \),
where \( Q \) is the heat added, \( m \) is the mass of the substance, and \( \triangle T \) is the change in temperature. Different materials have varying specific heat capacities. For example, water has a high specific heat capacity, meaning it can absorb a lot of heat before it begins to get hot. This is why water is often used as a coolant in various applications.

When a substance changes from solid to liquid, it undergoes a phase transition at its melting point. The heat of fusion—also known as the enthalpy of fusion—is the amount of energy needed to change a substance from solid to liquid at constant temperature. It's a latent heat, which means it's energy absorbed or released during a change of phase at constant temperature and pressure, without changing the temperature of the substance. For water, this occurs at 0°C. In calculations, it's denoted as

\( \triangle H_{\text{fusion}} \) and often given in units of joules per mole (J/mol). This energy breaks the molecular bonds holding the solid together without increasing its temperature, which is why temperatures remain constant during this phase change.

The heat of vaporization is the counterpart of the heat of fusion, but for the liquid-gas phase transition. It defines the energy needed for a liquid to become a gas at its boiling point. Similar to the heat of fusion, this is also a latent heat and is denoted as

\( \triangle H_{\text{vaporization}} \).
For water, this process occurs at 100°C under standard atmospheric pressure. The high heat of vaporization for water makes it an effective substance for transferring heat in systems like steam engines or evaporation coolers. Even in the human body, sweating leverages this property to help regulate temperature as the body heat is used to evaporate the sweat, thereby cooling the skin.

The heating curve of water graphically represents the temperature change of water as energy is added at a constant rate. There are several key regions in this curve. Starting with ice below 0°C, the temperature rises linearly as heat is added, illustrating the effect of specific heat. Once the ice reaches 0°C, the temperature plateaus as the heat of fusion is absorbed to convert the ice to liquid water.

The temperature then climbs again as the water heats up, once more showing a linear relationship. Upon reaching 100°C, another plateau represents the heat of vaporization being absorbed to turn water into steam. Finally, as the gas phase is heated, the temperature will rise once again. These plateaus and linear portions of the heating curve perfectly demonstrate the interplay between specific heat and latent heats (fusion and vaporization) during phase transitions.