The latent heat of fusion for ice \(I\) is \(3.34 \times 10^{5} \mathrm{~J} / \mathrm{kg}\) at \(0^{\circ} \mathrm{C}\) and atmospheric pressure. If the change in specific volume on melting is \(-9.05 \times 10^{-5} \mathrm{~m}^{3} / \mathrm{kg}\), then calculate the change of melting temperature due to change of pressure.

The latent heat of fusion is the amount of energy required to change a substance from a solid to a liquid at its melting point without changing its temperature. In context, for ice, this value is given as \(\boxed{3.34 \times 10^{5} \mathrm{J} / \mathrm{kg}}\). This means that to melt 1 kg of ice at 0°C, you need 334,000 Joules of energy. The concept is essential because it explains how much energy is involved in phase changes.

It's interesting to note that during the melting process, the temperature remains constant at 0°C until all the ice has turned into water. This property is critical in problems involving heat transfer, such as this one where we're determining the effect of pressure on the melting point.

• The latent heat of fusion is unique to each substance.
• For ice, it's a high value, indicating strong bonds between water molecules in the solid state.
• Understanding this helps in calculating the energy requirements for melting or freezing processes.

The melting point of ice is the temperature at which it changes from solid to liquid phase under a constant pressure, typically atmospheric pressure. For pure water ice, the standard melting point is 0°C (32°F). This point is where thermal energy overcomes the molecular bonds keeping ice solid.

However, the melting point can vary slightly depending on pressure. For example, if you apply more pressure, it can lower the melting point—a phenomenon easily explained by considering the molecular dynamics involved.

In the problem, we are asked to determine how a pressure change affects this melting point. This is where the Clausius-Clapeyron equation becomes handy, offering a way to quantify this change through established relationships between latent heat, volume change, and pressure.

Understanding the melting point is crucial since:

• It's a fundamental concept in thermodynamics and physical chemistry.
• Changes in melting points can indicate changes in molecular interactions.
• It has practical implications in areas such as climate science and engineering.

The Clausius-Clapeyron equation describes how the phase boundary between two states (like solid and liquid) shifts with changes in pressure and temperature. The equation is important because it allows us to understand and predict how changes in external conditions impact the melting or boiling points of substances. For phase transitions, it can be written as:

\[ \frac { \mathrm{d} T }{ \mathrm{d} P } = \frac{ L / T }{ \Delta V } \]

Here, \(\frac { \mathrm{d} T }{ \mathrm{d} P } \) represents the rate of change of temperature with respect to pressure, \(L\) is the latent heat, and \(\Delta V\) represents the change in specific volume during melting.

In practical terms, for ice:

• When pressure increases, the melting point of ice decreases.
• The specific volume change \(\Delta V\) is negative because water expands when it freezes.

It's vital for students to grasp this relationship because it provides a mathematical way to predict how materials will behave under different conditions. Whether working in a lab or addressing large-scale environmental issues, understanding pressure-temperature relationships through the Clausius-Clapeyron equation is invaluable.