Water at its freezing point \(\left(T_{i}, P_{i}\right)\) completely fills a strong steel container. The temperature is reduced to \(T_{f}\) at constant volume, with the pressure rising to \(P_{f}\). (a) Show that the fraction \(y\) of water that freezes is given by $$ y=\frac{v_{f}^{\prime \prime}-v_{f}^{n}}{v_{f}^{\prime \prime}-v_{f}^{\prime}} $$ (b) State explicitly the simplifying assumptions that must be made in order that \(y\) may be written $$ y=\frac{v^{\prime \prime}\left[\beta^{\prime \prime}\left(T_{f}-T_{i}\right)-\kappa^{\prime \prime}\left(P_{f}-P_{i}\right)\right]}{v_{f}^{\theta}-v_{f}^{\prime}} $$ (c) Calculate \(y\) for \(i=0^{\circ} \mathrm{C}, \quad 1.01 \times 10^{5} \mathrm{~Pa} ; \quad f=-5^{\circ} \mathrm{C}, \quad 5.98 \times 10^{7} \mathrm{~Pa}\); \(\beta^{\prime \prime}=-67 \times 10^{-6} \mathrm{~K}^{-1} ; \kappa^{\prime \prime}=12.04 \times 10^{-11} \mathrm{~Pa}^{-1} ; v_{f}^{\prime \prime}-v_{f}^{\prime}=-1.02 \times 10^{-4} \mathrm{~m}^{3} / \mathrm{kg}\)

Thermodynamics is the study of energy transformations and how they relate to the properties of matter. It involves principles that govern the behavior of physical systems in terms of heat, work, temperature, and energy. This field is crucial when analyzing processes such as phase transitions, compressibility, and thermal expansion.

In the context of the given problem, the thermodynamic principles help us understand changes in state variables such as temperature and pressure. As the water in the steel container undergoes cooling, the thermodynamic behavior of water needs to be understood to analyze the freezing process. The fundamental laws of thermodynamics provide a framework for interpreting these energy exchanges, ensuring that the outcomes align with the conservation of energy and matter principles.

Keywords related to thermodynamics such as 'enthalpy,' 'entropy,' and 'Gibbs free energy' are essential, but for this problem, the focus remains on temperature, pressure, and volumes as primary state variables.

Phase transitions refer to changes in the state of matter, such as the transition from liquid water to ice. These changes are characterized by alterations in structure and energy states of the material. Phase transitions typically occur due to changes in temperature and pressure, leading to different physical properties.

For water in a steel container, reducing the temperature below the freezing point at constant volume causes water to convert to ice. Since this process also produces significant pressure changes due to the expansion or contraction of water and ice, it results in a measurable phase transition.

Key aspects of phase transitions include:

• Latent Heat: The energy required for the transition.
• Equilibrium States: The balance between phases at given conditions.
• Specific Volume Changes: Differences in volumes between phases.

Understanding phase transitions helps us predict and quantify the fraction of water that transitions to ice, using the relationship between pressure, temperature, and volume.

Compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure change. It is denoted by \(ewline κ = -\frac{1}{v}\frac{∂v}{∂P}ewline \) and typically has units of inverse pressure. Compressibility is crucial for understanding how materials respond under pressure.

In our problem, compressibility of water and ice play a role as pressure increases while the volume is held constant. This change in pressure affects the fraction of water turning into ice, and is accounted for in the equation used to determine the fraction \( f\).

Specific formulas are often employed to incorporate compressibility into calculations: \(ewline v_f'' \beta''=-κ''ewline \). Changes in pressure occur when the temperature drops and water contracts. Thus, understanding compressibility allows us to quantify these changes accurately.

Thermal expansion is described by how a material's volume changes in response to temperature variations. It is quantified using the coefficient of thermal expansion, \(ewline \beta = \frac{1}{v}\frac{∂v}{∂T}ewline \), with units of per degree temperature.

In examining water's behavior in the problem, the thermal expansion plays a role when temperature changes from the initial to the final state. Since steel constrains the volume, we use the coefficient of thermal expansion to determine what fraction of the liquid water freezes.

Key elements related to thermal expansion:

• Material Properties: Each material has a unique coefficient.
• Temperature Range: Effects can vary significantly across different temperatures.
• Volume Change: Direct impact on the specific volume calculations.

Including thermal expansion provides a comprehensive understanding of water’s volume change as it freezes under constant volume conditions.

Specific volume is the volume occupied by a unit mass of a substance, usually denoted \(ewline v = \frac{V}{m}ewline \), with units of cubic meters per kilogram (m³/kg). It is a critical variable in thermodynamics and fluid mechanics for characterizing the state of a substance.

In the freezing water scenario, specific volumes for liquid water and ice feature prominently. The difference between these specific volumes ( v_{f}'' - v_{f}'•) relates directly to the phase transition process, indicating how much space ice occupies compared to liquid water.

When calculating the fraction y of water freezing:

• Initial and Final Specific Volumes: Critical for determining phase mix.
• Thermal and Pressure Effects: Governed by coefficients \(\beta \) and \( κ \).

Properly accounting for specific volume differences ensures accurate computation of the frozen fraction.