Question

One kilogram of saturated solid water at the triple point is heated to saturated liquid while the pressure is maintained constant. Determine the work and the heat transfer for the process, each in \(\mathrm{kJ}\). Show that the heat transfer equals the change in enthalpy of the water in this case.

Short answer

The heat transfer is 333.4 kJ/kg and the work done is 0 kJ. The heat transfer equals the change in enthalpy.

Step-by-step solution

Identify the Process and Conditions

The substance is initially at the triple point and starts as saturated solid water (ice). It is heated to a saturated liquid state while pressure is maintained constant.

Find Relevant Properties at Triple Point

At the triple point, water coexists in solid, liquid, and gas phases. The triple point temperature is 0.01°C and pressure is 0.0061 atm. From a steam table, locate the enthalpy values: \(\text{Enthalpy of Saturated Solid (Ice),} \, h_{f} = -333.4 \, \text{kJ/kg}\) \(\text{Enthalpy of Saturated Liquid (Water),} \, h_{f} = 0 \, \text{kJ/kg}\)

Calculate the Heat Transfer

Heat transfer (\text{Q}) can be found using the change in enthalpy between the initial and final states. \[ Q = h_{f, \text{liquid}} - h_{f, \text{solid}} \] Substitute the enthalpy values: \[ Q = 0 \, \text{kJ/kg} - (-333.4 \, \text{kJ/kg}) = 333.4 \, \text{kJ/kg} \]

Determine Work Done

In a phase change process at constant pressure, the volume change is very small, so the \text{work done} \(W\) is approximately zero. \[ W = 0 \, \text{kJ} \]

Verify Heat Transfer Equals Change in Enthalpy

Since no significant work is done, the heat transfer should equal the change in enthalpy for the process. \[ \text{Change in Enthalpy} \, \text{(ΔH)} = Q = 333.4 \, \text{kJ/kg} \]

Key concepts

Triple Point

The triple point of a substance is an incredibly important concept in thermodynamics. It represents the exact temperature and pressure at which a substance can coexist in three different phases: solid, liquid, and gas. For water, this specific point occurs at a temperature of 0.01°C and a pressure of 0.0061 atm. At this unique condition, ice, liquid water, and water vapor are in a stable equilibrium. This phenomenon is crucial because it serves as a fundamental reference point for calibration and understanding phase behavior. The triple point also ensures that all three phases of a substance have the same Gibbs free energy, which is an essential concept in determining the spontaneous processes in thermodynamic systems.

Enthalpy Change

Enthalpy is a measure of the total energy of a thermodynamic system, encompassing internal energy plus the product of pressure and volume. When it comes to phase changes, like transitioning from solid to liquid, the change in enthalpy (\( \text{ΔH} \)) is key. It quantifies the amount of heat required to change the phase at constant pressure. In the given problem, the change in enthalpy can be determined by subtracting the enthalpy of the initial phase (solid) from the enthalpy of the final phase (liquid). For water at its triple point, the enthalpy change when going from solid to liquid is \[ \text{ΔH} = h_{f, \text{liquid}} - h_{f, \text{solid}} \] Using the steam table values provided, the calculation yields \[ \text{ΔH} = 0 \text{ kJ/kg} - (-333.4 \text{ kJ/kg}) = 333.4 \text{ kJ/kg} \]. This positive value indicates that heat needs to be added to the system to melt the ice into water.

Phase Change

A phase change involves a transition between different states of matter: solid, liquid, and gas. This process occurs at specific temperatures and pressures characteristic to each substance. In the context of the given problem, the phase change refers to the melting of ice to form liquid water at the triple point. When water is heated at the triple point, it undergoes this phase transition without any change in temperature once the required heat is absorbed. The key aspects of phase change are:
• Heat Transfer: Heat must be provided or removed for a phase change to occur. For melting ice, heat is absorbed.
• Latent Heat: This is the energy needed for phase change without changing temperature, reflected as the enthalpy change. The latent heat of fusion for water at its triple point is 333.4 kJ/kg.
• Constant Pressure: The pressure remains constant during the phase change, ensuring that the process remains isobaric.
Phase changes are crucial in many engineering and natural processes, such as in refrigeration, meteorology, and material sciences, where controlling or utilizing these changes is essential.