Entropy Changes: Phase Changes — Compute entropy changes for phase changes and idealized paths

The concept of entropy arose from efforts to understand the fundamental limits of heat engines and the irreversibility of natural processes. In the early nineteenth century, engineers and physicists recognized that heat could not be converted entirely into work, and this realization demanded a new thermodynamic quantity to describe the 'lost' capacity for useful transformation. Phase changes—melting, boiling, sublimation—offered some of the cleanest experimental windows into this new quantity, because they occur at constant temperature and pressure, simplifying the mathematical treatment considerably. Understanding how entropy changes during these transitions became a cornerstone of classical thermodynamics and remains essential for predicting equilibrium, designing industrial processes, and connecting macroscopic measurements to molecular-level disorder.

The central question that motivated these developments is deceptively simple: how do we quantify the entropy change when a substance undergoes a phase transition, and how do we extend that calculation to processes that follow non-trivial paths through state space? Because entropy is a state function, we are free to devise any idealized reversible path connecting the initial and final states, compute ΔS along that path, and the result will be valid for any real process between the same endpoints. This powerful strategy is the focus of this lesson.

Before computing entropy changes for phase transitions, several foundational ideas must be firmly in place. The overarching principle is that entropy is a state function: its value depends only on the current thermodynamic state (T, P, phase, composition), not on the path by which that state was reached. This path-independence is what licenses us to replace a messy irreversible process with a convenient reversible one for calculation purposes. The Clausius definition provides the operational recipe: for a reversible process, the infinitesimal entropy change is dS = δq_rev / T. When integrated over a finite process, ΔS = ∫ δq_rev / T, and evaluating this integral under various constraints yields all the expressions used in this lesson.

The diagram above encodes the entire strategy for computing ΔS between any two states that span one or more phase boundaries. Starting from a temperature in the solid phase and ending in the gas phase, for example, the total entropy change is the sum of five additive contributions: heating the solid to T_fus, the isothermal entropy of fusion, heating the liquid from T_fus to T_vap, the isothermal entropy of vaporization, and finally heating the gas from T_vap to the final temperature. Each of these segments is evaluated independently and then summed, exploiting the additivity of entropy as a state function.

We now derive the key expressions used throughout this lesson. All derivations start from the Clausius equality for reversible processes: dS = δq_rev / T. For an isobaric (constant-pressure) process, δq_rev = dH = C_p dT (within a phase) or δq_rev = Δ_trsH (at a phase boundary).

The real power of the state-function property of entropy is unleashed when a phase change occurs under conditions that are irreversible. Consider a classic example: liquid water supercooled to −10 °C (263.15 K) at 1 atm suddenly crystallizes. This process is irreversible—it happens spontaneously and cannot be reversed by an infinitesimal change in conditions. Clausius's definition dS = δq_rev / T cannot be applied directly to the actual irreversible path. Instead, we construct a hypothetical multi-step reversible path connecting the same initial state (liquid at 263.15 K) and final state (ice at 263.15 K). Because ΔS is path-independent, the entropy change computed along this hypothetical path is exactly the entropy change of the irreversible process.

This strategy generalizes immediately. Whenever a real process involves a phase change at a temperature other than the equilibrium transition temperature, or involves heating across multiple phase boundaries, you decompose the path into segments where the entropy integral can be evaluated analytically. The only data you need are the heat capacities of each phase and the enthalpies of transition at the equilibrium temperatures. This approach is used extensively in materials science, chemical engineering, and geochemistry to compute absolute entropies from 0 K calorimetric data using the Third Law.

Calculate the molar entropy change of the system when supercooled liquid water at −10 °C (263.15 K) freezes irreversibly to ice at −10 °C at 1 atm. Use the following data: Δ_fusH = 6.01 kJ mol⁻¹ at 273.15 K, C_p(liquid) = 75.3 J mol⁻¹ K⁻¹, C_p(ice) = 38.0 J mol⁻¹ K⁻¹. Assume heat capacities are independent of temperature over this range.

The techniques developed in this lesson form the foundation for several more advanced thermodynamic treatments. In particular, the ability to compute ΔS along idealized paths feeds directly into the Third Law of Thermodynamics and the determination of absolute (Third-Law) entropies. By integrating C_p/T from near 0 K to any desired temperature—pausing to add Δ_trsH/T_trs at each phase boundary—we obtain S°(T), the standard molar entropy at temperature T. Furthermore, these entropy calculations underpin the Gibbs energy criterion for spontaneity: ΔG = ΔH − TΔS, which is the central quantity in chemical equilibrium, electrochemistry, and biochemistry.

Looking forward, when you study the Gibbs phase rule and phase diagrams, you will see that the entropy difference between coexisting phases is precisely what determines the slope of the phase boundary line. The Clausius–Clapeyron equation, dP/dT = Δ_trsS / Δ_trsV, directly employs the Δ_trsS values you are now learning to compute. Mastery of entropy calculations for phase changes therefore serves as the quantitative gateway to understanding all of chemical phase equilibrium.