Phase Changes

Phases, coexistence, and latent heat

A phase diagram maps the equilibrium state of a pure substance from pressure and temperature. Single-phase regions contain solid, liquid, or vapor. A boundary curve contains two coexisting phases at the stated pressure and temperature. Along the liquid--vapor curve, liquid and vapor have equal equilibrium chemical potential; the same condition defines the solid--liquid and solid--vapor curves. The triple point is the one pressure--temperature state where all three boundaries meet. The liquid--vapor boundary terminates at the critical point. Above that point, liquidlike and gaslike states connect continuously, so boiling and condensation no longer occur as a sharp coexistence transition.

The thermodynamic model must be selected before an energy calculation. Heating at fixed pressure may cross a melting or boiling boundary. Compression at fixed temperature can cross a vapor-condensation boundary. A sealed rigid container follows neither of those simple paths because both pressure and temperature may change. The boundary itself describes equilibrium. Superheated liquid, supercooled liquid, and delayed condensation can persist when nucleation is slow, but their persistence does not move the equilibrium curve. State the imposed pressure, the container condition, and whether the sample has time to reach equilibrium.

Latent heat and heating stages.

Within one phase, a moderate temperature change has energy transfer $Q=mc\Delta T$. At a coexistence temperature and pressure, added or removed energy can change the phase fraction while temperature remains fixed. The corresponding latent-heat term is $Q=mL$, where $L_f$ denotes latent heat of fusion and $L_v$ latent heat of vaporization. Melting and vaporization require positive heat transfer to the sample; freezing and condensation release energy from it. The temperature plateau does not indicate zero energy transfer. It indicates that the energy changes molecular arrangement, separation, and interaction energy rather than the thermal energy represented by a temperature rise.

A heating curve divides a process into stages. Ice below its melting temperature is warmed with $m{c}_{\mathrm{ice}}\Delta T$. At the melting temperature, the required term is $m{L}_f$. Liquid water is then warmed with $m{c}_{\mathrm{water}}\Delta T$. At its boiling temperature, vaporization requires $m{L}_v$, followed by heating of the vapor if energy continues to enter. The sequence reverses during cooling. A mass can remain partly solid and partly liquid, or partly liquid and partly vapor, at a plateau; the phase inventory is then an unknown determined by the available energy.

Calorimetry with a phase change.

An insulated calorimeter uses one energy balance for all objects inside the chosen boundary. With positive $Q$ defined as energy entering each object, ${\sum }_i{Q}_i=0$. A warm liquid cooling to a melting temperature can supply energy to warm an ice sample, melt some or all of it, and warm the resulting liquid. Each stage receives its own term. Apply this sequence: bring every candidate phase to the relevant coexistence temperature, compare the available energy with the latent heat needed for complete conversion, then determine the final phase inventory or continue to a final temperature.

For example, if insufficient energy is available to melt all added ice, the final state contains ice and liquid at the melting temperature. A temperature below that point from an equation that assumed complete melting is physically inconsistent. If enough energy melts all the ice, the remaining energy raises the liquid temperature. The calorimeter cup, thermometer, and any stirrer inside the boundary also store energy and require heat-capacity terms. Heat leakage, evaporation, and unknown initial phase fraction are separate model assumptions that must be reported.

Latent heat depends on the material and on the transition conditions. The tabulated normal boiling point applies at one specified pressure, not at every pressure. A pressure change can shift the coexistence temperature and alter the appropriate energy path. Calorimetry results should state the pressure, phase labels, masses, initial temperatures, and whether the final state is single phase or coexistence.

Microscopic and thermodynamic phase models

Temperature records the thermal state of a sample, but it does not measure all of its internal energy. In a simple molecular model, raising temperature increases the average kinetic energy associated with molecular motion. At a phase boundary, added energy can instead change the relative arrangement of molecules while the two phases remain at the same equilibrium temperature. During melting, a crystal loses part of its ordered binding structure. During vaporization, molecules separate much farther against cohesive attraction. The associated increase in intermolecular energy is a major part of the latent heat. The plateau persists while both phases coexist under the specified pressure and energy continues to convert one phase into the other.

An intermolecular potential has a steep repulsive region at very short separation, an attractive well at moderate separation, and approaches zero for widely separated molecules. The minimum identifies a preferred separation in the simplified model. Heating a solid or liquid changes both kinetic energy and the distribution of separations. A phase conversion changes that distribution substantially. The potential curve is a molecular-pair model, not a complete latent-heat calculation: real samples have many-body interactions, rotational and vibrational modes, defects, and pressure-dependent volume changes. It nevertheless explains why an energy input can be large even when a thermometer reading is fixed.

At constant pressure, the heat absorbed during a reversible phase conversion is the enthalpy change, $Q_p=\Delta H=mL$. For vaporization, part of that energy can appear as expansion work $p\Delta V$ as well as an internal-energy change. At fixed volume, the energy account uses $\Delta U$ rather than automatically using a tabulated constant-pressure latent heat. The usual calorimetry formula therefore carries a condition: the sample follows the pressure and phase path for which $L$ was measured. Boiling in an open vessel, evaporation from an exposed surface, and heating a sealed rigid vessel do not share the same boundary conditions or energy terms.

Ideal phase-change calorimetry assumes rapid internal equilibration, negligible heat transfer through the container wall, a known pressure, and tabulated heat capacities and latent heats that remain appropriate over the stated range. It also treats the phase transition as complete equilibrium rather than a kinetic process. Nucleation barriers can allow supercooling or superheating, so a sample can pass the nominal transition temperature before the new phase appears. Evaporation can remove mass and energy from an open vessel. Dissolved material shifts transition conditions. These effects change the phase inventory or the energy balance; they are not small algebra corrections when the final state lies near a coexistence boundary. Pressure control and phase identification remain experimental requirements throughout the calculation.

The experimental check is a final-state check. Measure the final temperature, mass, and visible phases, then compare them with the staged balance. If ice remains in an ice--water mixture at the stated pressure, the mixture should be at its melting temperature within measurement uncertainty. If a computed final state requires a temperature on the wrong side of a coexistence boundary while both phases were assumed present, the assumed phase path or an omitted energy term is inconsistent.

Clausius--Clapeyron relation and vapor pressure.

The slope of an equilibrium coexistence curve follows from the equality of the two phases' molar Gibbs free energies. A transition between phases 1 and 2 obeys the Clapeyron relation

where $\Delta h$ and $\Delta v$ are the molar enthalpy and molar-volume changes from phase 1 to phase 2. The numerator is positive for ordinary melting and vaporization. For liquid--vapor coexistence, the vapor molar volume exceeds the liquid molar volume, so vapor pressure rises with temperature. The solid--liquid slope depends on the sign of the volume change. Water contracts on melting near ordinary pressure, giving its solid--liquid boundary a slope opposite to that of many substances.

Vapor pressure is the equilibrium pressure of vapor above its condensed phase at a specified temperature. Measure it in a sealed cell containing both liquid and vapor, hold the cell at a uniform temperature, allow time for equilibration, and record the pressure after subtracting or excluding noncondensable gas. A vessel containing only vapor after all liquid has evaporated does not necessarily report the saturated vapor pressure; its pressure also depends on total amount and container volume. Temperature gradients between the liquid surface and the pressure sensor produce a reading that does not correspond to one equilibrium state.

For liquid--vapor data away from the critical point, take the vapor as ideal, neglect the liquid molar volume relative to the vapor molar volume, and treat the enthalpy of vaporization as approximately constant. The Clapeyron relation then becomes

Thus a plot of $\ln p_{\mathrm{sat}}$ against $1/T$ is approximately linear, with slope $-\Delta h_{\mathrm{vap}}/R$. Use absolute temperature and consistent pressure units in the logarithm ratio. The relation estimates a vaporization enthalpy from pressure data or predicts a pressure ratio over a limited temperature interval; it does not replace direct measurements when the assumptions fail.

Data reduction starts with absolute pressure. A gauge pressure must be increased by the contemporaneous ambient pressure before it is used in a saturation-pressure ratio. Temperature sensors require calibration on an absolute-temperature scale; an offset in temperature distorts the inverse-temperature axis most strongly at the lower end of a data set. Record the uncertainty of each pressure and temperature reading, the equilibration time, and whether liquid remained visible. A weighted linear fit to the logarithmic plot estimates the slope only when its residuals have no systematic curvature. Repeated points at the same temperature test drift and sensor hysteresis. Measurements taken while the bath temperature is changing rapidly can lag the liquid surface temperature and bias the inferred enthalpy. The reported temperature interval belongs with every fitted vaporization enthalpy because the constant-enthalpy approximation is local rather than universal.

Several limits require the full relation or measured property data. Enthalpy of vaporization varies with temperature and approaches zero at the critical point. Vapor nonideality grows at high density, and liquid volume can no longer be ignored at high pressure. Near a triple point, solid, liquid, and vapor branches require the appropriate phase pair and volume change. Metastable superheating or supercooling can delay nucleation, so a measured pressure may temporarily differ from the equilibrium saturation value. Calibration error, trapped gas, sensor lag, and a leaking seal are experimental errors rather than thermodynamic corrections.

Mixtures, measurement, and calorimetry

In a gas mixture, the total pressure is the sum of partial pressures, $p={\sum }_i{p}_i$. Air above liquid water contains dry-air components and water vapor. At equilibrium with pure liquid water, the water-vapor partial pressure equals the saturation pressure at the liquid temperature, while the dry-air partial pressures add to the total. Relative humidity is $H=p_{\mathrm{water}}/p_{\mathrm{sat}}(T)$, expressed as a fraction or percentage. It is not the mass fraction of water in air. Cooling a sample of humid air at fixed water-vapor partial pressure raises $H$ because saturation pressure decreases. The dew-point temperature $T_d$ satisfies $p_{\mathrm{sat}}(T_d)=p_{\mathrm{water}}$; below that temperature, excess water vapor condenses if nucleation sites are available.

Boiling is set by a pressure condition. A liquid boils when its equilibrium vapor pressure equals the external pressure acting on the liquid surface. Lower external pressure intersects the vapor-pressure curve at a lower temperature; higher external pressure requires a higher temperature. A pressure cooker raises the boiling temperature by raising the external pressure. The statement applies to a specified liquid composition. At a given external pressure, a nonvolatile solute lowers the solvent activity and reduces the solvent vapor pressure. More heating is then needed to reach the boiling condition, producing boiling-point elevation.

An ideal solution with a volatile component $i$ obeys Raoult's law, $p_i=x_i{p}_i^{\mathrm{sat}}(T)$, where $x_i$ is its liquid mole fraction. The total vapor pressure is the sum of the component partial pressures. Vapor composition generally differs from liquid composition because the more volatile component contributes a larger fraction of the vapor. Distillation, evaporation of a mixed solvent, and boiling of an alcohol--water mixture therefore change composition as material leaves the liquid. A pure-substance boiling table cannot be used without a composition model for such systems.

Mixture measurements require composition and phase labels at every observation. Mass fraction, mole fraction, and relative humidity are different quantities and cannot be substituted for one another. A humidity sensor may report relative humidity directly, but the water-vapor partial pressure requires the simultaneous air temperature and a saturation-pressure relation. Dew-point instruments infer water-vapor partial pressure from the temperature of first condensation; surface contamination and optical detection threshold affect that reading. For boiling-point data, measure external absolute pressure, liquid composition, sample mass, and any composition change caused by evaporation.

Uncertainty grows near a phase boundary. A small temperature error can produce a large relative error in saturation pressure where the vapor-pressure curve is steep. Pressure-sensor offset shifts an inferred boiling point. Nonideal solution behavior, dissolved gases, and incomplete mixing can dominate the error in a mixture model. Report whether the measured value is a vapor partial pressure, total pressure, relative humidity, dew point, or boiling temperature; each constrains a different thermodynamic quantity.

A controlled boiling measurement holds the external pressure and composition as nearly constant as practical. Stirring reduces temperature gradients but can increase evaporation, so mass should be checked before and after the run. A volatile mixture requires liquid and vapor compositions to be sampled, or its composition assumption to be stated. Relative-humidity measurements require sensor equilibration time and airflow conditions. A sensor placed near a cold wall can report a local humidity that does not represent the bulk gas. Repeated measurements during warming and cooling identify hysteresis, condensation on the sensor, and incomplete thermal equilibrium.

Free energy and phase-change cycles

A pure substance has chemical potential equal to molar Gibbs free energy, $\mu =g_m$. At fixed temperature and pressure, the stable phase has the lower chemical potential. Two phases can coexist only when their chemical potentials are equal:

At that equality, converting a small amount of material from one phase to the other does not change the total Gibbs free energy to first order. Away from the equality, the phase with lower $\mu$ reduces $G$ when it grows. This criterion applies to solid--liquid, liquid--vapor, and solid--vapor boundaries. For mixtures, each component has its own chemical potential, so phase equilibrium requires equality of the chemical potential of every component that can transfer between phases.

The temperature dependence at fixed pressure follows $\mathbf{d}\mu =-s_m\mathbf{d}T+v_m\mathbf{d}p$. Applying this relation to two coexisting phases and differentiating their equality gives the Clapeyron slope. The entropy and volume differences determine how the boundary moves through the pressure--temperature plane. A phase diagram expresses the free-energy condition for coexistence rather than a catalogue of observed appearances. A phase label based on temperature alone is incomplete unless pressure and composition are also fixed.

Metastable states require a separate statement. A supercooled liquid below its equilibrium freezing temperature can have a higher chemical potential than the solid and still persist because a new solid nucleus must form. Surface energy creates a nucleation barrier for small clusters. Dust, scratches, dissolved particles, and mechanical agitation can lower that barrier and trigger rapid conversion. A superheated liquid or supersaturated vapor has the analogous history dependence. Such states are suitable experimental preparations, but they are not evidence that the equilibrium coexistence condition has moved.

Temperature and pressure control determine which free-energy comparison is relevant. A thermostatic bath fixes the sample temperature only after internal gradients have relaxed. A piston, pressure regulator, or vapor reservoir can impose pressure, but a hydrostatic column produces a pressure variation with height. Mechanical equilibrium also requires matching pressure across a flat phase boundary; a curved interface can have an additional pressure difference from surface tension. Finite samples can have wall interactions and confinement that shift observed transition behavior relative to bulk tabulated values.

Equilibrium measurements need a protocol that separates controlled variables from observed variables. Set the bath temperature, allow the sample and sensor to settle, record absolute pressure, and verify the phases by imaging or an independent density or volume measurement. Repeat after approaching the target from higher and lower temperature. A reproducible transition offset between the two directions indicates hysteresis or sensor lag. The uncertainty statement should include bath stability, sample temperature gradient, pressure calibration, composition, equilibration time, and the criterion used to identify the phase boundary.

Finite samples round an ideal boundary in practical measurements. A thermometer has finite response time, a pressure regulator has finite resolution, and a sample can contain a temperature gradient while it is changing phase. The reported coexistence temperature should therefore be an interval or fitted value with a stated pressure, not an isolated display reading. Repeating the measurement with different sample sizes and wall materials helps separate bulk equilibrium behavior from container effects and nucleation history.

Heat engines, refrigeration, and phase-change cycles.

Phase-change devices use a working fluid that repeatedly evaporates, compresses, condenses, and expands. In an evaporator, low-pressure liquid--vapor mixture absorbs energy from a cold region while liquid converts to vapor. The latent heat permits a large energy transfer at a nearly fixed working-fluid temperature. A compressor then raises the pressure of the vapor and requires work input. In a condenser, the higher-pressure vapor rejects energy and condenses. An expansion valve reduces the pressure before the fluid returns to the evaporator. The valve is commonly modeled as a throttling device with nearly constant enthalpy, while the compressor, heat exchangers, and piping require separate control-volume energy balances.

The same sequence can serve refrigeration, heat pumping, or power production depending on the direction of work and the selected useful output. A refrigerator uses work input $W_{\mathrm{in}}$ to remove energy $Q_c$ from its cold region and rejects $Q_h=Q_c+W_{\mathrm{in}}$ to a warmer region. Its coefficient of performance is

A heat pump delivers the warm-side energy, with ${\mathrm{COP}}_{\mathrm{HP}}=Q_h/W_{\mathrm{in}}={\mathrm{COP}}_{\mathrm{R}}+1$. A COP can exceed one because it is a heat-transfer ratio, not a conversion efficiency. A heat engine reverses the useful-work convention: for a complete cycle, $W_{\mathrm{out}}=Q_h-Q_c$.

Working-fluid state labels matter in measurements. The evaporator outlet should be mostly vapor before the compressor, because liquid entering a compressor can damage it. The condenser outlet is usually liquid or subcooled liquid. Pressure readings give saturation temperatures only when the working fluid is at saturation and its composition is known. Superheated vapor and subcooled liquid require temperature as well as pressure to identify the state. Refrigerants can be mixtures, so leakage or fractionation changes the composition and shifts the pressure--temperature relation.

Cycle performance requires steady repeating operation. Over one period, the working fluid returns to its initial state, so its net internal-energy change is zero. The measured electric power to a compressor includes motor and controller losses; it is not automatically the shaft work delivered to the fluid. Fan power, pumps, defrost heaters, and heat leakage through insulation belong inside or outside the device boundary according to the stated COP definition. A test records inlet and outlet temperatures, pressures, mass flow, electrical power, and heat-exchanger energy transfer over the same averaged interval.

Ideal cycle comparisons omit finite temperature differences in heat exchangers, pressure drops in piping, nonideal compression, throttling irreversibility, and parasitic heat loads. These effects lower the observed COP and shift the working-fluid states away from ideal saturation points. A phase-change cycle remains an energy balance with specified control volumes; latent heat is one transfer mechanism within that balance, not a separate source of energy.

Heat-transfer measurements require the same boundary discipline as the cycle model. An evaporator load may be inferred from the cooled-air mass flow and temperature change, from a secondary liquid loop, or from working-fluid enthalpy data; those methods have different sensor and property uncertainties. Electrical input should include every component placed inside the reported device boundary. A short cycling interval can store energy in heat exchangers and cabinet walls, so an instantaneous ratio of heat flow to power need not equal the period-averaged COP. State the averaging interval, operating pressures, ambient conditions, and defrost status with any reported cycle performance.

Critical phenomena and quantitative checks

The liquid--vapor coexistence curve ends at a critical temperature and pressure. Approaching that point from below, liquid and vapor densities converge, the interface becomes diffuse, and the latent heat of vaporization approaches zero. Above the critical point there is one supercritical fluid region; a path can connect states that would be called liquidlike and gaslike without crossing a coexistence boundary. The critical point is distinct from the triple point, where three phases coexist. Near critical conditions, density fluctuations become large and simple ideal-gas or constant-latent-heat approximations lose accuracy.

Supercooling and superheating occur when a system crosses an equilibrium boundary without immediately forming the stable phase. A liquid cooled below its freezing temperature remains metastable until a solid nucleus forms. A liquid heated above its normal boiling temperature can remain liquid when bubble formation is inhibited. These observations are kinetic: the stable phase has lower Gibbs free energy under the imposed conditions, but conversion requires an interface and an initial cluster. The waiting time depends on sample volume, impurities, surface roughness, agitation, and the rate at which temperature or pressure is changed.

Classical homogeneous nucleation models a spherical nucleus with free-energy change

where $\gamma$ is interfacial energy and $\Delta g_v$ is the positive bulk free-energy driving scale. The surface term opposes small nuclei, whereas the volume term favors growth. The maximum occurs at $r^{\ast }=2\gamma /\Delta g_v$. Nuclei smaller than this critical size tend to disappear; larger nuclei reduce free energy by growing. A surface, dust particle, or scratch can lower the barrier through heterogeneous nucleation and substantially shorten the observed waiting time.

Measurement-rate dependence follows from this barrier. Fast cooling can carry a liquid farther below its equilibrium freezing point before a nucleus appears. Slow cooling, seeding, or rough container walls usually reduce the observed supercooling. Record the cooling rate, sample volume, stirring condition, surface preparation, and number of trials. A single observed freezing temperature is a random nucleation event, not a complete equilibrium boundary measurement. Repeated transition temperatures form a distribution whose width can be larger than thermometer resolution.

A constant nucleation rate $J$ per unit volume gives a simple independent-event model with probability of no nucleation over time $t$ as $\exp (-JVt)$. The probability of observing a transition therefore increases with sample volume and observation time even at fixed temperature. Real experiments rarely maintain constant $J$: cooling changes the driving force, and surfaces can become activated or deactivated. Still, the relation explains why transition temperatures must be reported with cooling protocol and sample size. Comparing only the mean transition temperature discards information carried by the spread and timing of the events.

Classical nucleation uses a bulk interfacial energy, a spherical cluster, and a well-defined bulk driving force. It becomes unreliable for molecular-scale clusters, strong confinement, complex mixtures, and the critical region where interfaces and bulk phases lose their sharp distinction. Use measured transition statistics and appropriate microscopic or mixture models when those assumptions control the result. Experimental nucleation rates require repeated trials, a defined detection threshold, and a stated observation window rather than one visually identified transition.

Quantitative calorimetry and uncertainty propagation.

Phase-change calorimetry begins with a system boundary and a sign convention. Let positive $Q$ denote energy entering each object inside an insulated calorimeter. A general final-state balance can be written as

The first sum contains sensible-heat terms for objects that remain in one phase. $C_{\mathrm{cal}}$ is the heat capacity of the cup, thermometer, stirrer, and other calorimeter components that change temperature. The latent term represents a mass $m_{p}f$ converted at the coexistence temperature. Its sign is fixed by the selected phase direction: melting and vaporization require energy entering the phase-changing sample, while freezing and condensation release energy. $Q_{\mathrm{loss}}$ represents energy crossing the stated boundary; it is zero only under an explicit insulation approximation.

The phase fraction $f$ must satisfy $0\le f\le 1$. An algebraic solution outside that range identifies the wrong final-state assumption. If a computed melting fraction is less than one, solid and liquid coexist at the melting temperature and no subsequent liquid-warming term belongs in the balance. If it exceeds one, set $f=1$, include the complete latent term, and use the remaining energy to determine a final temperature. The same test applies to partial freezing, boiling, and condensation. A staged calculation prevents a temperature equation from continuing through a transition that has not yet completed.

Calorimeter heat capacity requires calibration under the same practical conditions as the phase-change experiment. Mix known masses of one liquid at two measured temperatures, solve the no-phase-change energy balance for $C_{\mathrm{cal}}$, and repeat over the relevant temperature range. A calibration value includes the immersed sensor and stirrer only if they are present in both runs. Changing fill level, lid position, or probe depth can change the effective heat capacity and heat-loss rate. A tabulated cup material value is not a substitute for calibration of the assembled instrument.

Heat loss and sensor lag bias the inferred latent heat in opposite directions in some protocols. A simple heat-leak model uses an integrated loss proportional to the temperature difference between calorimeter and surroundings, but the coefficient must be measured from a blank cooling or warming run. A sensor with time constant $\tau$ reports a delayed temperature during rapid stirring or phase conversion. Record the time series and fit an equilibrium temperature after the contents have settled; selecting the first displayed plateau can bias both $T_f$ and $f$.

At a coexistence final state, the inferred phase fraction often has the form $f=Q_{\mathrm{avail}}/(m_{p}L)$ after all sensible terms have been evaluated at the transition temperature. Its relative uncertainty is sensitive to the available-energy estimate, sample mass, and latent heat. A small difference between two large sensible terms can make $Q_{\mathrm{avail}}$ poorly determined even when each temperature reading appears precise. Report the sign and magnitude of every contribution before forming the residual latent term. The final temperature and visible phase inventory provide independent checks: a nonzero inferred ice fraction and a final liquid temperature above the melting point cannot both describe equilibrium at the stated pressure.

For independent input estimates $x_k$, first-order propagation uses

Mass, temperature, heat-capacity, latent-heat, and calibration uncertainties enter the inferred phase fraction or latent heat through this relation. Shared thermometer calibration, common mass-scale bias, and a fitted heat-loss coefficient create correlations, so independent-error addition can understate or overstate the result. Report the model, measured final phases, calibration data, time window, heat-loss correction, and uncertainty method with every calorimetric latent-heat value. Blank calorimeter runs quantify drift that a single final-temperature reading cannot separate from phase-change energy.

Data tables, phase fractions, and worked calorimetry checks.

A staged calculation starts with the possible final states, not with one universal temperature equation. List every initial phase, its mass, initial temperature, specific heat, and coexistence temperature at the stated pressure. Identify the energy required to bring each candidate material to a phase boundary. Next compare the available energy with the latent energy for complete conversion. Only after that comparison is it valid to include a temperature change of the newly formed phase. This order enforces the physical bounds on phase fraction and prevents a solution from warming liquid that has not yet formed.

The numerical result has several immediate checks. The final temperature lies at the melting point because two phases remain. The inferred fraction lies between zero and one. The energy released by cooling the warm components has the same magnitude as the energy absorbed in melting. A reported final temperature below zero with ice and liquid at ordinary pressure, or above zero with unmelted ice in equilibrium, signals a missing term, an invalid pressure assumption, or a sign error. The calorimeter capacity must be included because omitting its $1.20\mathrm{kJ}$ energy release would reduce the inferred melt fraction appreciably.

Validation limits arise from the model assumptions. The ice is treated as initially at its melting temperature; colder ice requires a warming term before melting. The water heat capacity and latent heat are treated as known constants over the stated range. Heat leakage, evaporation, dissolved solute, incomplete stirring, and sensor lag shift the available energy. If the final phases cannot be observed directly, the phase-fraction inference should include a confidence interval from masses, temperatures, calibration uncertainty, and heat-loss correction rather than a single exact decimal.

A data table with repeated runs retains raw masses, initial temperatures, calorimeter condition, final phase observation, elapsed time, and calculated energy terms. Plot residual energy against elapsed time and initial temperature. A systematic trend indicates heat loss or an uncalibrated heat capacity; random scatter supports the uncertainty model but does not prove that the assumed phase path was correct.

A rigorous workflow records each branch decision before numerical substitution. Test the two-phase endpoint first whenever a supplied energy source or sink can reach a coexistence temperature. State the amount of energy available at that endpoint, the latent energy required for complete conversion, and the resulting inequality. If complete conversion is possible, carry the residual energy into the next sensible stage. If conversion is incomplete, stop at coexistence and report the remaining mass of each phase. This branch structure makes independent checking possible: a reader can locate whether a disagreement arose from a mass, a sign convention, a calorimeter calibration, or the selected final-state hypothesis.

Numerical precision should follow the input data and model resolution. A temperature probe reading to one tenth of a kelvin does not support a phase fraction reported to six significant digits when the heat-loss correction is uncertain. Preserve extra digits during intermediate subtraction, then round the final phase fraction and latent heat according to propagated uncertainty. Recalculate the balance from the reported rounded values as a final check; the residual should be consistent with the stated uncertainty rather than hidden by rounding.

Pressure control and experimental conditions

Phase-change data are functions of pressure, composition, and sample history. The melting point of a pure substance is often treated as fixed in a classroom calorimeter because ordinary atmospheric-pressure variations make a small change for many materials. Boiling measurements are more sensitive. A reduced ambient pressure lowers boiling temperature, and a sealed vessel can build vapor pressure far above the room value. Record barometric pressure or vessel pressure whenever a temperature is compared with a tabulated boiling or saturation value.

Composition changes the phase boundary. Dissolved solute shifts the chemical potential of a liquid and can depress the freezing temperature or elevate the boiling temperature. Evaporation from an open sample can change concentration during a long heating run. A nominally pure water--ice exercise therefore differs from a salt solution, a mixture of volatile liquids, or a material with trapped gas. Sample preparation, lid condition, stirring, and elapsed time belong in the experimental record when they can alter composition or pressure.

The sample may also retain a thermal gradient. A thermometer near a heater can read above the bulk temperature before stirring has mixed the calorimeter. A probe embedded in an ice cluster can read a local coexistence temperature while surrounding liquid remains warmer. Measure at multiple locations or stir under a declared protocol, then wait for the temperature record to approach a stable value. The equilibrium calculation applies to the equilibrated state, not to an instantaneous local sensor value during heat transfer.

Pressure, composition, and thermal uniformity provide physical checks on a phase identification. A measured plateau that shifts with pressure may be a true boiling or melting boundary. A drifting plateau during evaporation may reflect changing composition or heat loss. Pairing the temperature trace with a phase observation, mass record, and pressure record distinguishes those mechanisms and keeps the latent-heat model attached to the state that was actually measured.