Phase diagram Study Guide

📖 Core Concepts Phase diagram – chart of pressure, temperature (or other variables) showing where distinct phases coexist at equilibrium. Phase boundary – line separating single‑phase regions; crossing it triggers a phase transition. Triple point – intersection of three phase boundaries; three phases coexist in stable equilibrium. Solidus / Liquidus – temperatures bounding the solid‑liquid coexistence region; below solidus → all solid, above liquidus → all liquid. Critical point – end of the liquid‑gas boundary; beyond it liquid and gas become a single super‑critical fluid. Clausius–Clapeyron (fusion) – relates slope of solid‑liquid boundary to heat of fusion and volume change: $$\frac{dP}{dT}= \frac{\Delta H{\text{fus}}}{T\,\Delta V{\text{fus}}}$$ Gibbs phase rule – degrees of freedom \(F = C - P + 2\) ( \(C\)= components, \(P\)= phases). 📌 Must Remember Positive solid‑liquid slope ⇔ solid denser than liquid; negative slope (e.g., water) ⇔ solid less dense. \(\Delta H{\text{fus}} > 0\) always; sign of slope set by sign of \(\Delta V{\text{fus}}\). At a triple point, \(F = 0\) (no degrees of freedom). Critical point marks disappearance of distinct liquid‑gas boundary. Binary eutectic: single temperature where liquid → two solid phases simultaneously. Peritectic reaction: \(L + \alpha \rightarrow \beta\) on cooling. Ideal liquid solution → Raoult’s law; ideal gas → Dalton’s law. Working‑fluid classification (dry, wet, isentropic, isenthalpic) is based on P–T curve shape. 🔄 Key Processes Reading a P–T diagram Locate region → identify stable phase. Follow a path (e.g., heating at constant pressure) → note phase‑boundary crossings. Using Clausius–Clapeyron for solid‑liquid slope Determine \(\Delta V{\text{fus}}\) sign → predict slope direction. Plug \(\Delta H{\text{fus}}\) and \(T\) to compute quantitative slope if needed. Constructing a binary eutectic diagram Plot temperature vs. composition. Draw liquidus lines from each pure component meeting at eutectic point. Connect solidus lines (two solid phases) to eutectic. Applying Gibbs phase rule Count components \(C\) and phases \(P\). Compute \(F\); if \(F=0\) you are at an invariant point (e.g., triple point, eutectic). 🔍 Key Comparisons Solid‑liquid slope: Positive vs. Negative → solid denser vs. less dense than liquid. Eutectic vs. Peritectic: Eutectic – liquid → two solids at one temperature. Peritectic – liquid + solid α → new solid β on cooling. Ideal solution (Raoult) vs. Real solution: Raoult – vapor pressure proportional to mole fraction of each component. Real – deviations (positive/negative) due to interactions. ⚠️ Common Misunderstandings Assuming all solids are denser than liquids – water, antimony, bismuth are exceptions → negative slope. Confusing triple point with eutectic point – triple point involves three phases of one component; eutectic involves one component’s liquid turning into two solid phases. Thinking critical point eliminates all phase boundaries – only the liquid‑gas boundary disappears; solid‑liquid line remains. 🧠 Mental Models / Intuition “Slope tells density” – picture melting: if solid expands (ΔV > 0), higher pressure pushes melting point down → negative slope. “Degrees of freedom = knobs you can turn” – each independent variable (P, T, composition) is a knob; Gibbs rule tells how many you can adjust without leaving equilibrium. “Triple point = fixed lock” – no freedom; any deviation forces a phase change. 🚩 Exceptions & Edge Cases Negative solid‑liquid slopes only for substances whose solid is less dense (e.g., H₂O, Sb, Bi). Polyamorphism: multiple amorphous states can appear, not captured by simple solid‑liquid‑gas diagrams. Non‑ideal solutions: Raoult’s law fails; need activity coefficients. 📍 When to Use Which Clausius–Clapeyron → estimate slope of a phase boundary when \(\Delta H\) and \(\Delta V\) are known. P–T diagram → quick identification of phase stability under changing pressure/temperature. T–s, h–s, p–h diagrams → analysis of thermodynamic cycles (Carnot, Rankine, vapor‑compression). Binary composition‑temperature diagram → design alloy solidification, locate eutectic/peritectic points. Ternary Gibbs triangle → visualize composition of three‑component systems, read isothermal sections. 👀 Patterns to Recognize Straight‑line solid‑liquid boundaries → approximate constant \(\Delta H{\text{fus}}\) and \(\Delta V{\text{fus}}\). “V‑shaped” liquidus in eutectic systems – two arms meeting at a low‑temperature eutectic point. Horizontal liquid‑gas line ending at critical point – indicates supercritical region above. Tie‑lines in binary vapor‑liquid diagrams – connect coexisting liquid and vapor compositions; they are nearly horizontal at low pressures. 🗂️ Exam Traps Choosing sign of slope – forgetting that a negative slope implies \(\Delta V{\text{fus}}<0\). Mixing up eutectic vs. peritectic diagrams – a peritectic has three phases on one side of the reaction line; eutectic has only liquid on one side. Assuming triple point always at 0 °C, 1 atm – only true for water; each substance has its own triple‑point coordinates. Applying Gibbs phase rule incorrectly – forgetting the “+2” accounts for P and T; for fixed‑pressure/composition problems, adjust accordingly. Treating critical point as a single temperature – it’s a specific pressure‑temperature pair; above both, fluid is supercritical.