[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665588-105":59,"doc-detail-665588-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","chapter-8-phase-diagrams-degrees-of-freedom-key-concepts","Chapter 8 - Phase Diagrams - Degrees of Freedom - Key Concepts","","Phase Diagrams describe thermodynamically distinct phases in a material system and the equilibrium conditions between them as temperature, pressure, and composition vary. The notes define phases, illustrate solid–liquid–vapour behavior in water, and present the Gibbs phase rule using variables P, C, and degrees of freedom F. They also show how to construct phase diagrams using cooling curves, then extend the framework to binary systems, including isomorphous alloys and the liquidus/solidus concepts, culminating in the lever rule for phase fractions.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/chapter-8-phase-diagrams-degrees-of-freedom-key-concepts/665588/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/chapter-8-phase-diagrams-degrees-of-freedom-key-concepts/665588.png","ImageObject",300,407,{"name":92,"@type":93},"cimolpedes","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is a phase, and why is its definition important for phase diagrams?","Question",{"text":112,"@type":113},"A phase is a region with distinct microstructure or composition. Phase diagrams track equilibrium between such thermodynamically distinct phases, so clear phase definitions determine what regions on the graph mean.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is the number of degrees of freedom determined using the Gibbs phase rule?",{"text":117,"@type":113},"For a closed system at equilibrium, the Gibbs phase rule is F + P = C + 2. Here F is degrees of freedom (variance), P is the number of phases, and C is the number of components.",{"name":119,"@type":110,"acceptedAnswer":120},"How do cooling curves help construct phase diagrams?",{"text":121,"@type":113},"Cooling curves record temperature versus time as a material cools from molten state through solidification to room temperature at constant pressure. Plateau regions (thermal arrest) indicate phase transition temperatures, while the later drop shows when solidification completes.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665588,1793305055,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},962094243610,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Chapter 8  \nPhase Diagrams  \nA phase in a material is a region that differ in its microstructure and or composition from another region  \nAl2CuMg  \nAl  \nH2 O(solid, ice) in H2 O (liquid) ⇒ 2 phases  \n• homogeneous in crystal structure and atomic arrangement  \n• have same chemical and physical properties throughout  \n• have a definite interface and able to be mechanically separated from its surroundings  \nChapter 8 in Smith & Hashemi  \nAdditional resources: Callister, chapter 9 and 10  \nChapter 8 1  \nPhase diagram and “degrees of freedom”  \nA phase diagrams is a type of graph used to show the equilibrium conditions between the thermodynamically-distinct phases; or to show what phases are present in the material system at various T, p, and compositions  \n• “equilibrium” is important: phase diagrams are determined by using slow cooling conditions ⇒ no information about kinetics  \nDegree of freedom (or variance) F is the number of variables (T, p, and composition) that can be changed independently without changing the phases of the system  \nPhase diagram of CO2 Chapter 8 2  \n8.1 Phase Diagram of Water  \n3 phases: solid, liquid, vapour Triple point:  \n4.579 Torr (~603Pa), 0.0098oC  \n• Field – 1 phase  \n• Line – phase coexistence, 2 phases  \n• Triple point – 3 phases  \n3  \n8.2 Gibbs Phase Rule  \nGibbs' phase rule describes the possible \\# of degrees of freedom (F) in a closed system at equilibrium, in terms of the number of separate phases (P) and the number of chemical components ( C) in the system (derived from thermodynamic principles by Josiah W. Gibbs in the 1870s)  \nF + P = C + 2  \nF is \\# of degrees of freedom or variance  \nP is \\# of phases  \nC is \\# of components Component is the minimum \\# of species necessary to define the composition of the system H2O C=1  \n(i) P=1, F=2;  \n(ii) P=2, F=1;  \n(iii) P=3, F=0  \nChapter 8 4  \n8.3 How to construct phase diagrams? -Cooling curves  \nCooling curves:  \n• used to determine phase transition temperature  \n• record T of material vs time, as it cools from its molten state through solidification and finally to RT (at a constant pressure!!!)  \nThe cooling curve of a pure metal  \nBC: plateaue or region of thermal arrest; in this region material is in the form of solid and liquid phases  \nCD: solidification is completed, T drops  \nChapter 8 5  \nCooling curve for pure iron @ 1atm  \nAs T ⇓ : melted iron (liquid) ⇒ bcc Fe, δ (solid) ⇒ fcc Fe, γ (solid) ⇒ bcc Fe, α (RT)  \nChapter 8 6  \n8.4 Binary systems (C = 2)  \nF + P = C + 2 = 4 ⇒ F = 4-P  \nDegrees of freedom (F): p, T, composition  \np  \nAt p = const (or T=const)  \nT F = 3-P  \nT  \ncomposition  \n0 weight % of B 100%   \n100%  \nA  \n100%  \nB  \n1. Two components are completely mixable in liquid and solid phase (form a solid state solution), and don’t react chemically  \n2. Two components (A and B) can form stable compounds or alloys (for example: A, A2 B, A3 B, B) Chapter 8 7  \nBinary Isomorphous Alloy System (C=2)  \nIsomorphous: Two elements are completely soluble in each other in solid and liquid state; substitutional solid state solution can be formed; single type of crystal str. exist Reminder: Hume-Rothery rules: (1) atoms have similar radii ; (2) both pure materials have same crystal structure; (3) similar electronegativity (otherwise may form a compound instead);  \n(4) solute should have higher valence  \nExample: Cu-Ni phase diagram (only for slow cooling conditions)  \nLiquidus line: the line connecting Ts at which liquid starts to solidify under equilibrium conditions Solidus: the temperature at which the last of the liquid phase solidifies  \nBetween liquidus and solidus: P =2  \nChapter 8 8  \n53 wt% Ni – 47 wt% Cu at 1300oC  \nP = 1  \nF = 3 – P = 2  \nP = 2 ; F = 3 – P = 1  \n• contains both liquid and solid phases ⇒ neither of these phases can have average composition 53 wt% Ni – 47 wt% Cu  \n• draw a tie line at 1300oC ⇒ from the graph: composition of liquid phase w L = 45% and solid phase wS = 58% at 1300oC  \nChapter 8 9  \n8.5 The Lever Rul","cbCaikJrAxoYMxss","https://ap.wps.com/l/cbCaikJrAxoYMxss","pdf",2316481,27,"English","# Phase diagram and degrees of freedom\n## Phase diagram of water\n## Gibbs phase rule\n## How to construct phase diagrams - Cooling curves\n## Binary systems (C = 2)\n## Binary isomorphous alloy system\n## The lever rule","[{\"question\":\"What is a phase, and why is its definition important for phase diagrams?\",\"answer\":\"A phase is a region with distinct microstructure or composition. Phase diagrams track equilibrium between such thermodynamically distinct phases, so clear phase definitions determine what regions on the graph mean.\"},{\"question\":\"How is the number of degrees of freedom determined using the Gibbs phase rule?\",\"answer\":\"For a closed system at equilibrium, the Gibbs phase rule is F + P = C + 2. Here F is degrees of freedom (variance), P is the number of phases, and C is the number of components.\"},{\"question\":\"How do cooling curves help construct phase diagrams?\",\"answer\":\"Cooling curves record temperature versus time as a material cools from molten state through solidification to room temperature at constant pressure. Plateau regions (thermal arrest) indicate phase transition temperatures, while the later drop shows when solidification completes.\"}]","Chapter 8 - Phase Diagrams - Degrees of Freedom - Key Concepts | PDF",68]