[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665337-105":59,"doc-detail-665337-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-6-phase-transitions-phase-transitions","Chapter 6 - Phase transitions - Phase transitions","","Phase transitions are presented as distinct states of matter characterized by macroscopic properties and separated by phase boundaries where microscopic properties change. The chapter explains first-order and second-order transitions in the pressure–temperature phase diagram, using Gibbs enthalpy to show continuity of G but discontinuities in entropy and volume. Critical and triple points for water are discussed alongside latent heat and the condition for phase coexistence, culminating in the Clausius–Clapeyron relation and Ehrenfest classification.",{"@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":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/chapter-6-phase-transitions-phase-transitions/665337/",{"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-6-phase-transitions-phase-transitions/665337.png","ImageObject",300,407,{"name":92,"@type":93},"kopisore","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What distinguishes phases and how do phase boundaries affect microscopic states?","Question",{"text":112,"@type":113},"Phases are states of matter with distinct macroscopic properties. At the phase boundary, microscopic properties change by definition, leading to different behaviors for first- versus second-order transitions.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the Gibbs enthalpy behave across a first-order phase boundary?",{"text":117,"@type":113},"Gibbs enthalpy G changes continuously across the phase boundary for a first-order transition, while entropy S and volume V (derived from derivatives of G) are discontinuous.",{"name":119,"@type":110,"acceptedAnswer":120},"How is the Clausius–Clapeyron equation used in this chapter?",{"text":121,"@type":113},"The chapter derives a slope relation along the equilibrium transition line in the P–T diagram. It can be used as a consistency check or to determine latent heat by measuring temperature, volume change, and the slope dP/dT.","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},665337,1793298828,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"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":41},962090880963,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Chapter 6  \nPhase transitions  \n6.1 Concept of phase  \nPhases are states of matter characterized by distinct macroscopic properties. Typical phases we will discuss in this chapter are liquid, solid and gas. Other important phases are superconducting and magnetic states.  \nFirst and second order phase transitions. States of matter come with their stability regions, the phase diagram. The properties of the microscopic state change by definition at the phase boundary. This change is  \ndiscontinuous continuous  \n􀀁  \nfor a  \n􀀂  \nfirst order second order  \n􀀁 phase transition  \nThe appropriate variables for phase diagram of water are the pressure P and the temperature T.  \ncritical point : The first-order phase boundary between gas and liquid becomes second order right at the critical point. The two phases have then equal densities and specific entropies (entropy per particle) .  \n􀀁 There is no critical point for the liquid-solid transition.  \ntriple point : The point at which gas, liquid and solid coexist.  \n64 CHAPTER 6 . PHASE TRANSITIONS  \nSkating on ice. The melting curve of water has negative slope. Ice does hence melt at constant temperature T \u003C T0 when increasing the pressure P. This happens during skating on ice.  \n6.2 First-order phase transition  \nWhen water starts boiling, it undergoes a phase transition from a liquid to a gas phase. For both phases independently, the equation of state is a well-defined regular function, continuous, with continuous derivatives. However, while going from liquid to gas one function “abruptly” changes to the other function. Such a transition is of first-order.  \nGibbs enthalpy. Phase transitions in the P − T phase diagram are described by the Gibbs enthalpy G(T, P, N), as defined by (5.11), which is itself a function of the pressure P and of the temperature T. G (T, P, N) changes continuously across the phase boundary when the transition is of first order. The entropy S and volume V , which are given by the derivatives  \nS = − 􀀃GT􀀄 P , V = 􀀃 GP􀀄 T , dG = −SdT + V dP + µdN of the Gibbs potential, are in contrast discontinuous.  \nLatent heat. Let us consider an instead of the P − T the P − V diagram, which is a projection of the equation of state for water.  \nTwo phases 1 and 2 coexisting at a temperature T0 have differententropies S1 and S2 . The system must therefore absorb or release heat, the latent heat ΔQL ,  \nΔQL = T0 (S2 − S1 ) ,  \nduring a phase transition of first order.  \n6.2.1 Condition for phase coexistence  \nWe consider a system composed by one species of particles at given conditions of P and T (constant) . Let us assume coexistence of two phases in our system. In this case, the  \n6.2. FIRST-ORDER PHASE TRANSITION 65  \nGibbs potential is the sum of Gibbs potentials of the two phases,  \nG (T, P, N) = G1 (T, P, N1 ) + G2 (T, P, N2 )= N1 µ1 (T, P) + N2 µ2 (T, P) ,  \nwhere we have used the Gibbs-Duhem relation (5.17) . Ni and µi are the number of particles and the chemical potential in phase i and N = N1 + N2 .  \nPrinciple of minimal Gibbs energy. The principle of minimal Gibbs energy discussed in Sect. 5.5.2 states that Gibbs potential has to be at its minimum, i.e. dG = 0, when P , T and N = N1 + N2 are fixed. G has to be therefore in equilibrium with respect toparticle transfer from one phase to the other. This implies that  \ndG = µ1 dN1 + µ2 dN2 = 0 ,  \nµ1 (T, P) = µ2 (T, P)  \n,  \n(6.1)  \nwith dN1 = −dN2 . The condition of coexistence (6.1), tells us that the two phases 1 and 2 must have identical chemical potentials.  \n6.2.2 Clausius-Clapeyron equation  \nWe denote the discontinuities across the phase boundary with  \nΔG = G2 (T, P, N1 ) − G1 (T, P, N2 ) (6.2)  \nΔS = S2 − S1 = − 􀀃∂~~ ~~(G)􀀄 P , ΔV = V2 − V1 = 􀀃 ∂~~ ~~) 􀀄 T , where we assume that S2 > S1 .  \nCyclic chain rule. The discontinuities ΔG, ΔS and ΔV are functions of V , T and P , which are in˜ turn related by the equation of state f(P, V, T) = 0 . There must hence exist  \na function f such that  \n˜f(ΔG, T, P) = 0 .  \nThis conditio","cbCaifvIYNsmWQ2i","https://ap.wps.com/l/cbCaifvIYNsmWQ2i","pdf",476098,12,"English","# Chapter 6: Phase transitions\n## 6.1 Concept of phase\n## 6.2 First-order phase transition\n### 6.2.1 Condition for phase coexistence\n### 6.2.2 Clausius-Clapeyron equation\n## 6.3 Ehrenfest classification of phase transitions","[{\"question\":\"What distinguishes phases and how do phase boundaries affect microscopic states?\",\"answer\":\"Phases are states of matter with distinct macroscopic properties. At the phase boundary, microscopic properties change by definition, leading to different behaviors for first- versus second-order transitions.\"},{\"question\":\"How does the Gibbs enthalpy behave across a first-order phase boundary?\",\"answer\":\"Gibbs enthalpy G changes continuously across the phase boundary for a first-order transition, while entropy S and volume V (derived from derivatives of G) are discontinuous.\"},{\"question\":\"How is the Clausius–Clapeyron equation used in this chapter?\",\"answer\":\"The chapter derives a slope relation along the equilibrium transition line in the P–T diagram. It can be used as a consistency check or to determine latent heat by measuring temperature, volume change, and the slope dP/dT.\"}]","Chapter 6 - Phase transitions - Phase transitions | PDF"]