Gibbs Helmholtz equation, Vant Hoff isotherm and isochore derivation in detail with each and every steps.<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n Sure! Let me provide detailed derivations and explanations of the Gibbs-Helmholtz equation<\/strong>, van’t Hoff isotherm<\/strong>, and isochore<\/strong> step-by-step.<\/p>\n\n\n\n Purpose:<\/strong> Relates the temperature dependence of the Gibbs free energy GG to the enthalpy HH.<\/p>\n\n\n\n Starting point:<\/strong><\/p>\n\n\n\n Gibbs free energy is defined as G=H\u2212TSG = H – TS<\/p>\n\n\n\n Differentiating GG with respect to temperature TT at constant pressure PP: (\u2202G\u2202T)P=(\u2202H\u2202T)P\u2212(S+T\u2202S\u2202T)P\\left(\\frac{\\partial G}{\\partial T}\\right)_P = \\left(\\frac{\\partial H}{\\partial T}\\right)_P – \\left( S + T\\frac{\\partial S}{\\partial T} \\right)_P<\/p>\n\n\n\n Using the thermodynamic relation (\u2202G\u2202T)P=\u2212S\\left(\\frac{\\partial G}{\\partial T}\\right)_P = -S<\/p>\n\n\n\n and recognizing (\u2202H\u2202T)P=CP\\left(\\frac{\\partial H}{\\partial T}\\right)_P = C_P (heat capacity at constant pressure),<\/p>\n\n\n\n We rewrite the Gibbs free energy as: G=H\u2212TS\u2005\u200a\u27f9\u2005\u200aGT=HT\u2212SG = H – TS \\implies \\frac{G}{T} = \\frac{H}{T} – S<\/p>\n\n\n\n Differentiating GT\\frac{G}{T} with respect to TT gives: \u2202\u2202T(GT)=1T\u2202G\u2202T\u2212GT2=1T(\u2212S)\u2212GT2=\u2212ST\u2212GT2\\frac{\\partial}{\\partial T} \\left( \\frac{G}{T} \\right) = \\frac{1}{T} \\frac{\\partial G}{\\partial T} – \\frac{G}{T^2} = \\frac{1}{T}(-S) – \\frac{G}{T^2} = -\\frac{S}{T} – \\frac{G}{T^2}<\/p>\n\n\n\n Recall G=H\u2212TS\u2005\u200a\u27f9\u2005\u200aS=H\u2212GTG = H – TS \\implies S = \\frac{H-G}{T}, substitute into above: \u2202\u2202T(GT)=\u22121T\u22c5H\u2212GT\u2212GT2=\u2212HT2+GT2\u2212GT2=\u2212HT2\\frac{\\partial}{\\partial T} \\left( \\frac{G}{T} \\right) = -\\frac{1}{T} \\cdot \\frac{H-G}{T} – \\frac{G}{T^2} = -\\frac{H}{T^2} + \\frac{G}{T^2} – \\frac{G}{T^2} = -\\frac{H}{T^2}<\/p>\n\n\n\n Final Gibbs-Helmholtz equation:<\/strong> (\u2202\u2202T(GT))P=\u2212HT2\\boxed{ \\left( \\frac{\\partial}{\\partial T} \\left( \\frac{G}{T} \\right) \\right)_P = – \\frac{H}{T^2} }<\/p>\n\n\n\n The van’t Hoff equation relates the change in the equilibrium constant KK with temperature.<\/p>\n\n\n\n Start from Gibbs free energy change at equilibrium: \u0394G=\u0394G\u2218+RTln\u2061K\\Delta G = \\Delta G^\\circ + RT \\ln K<\/p>\n\n\n\n At equilibrium, \u0394G=0\\Delta G = 0, so: 0=\u0394G\u2218+RTln\u2061K\u2005\u200a\u27f9\u2005\u200a\u0394G\u2218=\u2212RTln\u2061K0 = \\Delta G^\\circ + RT \\ln K \\implies \\Delta G^\\circ = -RT \\ln K<\/p>\n\n\n\n Since \u0394G\u2218=\u0394H\u2218\u2212T\u0394S\u2218\\Delta G^\\circ = \\Delta H^\\circ – T \\Delta S^\\circ<\/p>\n\n\n\n We substitute: \u2212RTln\u2061K=\u0394H\u2218\u2212T\u0394S\u2218\u2005\u200a\u27f9\u2005\u200aln\u2061K=\u2212\u0394H\u2218RT+\u0394S\u2218R-RT \\ln K = \\Delta H^\\circ – T \\Delta S^\\circ \\implies \\ln K = -\\frac{\\Delta H^\\circ}{RT} + \\frac{\\Delta S^\\circ}{R}<\/p>\n\n\n\n Taking derivative w.r.t temperature TT at constant pressure: dln\u2061KdT=\u0394H\u2218RT2\\frac{d \\ln K}{dT} = \\frac{\\Delta H^\\circ}{RT^2}<\/p>\n\n\n\n This is the van’t Hoff equation:<\/strong> dln\u2061KdT=\u0394H\u2218RT2\\boxed{ \\frac{d \\ln K}{dT} = \\frac{\\Delta H^\\circ}{RT^2} }<\/p>\n\n\n\n An isochore<\/strong> is a path of constant volume in a thermodynamic system. The derivation involves relating pressure PP, temperature TT, and internal energy UU at constant volume VV.<\/p>\n\n\n\n From the first law of thermodynamics: dU=\u03b4Q\u2212PdVdU = \\delta Q – PdV<\/p>\n\n\n\n At constant volume (dV=0dV = 0): dU=\u03b4QVdU = \\delta Q_V<\/p>\n\n\n\n Define heat capacity at constant volume: CV=(\u2202U\u2202T)VC_V = \\left(\\frac{\\partial U}{\\partial T}\\right)_V<\/p>\n\n\n\n For an ideal gas, internal energy depends only on temperature: U=U(T)U = U(T)<\/p>\n\n\n\n Use thermodynamic identity for pressure change at constant volume: (\u2202P\u2202T)V=\u03b1\u03baT\\left(\\frac{\\partial P}{\\partial T}\\right)_V = \\frac{\\alpha}{\\kappa_T}<\/p>\n\n\n\n where \u03b1\\alpha is the coefficient of thermal expansion and \u03baT\\kappa_T is the isothermal compressibility.<\/p>\n\n\n\n For an ideal gas: PV=nRT\u2005\u200a\u27f9\u2005\u200aP=nRTVP V = nRT \\implies P = \\frac{nRT}{V}<\/p>\n\n\n\n At constant VV: (\u2202P\u2202T)V=nRV\\left(\\frac{\\partial P}{\\partial T}\\right)_V = \\frac{nR}{V}<\/p>\n\n\n\n Thus the isochore is described by: P=constant\u00d7T(for ideal gases)\\boxed{ P = \\text{constant} \\times T \\quad \\text{(for ideal gases)} }<\/p>\n\n\n\n The Gibbs-Helmholtz equation<\/strong> relates the temperature dependence of Gibbs free energy to enthalpy. Starting from the definition G=H\u2212TSG = H – TS, differentiating G\/TG\/T with respect to TT yields (\u2202(G\/T)\u2202T)P=\u2212H\/T2\\left(\\frac{\\partial (G\/T)}{\\partial T}\\right)_P = -H\/T^2. This relation helps predict how the Gibbs free energy changes with temperature, essential in chemical thermodynamics.<\/p>\n\n\n\n The van’t Hoff isotherm<\/strong> relates the temperature dependence of the equilibrium constant KK to the reaction enthalpy. By expressing the Gibbs free energy at equilibrium, \u0394G\u2218=\u2212RTln\u2061K\\Delta G^\\circ = -RT \\ln K, and differentiating, one obtains dln\u2061KdT=\u0394H\u2218RT2\\frac{d \\ln K}{dT} = \\frac{\\Delta H^\\circ}{RT^2}. This equation predicts how equilibrium shifts with temperature changes and is widely used to analyze reaction spontaneity.<\/p>\n\n\n\n The isochore<\/strong> represents a process at constant volume. Using the ideal gas law P=nRT\/VP = nRT\/V, the pressure increases linearly with temperature when volume is fixed. Heat capacity at constant volume CVC_V relates to internal energy changes since U=U(T)U = U(T) for ideal gases. Isochore paths are crucial in thermodynamic cycles and processes involving confined gases.<\/p>\n\n\n\n These derivations underscore core thermodynamic principles linking energy, entropy, and equilibrium, vital for understanding chemical and physical systems<\/p>\n","protected":false},"excerpt":{"rendered":" Gibbs Helmholtz equation, Vant Hoff isotherm and isochore derivation in detail with each and every steps. The correct answer and explanation is : Sure! Let me provide detailed derivations and explanations of the Gibbs-Helmholtz equation, van’t Hoff isotherm, and isochore step-by-step. 1. Gibbs-Helmholtz Equation Purpose: Relates the temperature dependence of the Gibbs free energy GG […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-11036","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/11036","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=11036"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/11036\/revisions"}],"predecessor-version":[{"id":11037,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/11036\/revisions\/11037"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=11036"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=11036"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=11036"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\n1. Gibbs-Helmholtz Equation<\/h3>\n\n\n\n
\n\n\n\n2. van’t Hoff Isotherm Derivation<\/h3>\n\n\n\n
\n\n\n\n3. Isochore Derivation<\/h3>\n\n\n\n
\n\n\n\nSummary (\u2248300 words)<\/h3>\n\n\n\n