Physical Chemistry Lecture 27 Temperature Dependence of ∆Gθ and Ka Energetics of a reaction 1 N2 (g) + 2 3 H 2 ( g ) → NH 3 ( g ) 2 Enthalpy diagram for ammonia synthesis Energy determines reactivity Must relate energetics and measures of reaction. Free-energy change is most significant. Obtaining ΔGθ at reaction temperature ∆Gθ (T ) ln K a (T ) = − RT Some data tables for temperatures other than 298.15 K exist Solve for ΔGθ(T) as done at 298.15 K More likely that one has to calculate how ΔGθ changes with T May be able to calculate ΔHθ(T) and ΔSθ(T) at the temperature by thermodynamic cycles involving a temperature at which these are known Calculate ΔGθ(T) by the simple equation ∆Gθ (T ) = ∆H θ (T ) − T∆S θ (T ) Temperature and equilibrium For a chemical reaction, at equilibrium ∆Gθ (T ) ln K a (T ) = − RT Must find ΔGθ(T) to evaluate Ka(T), and vice versa Relationship of temperature derivatives ∂ ln K a (T ) P ∂T 1 ∂ = − R ∂T ∆Gθ (T ) T P Temperature dependence of the equilibrium constant By definition of the standard Gibbs energy and the distribution of the derivative ∂ K T ln ( ) a ∂T P 1 ∂ = − R ∂T = = ∆H θ (T ) RT 2 ∆H θ (T ) RT 2 ∆H θ (T ) − T∆S θ (T ) T P − ∆C Pθ (T ) RT + ∆C Pθ (T ) RT A very simple result Must know how ∆Hθ(T) depends on T to determine Ka(T) Temperature dependence of ΔHθ, ΔSθ, and ΔGθ Determining temperature dependence involves evaluation of change for both reactants and products T θ ∆ C ∫ P (T )dT θ θ ∆H reaction (T0 ) + (T ) = ∆H reaction T0 θ ∆ C θ θ P (T ) ∆S reaction dT (T ) = ∆S reaction (T0 ) + ∫ T T0 The Gibbs energy change at the temperature is T ∆Gθ (T ) = ∆H θ (T ) − T∆S θ (T ) θ ∆ C P = ∆Gθ (T0 ) − (T − T0 )∆S θ (T0 ) + ∫ ∆C Pθ dT ' − T ∫ dT ' T' T0 T0 T T Temperature dependence of reaction thermodynamics H2 (g) + ½ O2 (g) → H2O (l) 5 2.8 10 80 ∆H( Tr ) 75 ∆S ( Tr ) ∆G( Tr ) 70 65 300 5 3 10 5 3.2 10 350 Tr 350 Tr 140 ln K a ( Tr ) 300 120 100 300 350 Tr Determining equilibrium constants experimentally Example: 1 3 O2 ( gas ) → NH 3 ( gas ) N 2 ( gas ) + 2 2 Equilibrium expression Ka = aNH 3 a1N/22 aH3 /22 Have to find expression for the activities in terms of concentration measures to use as a means to find equilibrium concentrations Using ideal activities in expressions for Ka •Ideal gases: ai = Pi/Pθ •Pure solids and liquids near standard pressure: Example: ammonia synthesis reaction Ka = ( PNH 3 / Pθ ) ( PN 2 / Pθ )1 / 2 ( PH 2 / Pθ )3 / 2 = PNH 3 ( PN 2 )1 / 2 ( PH 2 )3 / 2 = K P Pθ Example: Boudouard reaction Ka = ( PCO / Pθ ) 2 aC ( PCO2 / Pθ ) = KP Pθ = 2 PCO 1 θ (1)( PCO2 ) P Pθ ai = 1 Nonideal gas-phase reactions Must correct for nonideality: ai = γi ai, ideal Include activity coefficients Example: Ammonia-synthesis reaction Ka γ NH ( PNH / Pθ ) = γ 1N/ 2 ( PN / Pθ )1/ 2 γ H3 / 2 ( PH / Pθ )3 / 2 3 2 2 3 2 2 = Kγ PNH 3 ( PN 2 )1/ 2 ( PH 2 ) 3 / 2 = K γ K P Pθ Ka is the equilibrium constant KP is not necessarily a constant because Kγ depends on pressure and temperature Must evaluate Kγ Pθ Temperature and pressure dependencies of KP KP for Ammonia Synthesis Reaction at 673K, 723K and 773K Measurement of KP allows extrapolation to obtain Ka in the limit of P=0 -3.5 -4.5 -5 -5.5 -6 0 200 400 600 800 1000 Pressure (atm) Kγ Pθ for the Ammonia Synthesis Reaction at 673K, 723K and 773K Knowledge of Ka(T) and KP allows the calculation of Kγ at any conditions 1.5 1 Kγ Pθ ln (KP) -4 0.5 0 0 200 400 600 Pressure (atm) 800 1000 Summary ΔG = 0 at equilibrium ΔGθ(T) and Ka(T) express the same quality of a chemical reaction Calculation of ΔGθ(T) and Ka(T) Directly from tabulations of ΔGθformation(T) at the reaction temperature Indirectly through calculation of ΔHθreaction(T) and ΔSθreaction(T) at the temperature of reaction from tables at T Calculation of ΔHθ(T) and ΔSθ (T) at the temperature of reaction from knowledge of ΔHθ(T0) and ΔSθ(T0) at a reference temperature and ΔCPθ(T) as a function of temperature Various approximations sometimes used in thermodynamic calculations Assume ΔHθ and ΔSθ are independent of temperature Assume ΔCPθ is independent of temperature Can never be totally correct Depends on the specific system how good the approximation is Do the full calculation unless you are sure an approximation applies Measurements of KP allow determination of Ka as a limiting value Nonidealities can be quantified if Ka and KP are known