Thermodynamic Stability: Free Energy and ... - McGill University

Free Energy -8- Chemistry 223and⎛ ∂ 2 G ⎞>0. (24)⎝ ∂ξ 2 ⎠ T ,P,NtotalBy using the differential form for the change in the free energy together with Eq. (22) we findthatwhich when used in Eq. (23) givesdG =−SdT + VdP + Σ rν i µ i dξ , (25)∆G ≡ ⎛ ∂G ⎞⎝ ∂ξ⎠ T ,P,Ntotali=1= rΣ ν i µ i = 0 (26)at equilibrium. ∆G is called the reaction Gibbs free energy. Since the µ i are the partial molarGibbs free energies, Eq. (26) is equivalent to ∆G = 0. At equilibrium the free energy changein the reaction per mole vanishes. (Indeed, this is the principle we applied in the "reaction"between graphite and diamond). From the definition of the Gibbs free energy (G = H -TS), it followsthati=1at equilibrium.∆S = ∆H TWhat happens if we change temperature or pressure by a small amount? Which way willthe equilibrium shift? To answer this, first note the following Maxwell relations:⎛ ∂µ i ⎞⎝ ∂T⎠ P,N j=− ⎛ ∂S ⎞=−S⎝ ∂N i ⎠ i (27a)P,T ,N j≠iand⎛ ∂µ i ⎞⎝ ∂P⎠ T ,N j= ⎛ ∂V ⎞= V⎝ ∂N i ⎠ i (27b)P,T ,N j≠iwhich follow from the Gibbs free energy. Thus the changes in the chemical potential associatedwith temperature or pressure are related to the partial molar entropies or volumes, respectively.Next consider⎛ ∂µd( Σν i µ i ) =i ⎞⎛ ∂µΣν i dT +i ⎞Σν⎝ ∂T ⎠ i dP + ⎛ ∂(Σν i µ i ) ⎞dξP,N j⎝ ∂P ⎠ T ,N j⎝ ∂ξ ⎠ T ,P,Ntotaliii=−∆SdT +∆VdP + ⎛ ∂∆G ⎞dξ , (28)⎝ ∂ξ ⎠ T ,P,Ntotalwhere, cf. Eqs. (27), ∆S ≡ Σν i S i and ∆V ≡ Σν i V i are the entropy and volume changes per moleof reaction. Equation (28) shows how the free energy change per mole of reaction changes whenwe change T, P,or ξ .What happens if we change, T or P in a system where chemical reaction is possible? Theprogress variable will change until Eq. (26) is again valid. Since both the initial and final statesFall Term, 2014