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

Free Energy -4- Chemistry 223One final point, note that the partial molar Gibbs free energy isG i = ⎛ ∂G ⎞= µ⎝ ∂N i ⎠ i , (16)T ,P,N j≠iwhere Eq. (14) was used. Hence, in a one component system, the Gibbs free energy per mole isjust the chemical potential. More generally, G i ,the partial molar quantity for the i’th componentis µ i ,and hence, from Euler’s theoremG =Σ N i µ i . (17)iAs we discussed earlier, inorder that Eqs. (17) and (14) be consistent, a Gibbs-Duhem relationmust hold; i.e.,0 = SdT − VdP +Σ N i dµ i , (18)iwhich shows that the changes in temperature, pressure and chemical potentials are not all independent.The various stability results are summarized in the following table.Criteria for Stable EquilibriumState Stable Equilibrium SimplestHeld Fixed Definition DifferentialFunction Criterion Physical ContentAdiabatic *(e.g., E,V,N)S∆S =∫ dQ revTdS = dQ revTmaximum -S,V,N E ∆E = Q + W dE = TdS − PdV + Σ µ i dN i minimum ∆E V ,N = Q V ,NS,P,N H H ≡ E + PV dH = TdS + VdP + Σ µ i dN i minimum ∆H P,N = Q P,NT,V,N A A ≡ E − TS dA =−SdT − PdV + Σ µ i dN i minimum −W T ,N ≤−∆A T ,NT,P,N G G ≡ H − TS dG =−SdT + VdP + Σ µ i dN i minimum −W non−PV ≤−∆G P,N* E,N,V implies adiabatic in systems where only PV or chemical work is allowed, the reverse is not true if other kinds orwork (e.g., electrical) are possible.iiii2. Examples of free energy calculationsFree energy calculations are carried out in much the same as enthalpy calculations. Thereare tables of standard free energies of formation of compounds. Elements in their standardstates are assigned zero astheir Gibbs free energy of formation.Consider the following chemical reaction:Fall Term, 2014