MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiReview of classical thermodynamicsFundamental Laws, Properties and Processes (3)Fundamental equationsThe Helmholtz Free EnergyThe Gibbs Free energy Changes in compositionChemical potentialThermodynamic relationsReading: Chapter 5.1 – 5.9 of Gaskellor the same material in any other textbook on thermodynamicsMSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiFundamental equationsCombined statement of 1stand 2ndLaws of Thermodynamics: dU = TdS – PdV This equation gives us¾ Relationship between the dependent variable U and independent variables V and S: U = U(S,V) or dU = (∂U/∂S)vdS + (∂U/∂V)sdV¾ The criteria for equilibrium: in a system of constant V and S, the internal energy has its minimum value, or, in a system of constant U and V, the entropy has its maximum value.The problem is that the pair of independent variables (V,S) is rather inconvenient – entropy is hard to measure or control. We want to have fundamental equations with independent variables that is easier to control. The two convenient choices are:P and T pair – the best choice from the practical point of view, easy to control/measure. For systems with constant pressure the best suited state function is the Gibbs free energy (also called free enthalpy) G = H - TSV and T pair – easy to examine in statistical mechanics. For systems with constant volume (and variable pressure), the best suited state function is the Helmholtz free energy A = U – TSAny state function can be used to describe any system (at equilibrium, of course), but for a given system some are more convenient than others.MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiThe Helmholtz Free EnergyA = U – TS⇒dA = dU – TdS – SdTCombining this equation with dU = TdS – PdV we getdA= –PdV–SdT - fundamental equationA = A(T,V) ⇒ dA = (∂A/∂T)VdT + (∂A/∂V)TdVComparing the equations we see that S = – (∂A/∂T)VP = – (∂A/∂V)TAt constant T and V the equilibrium states corresponds to the minimum of Helmholtz Free Energy (dA = 0). From A = U – TSwe see that low values of A are obtained with low values of Uand high values of S.Illustration by Gaskell on the criterion for equilibrium in a closed solid-vapor system at constant V and T. The transfer of one atom to vapor increases U by a sublimation energy, whereas increase of entropy is slowing down with increasing number of atoms in the vapor phase, nv.MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiThe Gibbs Free EnergyG = H – TS = U + PV - TS ⇒ dG = dU+PdV+VdP–TdS–SdTCombining this equation with dU = TdS – PdV we getdG = VdP – SdT - fundamental equationG = G(T,P) ⇒ dG = (∂G/∂P)TdP + (∂G/∂T)PdTComparing the equations we see that S = – (∂G/∂T)PV = (∂G/∂P)TFor an isothermal-isobaric system the equilibrium state corresponds to the minimum of the Gibbs Free Energy (dG=0). From G=H–TS we see that low values of G are obtained with low values of H and high values of S.Summary of the equations for a closed systemdU = δq - δw dU = TdS – PdVH = U + PV dH = TdS +VdPA = U – TS dA = – PdV – SdTG = H – TS dG = VdP – SdTMSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiGibbs Free Energy: Equilibrium is Trade-off Between Enthalpy and EntropyG = H – TSA change to a lower enthalpy state (ΔH < 0) usually decreases the randomness (ΔS < 0):¾ Freezing of a liquid ¾ Oxidation of a metalexothermic processesA change to a higher entropy state (ΔS > 0) usually increases the enthalpy (ΔH > 0) All of these processes are characterized by a lowering of the Gibbs free energy: ΔG = Δ(H - TS) < 0¾ Melting of a solid ¾ Evaporation of a liquid ¾ Dissolution of salt in water endothermic processesA crystal at equilibrium has its minimum Gibbs free energy:G = H - TS = minimum(if T = const and P = const)MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiVariation of number of particles and compositionIf the number of particles and/or composition of the system changes during a process, then two independent variables are notsufficient to describe the state of the system. Chemical reactions or exchange with surroundings can lead to the change in composition (number of moles of different species, ni, nj, nk, …).The Gibbs free energy (as well as other thermodynamic potentials) is an extensive property – depends on size of the system and on number of moles of different species, G = G(T, P, ni, nj, nk,…). i,...n,T,Pki1iidnnGVdPSdTdGi∑==⎟⎟⎠⎞⎜⎜⎝⎛∂∂++−=Wherei,...n,T,PijnGμ=⎟⎟⎠⎞⎜⎜⎝⎛∂∂is chemical potential of the species iiki1iidnVdPSdTdG∑==μ++−=.,...,,,...,,,,...,,,...,,etcdnnGdnnGdTTGdPPGdGjnnPTjinnPTinnPnnTkikjjiji+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂=If composition is not changing, dG = VdP – SdT and, therefore,S = – (∂G/∂T)P,ni,nj,…and V = (∂G/∂P)T,ni,nj,…Therefore:MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiThe Chemical PotentialThe chemical potential of the species i is the rate of increase of Gwith niwhen species i are added/generated in the system at constant T, P, and number of moles of all the other species.inTPijnGμ=⎟⎟⎠⎞⎜⎜⎝⎛∂∂,...,,ikiidnVdPSdTdG∑=μ++−=1This equations can be applied to open systems that exchange both matter and heat with their surroundings, as well as to closed systems where changes in composition are due to chemical reactions.Similar equations can be written for U, H, and A:,...,,,...,,,...,,,...,,jjjjnVTinPSinVSinTPiinAnHnUnG⎟⎟⎠⎞⎜⎜⎝⎛∂∂=⎟⎟⎠⎞⎜⎜⎝⎛∂∂=⎟⎟⎠⎞⎜⎜⎝⎛∂∂=⎟⎟⎠⎞⎜⎜⎝⎛∂∂=μikiidnPdVTdSdU∑=μ+−=1ikiidnVdPTdSdH∑=μ++=1ikiidnPdVSdTdA∑=μ+−−=1ikiidnVdPSdTdG∑=μ++−=1We can rewrite the 1stLaw for a closed system undergoing a reversible change in composition due to a chemical reaction:wqdU δ−δ= TdSq =δikiidnPdVw∑=μ−=δ1Where is the chemical work done by the systemikiidn∑=μ−1MSE 3050, Phase Diagrams and Kinetics, Leonid ZhigileiThermodynamic RelationsFrom the equations We can obtain the following thermodynamic