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Relating Chemical Potential to Equilibrium SVNA 11.2

Relating Chemical Potential to Equilibrium SVNA 11.2. Purpose of this lecture : To show how the chemical potentials of species in a mixture are used to express the state of thermodynamic equilibrium in a multi-phase system Highlights

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Relating Chemical Potential to Equilibrium SVNA 11.2

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  1. Relating Chemical Potential to Equilibrium SVNA 11.2 • Purpose of this lecture: • To show how the chemical potentials of species in a mixture are used to express the state of thermodynamic equilibrium in a multi-phase system • Highlights • The total change in the Gibbs free energy (or any other thermo property) of a multi-phase system is the sum of the changes occurring in each phase • In a multiphase system (e.g., Vapour-Liquid) existing at equilibrium, the chemical potential of any species is equal in all phases • Reading assignment: Sections 11.2 from textbook (eds. 7 or 6) Lecture 5

  2. Vap, k components Liq, k components @ uniform T,P 3. Relating Chemical Potential to Equilibrium SVNA 11.2 • Consider a two-phase system of k components: • The vessel as a whole (vap+liq) • is closed, as energy may be exchanged • with its surroundings but material • cannot. • Each phase, however, is an open • system, as it may exchange matter • with the other phase. Lecture 5

  3. Gibbs Energy Changes for a Closed System • For the total vessel contents (vapour+liquid phases), we can write the fundamental equation for a closed system at eq’m with its surroundings. • (6.6) (A) • where • n is the total number of moles of material; mole • G is the total molar Gibbs energy; J/mole • V is the molar Volume of the total system; m3/mole • P represents the system pressure; Pa • S is the total molar Entropy; J/moleK • T represents the system temperature; K • Note that because the composition of the entire system (vap + liq) cannot change, only changes in pressure and temperature can influence the Gibbs energy of the whole system. • Composition is invariant, so no chemical potential terms are included in the closed system expression. Lecture 5

  4. Gibbs Energy Changes for an Open System • Each phase can exchange not only energy, but material with the other. Therefore the vapour phase and the liquid phase are individual open systems. • For the vapour phase (superscript v refers to vapour): • (11.2) (B) • For the liquid phase (superscript l refers to liquid): • (11.2) (C) • These equations detail how the Gibbs energy of each phase is affected by changes in pressure, temperature, and composition. Lecture 5

  5. Back to the Overall System • The change in the Gibbs energy of the whole, two-phase system is the sum of the vapour and liquid changes. • For the whole system (vap + liq), the sum of equations B and C yields the total Gibbs energy change: • (B+C) • According to this equation, the Gibbs energy of the overall system is affected by changes in T, P and composition. • If we substract B+C-A, we get the following eq’n which is true at eq’m: Lecture 5

  6. Relating Chemical Potential to Equilibrium • We now have the tools needed to translate our Gibbs energy criterion for equilibrium into one based on chemical potential. • Note also, that at constant T and P, Equation (B+C) becomes: • (D) • so phase equilibrium also requires that Gibb’s energy is at its minimum value: • Since component i enters the vapour phase when it leaves the liquid: • (E) Lecture 5

  7. Relating Chemical Potential to Equilibrium • Eqn. (E) is true for any arbitrary dniv: • for all components, i • For a system of p phases, equilibrium exists at a given P and T if: • 11.6 • Chemical equilibrium calculations require expressions for mi as a function of T,P, and composition • For ideal gases in equilibrium with ideal liquid solutions eqn. 11.6 is equivalent to Raoult’s law. Lecture 5

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