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# chapter7 - Chapter 7 Fugacity of a Component in a Mixture...

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Chapter 7 Fugacity of a Component in a Mixture When two phases of a pure component are in equilibrium, then fugacities in each phase must be the same. The purpose of this chapter is to replace a pure component by a multicomponent mixture and investigate the conditions under which two phases of a multicomponent mixture are in equilibrium with each other. For this purpose, fi rst the fugacity de fi nition given in Chapter 5 will be extended to multicomponent mixtures, then equations to calculate fugacities of components making up the mixture will be developed. 7.1 FUNDAMENTAL EQUATIONS FOR A MULTICOMPONENT MIXTURE The combined law for a single-phase, single-component open system, Eq. (4.2-9), is given as dU = T dS P dV + e G dn (7.1-1) By de fi nition, Gibbs energy is expressed as G = U + PV TS (7.1-2) In di ff erential form, Eq. (7.1-2) becomes dG = dU + P dV + V dP T dS S dT (7.1-3) Substitution of Eq. (7.1-1) into Eq. (7.1-3) leads to dG = V dP S dT + e G dn (7.1-4) indicating that the Gibbs energy is dependent on pressure, temperature, and number of moles, i.e., G = G ( P, T, n ) (7.1-5) The total di ff erential of G is dG = μ ∂G ∂P T,n dP + μ ∂G ∂T P,n dT + μ ∂G ∂n P,T dn (7.1-6) Comparison of Eq. (7.1-6) with Eq. (7.1-4) yields V = μ ∂G ∂P T,n (7.1-7) S = μ ∂G ∂T P,n (7.1-8) e G = μ ∂G ∂n P,T (7.1-9) 169

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In the case of a single-phase, multicomponent open system, Gibbs energy of the phase not only depends on pressure and temperature, but also on the number of moles of each component existing in the phase, i.e., G = G ( P, T, n 1 , n 2 , ..., n k ) (7.1-10) Therefore, the total di ff erential of G is expressed in the form dG = μ ∂G ∂P T,n j dP + μ ∂G ∂T P,n j dT + k X i =1 μ ∂G ∂n i P,T,n j 6 = i dn i (7.1-11) The subscript n j in the fi rst two terms of Eq. (7.1-11) implies that the number of moles of all components are held constant during di ff erentiation. For constant number of moles, Eqs. (7.1-7) and (7.1-8) are still valid. On the other hand, the partial molar Gibbs energy is de fi ned by G i = μ ∂G ∂n i P,T,n j 6 = i (7.1-12) Thus, Eq. (7.1-11) becomes dG = V dP S dT + k X i =1 G i dn i (7.1-13) Historically, for a multicomponent system, the partial molar Gibbs energy has been called the chemical potential and designated by μ i , i.e., G i = μ i . The enthalpy is related to the Gibbs energy by H = G + TS (7.1-14) Di ff erentiation of Eq. (7.1-14) gives dH = dG + T dS + S dT (7.1-15) Substitution of Eq. (7.1-13) into Eq. (7.1-15) yields dH = T dS + V dP + k X i =1 G i dn i (7.1-16) The internal energy is related to the enthalpy by U = H PV (7.1-17) Di ff erentiation of Eq. (7.1-17) gives dU = dH P dV V dP (7.1-18) The use of Eq. (7.1-16) in Eq. (7.1-18) results in dU = T dS P dV + k X i =1 G i dn i (7.1-19) 170
The Helmholtz energy is related to the internal energy by A = U TS (7.1-20) Di ff erentiation of Eq. (7.1-20) gives dA = dU T dS S dT (7.1-21) Substitution of Eq. (7.1-19) into Eq. (7.1-21) yields dA = P dV S dT + k X i =1 G i dn i (7.1-22) Equations (7.1-13), (7.1-16), (7.1-19) and (7.1-22) are the fundamental equations for a single- phase, multicomponent system. From these equations, partial molar Gibbs energy can be expressed in di ff erent forms as G i = μ

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chapter7 - Chapter 7 Fugacity of a Component in a Mixture...

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