Page 270 - Advanced Thermodynamics for Engineers, Second Edition
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12.6 VARIATION OF GIBBS ENERGY WITH COMPOSITION            259




                  This can be written, by substituting for m from Eqns (12.49) and (12.50),as

                               0                     0                 0
                               a                     b                 c
                   G ¼ð1   εÞ m þ<Tlnpr a  þð1   εÞ m þ<Tlnpr b  þ 2ε m þ<Tlnpr c
                                                                                           (12.66)
                       h                        i     h                               i
                               0
                                         0
                     ¼ ð1   εÞm þð1   εÞm þ 2εm 0
                               a         b      c  þ<T ð1   εÞlnpr a  þð1   εÞlnpr b  þ 2εlnpr c
                  The partial pressures are defined by Eqn (12.60) as p i ¼ x i p and the mole fractions of the con-
               stituents are
                                            1   ε      1   ε       2ε
                                                ;           ;        ¼ ε:                  (12.67)
                                             2           2         2
                                       x a ¼       x b ¼      x c ¼
                  Substituting these terms in Eqn (12.66) gives

                            0         0     0                1   ε              1   ε
                G ¼ ð1   εÞm þð1   εÞm þ 2εm c  þ<T ð1   εÞln      p r þð1   εÞln    p r þ 2εlnεp r
                            a
                                      b
                                                               2                 2
                            0         0     0
                  ¼ ð1   εÞm þð1   εÞm þ 2εm
                            a         b     c  þ<T½ð1   εÞþð1   εÞþ 2εŠlnp r

                                  1   ε            1   ε
                    þ<T ð1   εÞln       þð1   εÞln       þ 2εlnε
                                    2                2
                                                                                           (12.68)
                  Equation (12.68) can be rearranged to show the variation of the Gibbs energy of the mixture as the
                                                                                 0
               reaction progresses from the reactants A and B to the product C by subtracting 2m from the left-hand
                                                                                 c
               side, giving

                                     0
                                          0
                          0

                    G   2m ¼ð1   εÞ m þ m   2m 0 c     þ 2<Tlnp r þ 2<T ð1   εÞln  1   ε  þ εlnε  (12.69)
                                          b
                                     a
                          c
                                                                            2
                  Equation (12.69) consists of three terms; the second one simply shows the effect of pressure and
               will be neglected in the following discussion. The first term is the difference between the standard
                                                         0
                                                             0
               chemical potentials of the separate components (m þ m ) before any reaction has occurred and the
                                                         a   b
                                                      0
               standard chemical potential of the mixture (2m ) after the reaction is complete. Since the standard
                                                      c
               chemical potentials are constant throughout this isothermal process, this term varies linearly with the
               fraction of reaction, ε. The third term defines the change in chemical potential due to mixing and is a
               function of the way in which the entropy of the mixture (not the specific entropy) varies as the reaction
               progresses. The manner in which the first and third terms might vary is shown in Fig. 12.2 and the sum
               of the terms is also shown. It can be seen that for this example, the equilibrium composition is at
               ε ¼ 0.78. This figure illustrates that the Gibbs energy of the mixture initially reduces as the compo-
               sition of the mixture goes from A þ B to C. The standard chemical potential of compound C is less than
               the sum of the standard chemical potentials of A and B, and hence the reaction will tend to go in the
               direction shown in Eqn (12.63). If the standard chemical potentials were the only parameters of
               importance in the reaction then the reactants A and B would be completely transformed to the product,
               C. However, as the reaction progresses the term based on the mole fractions varies non-monotonically,

                                           	        1   ε
               as shown by the line labelled <T 1   ε ln  þ 2εlnε , and this affects the composition of the
                                                     2
               mixture which obeys the law of mass action. If the two terms are added together then the variation of
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