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150    REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE


                                              Justification Box 4.3

                         We start with Equation (4.21) for a single phase, and write G = H − TS. Its differen-
                         tial is

                                                dG = dH − T dS − S dT                    (4.32)

                         From Chapter 2, we recall that H = U + pV , the differential of which is

                                                        dH = dU + p dV + V dp            (4.33)
                         Equation (4.35) com-
                         bines the first and       Substituting for dH in Equation (4.32) with the expres-
                         second laws of ther-     sion for dH in Equation (4.33), we obtain
                         modynamics: it derives
                         from Equation (3.5)        dG = (dU + p dV + V dp) − T dS − S dT  (4.34)
                         and says, in effect,
                         dU = dq + dw.The p dV    For a closed system (i.e. one in which no expansion
                         term relates to expan-   work is possible)
                         sion work and the T dS
                         term relates to the                   dU = T dS − p dV          (4.35)
                         adiabatic transfer of
                         heat energy.             Substituting for the dU term in Equation (4.34) with
                                                  the expression for dU in Equation (4.35) yields

                         The Gibbs–Duhem           dG = (T dS − p dV) + p dV + V dp − T dS − S dT
                         equation is also com-
                                                                                         (4.36)
                         monly (mis-)spelt
                         ‘Gibbs–Duheme ’.         The T dS and p dV terms will cancel, leaving the
                                                  Gibbs–Duhem equation, Equation (4.31).



                      We now come to the exciting part. By comparing the total differential of Equa-
                      tion (4.30) with the Gibbs–Duhem equation in Equation (4.31) we can see a pat-
                      tern emerge:


                                                      ∂G          ∂G
                                             dG =          dp +       dT
                                                      ∂p          ∂T
                                             dG = V        dp   −S    dT


                      So, by direct analogy, comparing one equation with the other, we can say

                                                            ∂G
                                                       V =                                 (4.37)
                                                            ∂p
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