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THE EFFECT OF PRESSURE ON THERMODYNAMIC VARIABLES      149

             apples, the total price will depend on the price of each item, and the amounts of each
             that we purchase. We could write it as

                           d(money) = (price of item 1 × number of item 1)

                                      + (price of item 2 × number of item 2)      (4.27)

             While more mathematical in form, we could have rewritten Equa-  We must use the sym-
             tion (4.27)                                                  bol ∂ (‘curly d’) in a dif-
                                                                          ferential when several
                            ∂(money)           ∂(money)                   terms are changing.
                d(money) =           × N(1) +           × N(2) + ···      The term in the first
                              ∂(1)               ∂(2)
                                                                  (4.28)  bracket is the rate of
                                                                          change of one variable
             where N is merely the number of item (1) or item (2), and each
                                                                          when all other variables
             bracket represents the price of each item: it is the amount of money  are constant.
             per item. An equation like Equation (4.28) is called a total differ-
             ential.
               In a similar way, we say that the value of the Gibbs function
                                                                          The value of G for
             changes in response to changes in pressure and temperature. We
                                                                          a single, pure mate-
             write this as
                                                                          rial is a function of
                                    G = f(p, T )                  (4.29)
                                                                          both its temperature
                                                                          and pressure.
             and say G is a function of pressure and temperature.
               So, what is the change in G for a single, pure substance as the
             temperature and pressure are altered? A mathematician would start  The small subscripted
             answering this question by writing out the total differential of G:  p on the first bracket
                                                                          tells us the differential
                                                                          must be obtained at
                                   ∂G           ∂G
                           dG =          dp +         dT          (4.30)  constant pressure. The
                                   ∂p           ∂T
                                       T            p                     subscripted T indicates
                                                                          constant temperature.
             which should remind us of Equation (4.28). The first term on the
             right of Equation (4.30) is the change in G per unit change in
             pressure, and the subsequent dp term accounts for the actual change in pressure.
             The second bracket on the right-hand side is the change in G per unit change in
             temperature, and the final dT term accounts for the actual change in temperature.
               Equation (4.30) certainly looks horrible, but in fact it’s simply a statement of the
             obvious – and is directly analogous to the prices of apples and sweets we started by
             talking about, cf. Equation (4.27).
               We derived Equation (4.30) from first principles, using pure mathematics. An alter-
             native approach is to prepare a similar equation algebraically. The result of the
             algebraic derivation is the Gibbs–Duhem equation:

                                         dG = V dp − S dT                         (4.31)
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