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                      The integral heat of solution per mole of B involves the addition of 1 mole of B        Section 9.5
                  to pure A to produce the solution, a process in which the B mole fraction changes from    Ideal Solutions
                  zero to its final value x . Suppose, instead, that we add (at constant T and P) 1 mole
                                      B
                  of B to an infinite volume of solution whose B mole fraction is x . The solution com-
                                                                         B
                  position will remain fixed during this process. The enthalpy change per mole of added
                  B when B is added at constant T and P to a solution of fixed composition is called the
                  differential heat of solution of B in A and is symbolized by  H diff,B . The quantity
                   H diff,B  is an intensive function of T, P, and solution composition. From the preceding
                  definitions, it follows that at infinite dilution the differential and integral heats of solu-
                  tion become equal:  H int,B    H diff,B  [see Figs. 9.10 and 9.12 and Eq. (9.38)].
                                               q
                                      q
                      Rather than imagine a solution of infinite volume, we can imagine adding at con-
                  stant T and P an infinitesimal amount dn of B to a solution of finite volume and with
                                                    B
                  composition x .If dH is the enthalpy change for this infinitesimal process, then
                               B
                   H diff,B     dH/dn at the composition x . When dn moles of pure B is added to the
                                                              B
                                 B
                                                    B

                  solution at constant T and P, the solution’s enthalpy changes by dH soln     H B  dn [this
                                                                                     B

                  follows from the definition H    ( H soln / n )  ] and the enthalpy of pure B changes
                                               0
                                                     0
                                           B
                                                       B T, P,n A
                  by dH*     H* dn (since H*    n H* ). The overall enthalpy change for this addi-
                                                    m,B
                                                 B
                               m,B
                                    B
                                            B
                        B


                  tion is then dH    H B  dn    H* dn , and ¢ H diff     dH/dn    H B     H* . Thus
                                            m,B
                                                 B
                                       B
                                                                               m,B
                                                                    B

                                             ¢H diff,B    H   H*                     (9.38)  Figure 9.12
                                                        B
                                                             m,B
                  The differential heat of solution of B equals the partial molar enthalpy of B in the  Integral heat of solution of H SO 4
                                                                                                                 2
                  solution minus the molar enthalpy of pure B. Figure 9.10 plots differential heats of  in water at 25°C and 1 atm versus
                  solution in H O–H SO solutions at 25°C; here, either H O or H SO can be regarded  H SO mole fraction.
                                                                                              2
                                                                                                 4
                                                                  2
                                                                         2
                                                                            4
                                      4
                             2
                                  2

                  as the solvent. As noted after Eq. (9.36), ¢H diff,B    H   H*  can be found from the
                                                                     m,B
                                                                B
                  intercept of a tangent line on the   mix H>n  versus x plot. Figure 9.13 plots   mix H>n  for
                                                            B
                  H O    H SO solutions at 25°C and 1 atm. This figure can be used to find  H diff  values
                    2
                          2
                              4
                  (Prob. 9.27).                                                                mix H/n     mix H/n  for
                      Some values of differential heats of solution (relative partial molar enthalpies) of  kJ/mol  H O   H SO 4
                                                                                                               2
                                                                                                         2
                  solutes in aqueous solution at infinite dilution at 25°C and 1 bar are:
                  Solute                NaCl K SO   4  LiOH CH COOH CH OH CO(NH )
                                                                                        2 2
                                                                3
                                                                           3
                                                2
                     q
                   1H   H* 2>1kJ>mol2    3.9    23.8   23.6      1.5       7.3      15.1
                           m,B
                     B
                  If B is a solid at 25°C, H* in this table refers to solid B. Dissolving a tiny amount of
                                       m,B
                  NaCl in water at 25°C is an endothermic process, whereas dissolving a tiny amount of
                  LiOH in water is exothermic.
                    9.5          IDEAL SOLUTIONS
                  The discussion in Secs. 9.1 to 9.4 applies to all solutions. The rest of this chapter deals  Figure 9.13
                  with special kinds of solutions. This section and the next consider ideal solutions.
                      The molecular picture of an ideal gas mixture is one with no intermolecular inter-    mix H/n for H O   H SO 4
                                                                                                            2
                                                                                                      2
                  actions. For a condensed phase (solid or liquid), the molecules are close together, and  solutions at 25°C and 1 atm.
                  we could never legitimately assume no intermolecular interactions. Our molecular
                  picture of a liquid or solid ideal solution (also called an ideal mixture) will be a
                  solution where the molecules of the various species are so similar to one another that
                  replacing molecules of one species with molecules of another species will not change
                  the spatial structure or the intermolecular interaction energy in the solution.
                      Consider a solution of two species B and C. To prevent change in spatial structure
                  of the liquids (or solids) on mixing B and C, the B molecules must be essentially the
                  same size and shape as the C molecules. To prevent change in the intermolecular in-
                  teraction energy on mixing, the intermolecular interaction energies should be essen-
                  tially the same for B-B, B-C, and C-C pairs of molecules.
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