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                    278                                          4. Adsorption and Ion Exchange


                         1

                       0.9
                       0.8
                                                                 0.12
                       0.7
                                                                  0.1
                       0.6
                                                                 0.08
                      erf(x)  0.5                                0.06

                       0.4
                                                                 0.04
                       0.3                                       0.02
                       0.2                                         0
                                                                    0   0.02 0.04 0.06 0.08  0.1
                       0.1
                         0
                          0  0.25 0.5 0.75  1  1.25 1.5 1.75  2  2.25 2.5 2.75
                                             x

                                  Figure 4.15  The erf(x) function. Note that erf(–  x )    erf(  x ).


                    responsible for the movement of ions. The error of Patterson’s approximation is below 2%
                    for the whole range of   U ( t ), compared to the exact solution.

                    Finite solution volume and liquid diffusion contr ol  General case  : The general solu-
                    tion for the Langmuir isotherm is gien by Perry and Green (1999): v
                                        b(1     ) La  ∞  1   (  2 cU t  ( ) b     d )(      )
                                                              d
                                                                 b
                                       1           ln
                                                              d
                                           2 c     d  (  2 cU t  ( ) b     d )(      )
                                                                 b

                                       1   La
                                              ln 1     ( ) bU t  cU    ( ) t  2
                                         2c                                            (4.59)
                                       3 kt  C
                                         f   o
                                        r  o  pe q
                    where
                                                     1  La
                                                  b          w
                                                     1  La                             (4.60)

                                                  c     1  La                          (4.61)
                                                     w

                                                  d      2  c  4  0.5                  (4.62)
                                                     b
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