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Uses of Radioactive  Tracers                 261


              isotopic  exchange  mechanism  is valid,  it could  be proposed  that  the  reaction  mechanism
              would be

                       Cr(III)*  +  X-  +  Cr(II)-~  [Cr*-X-Cr] 4+  ~  Cr(II)*  +  X-  +  Cr(III)

              Such  a  reaction  mechanism  would  be  fostered  by  the  presence  of  anions  that  form
              complexes  more readily than perchlorate ion.  If HC1 solutions are used rather than HCIO 4
              it is found that the exchange takes place more rapidly and the half time of exchange is only
              2  rain,  which  agrees  with  the  proposed  mechanism  since  chloride  ions  are  known  to  be
              more favorable to complex formation than perchlorate ions.  Without the use of radioactive
              (or  isotope  separated)  chromium to  label  one  of the original  oxidation  states  there  would
              be no  means  of identifying  the exchange.



              9.4.3.  Determination of equilibrium  constants

                Determination  of  cation  (M z+)  -  anion  (L-,  for  ligand)  solution  equilibria  can
              advantageously be done using radioactive metal tracers because of the ease with which  the
              metal  concentration  can be  measured,  as e.g.  in the determination  of solubility products

                                            Ks0  =  [MZ+][L-] z                    (9.21)

              or  complex formation  constants,  also named stability constants

                                        fin =  [MLnZ-n]/[M z+][L-]n                (9.22)

              At trace concentrations of the metal (i.e. [M]tot  ,~ [L~  the complexing anion concentration
              is unaffected by the metal ion concentration,  which allows easy calculation of the values of
              these  two  important  variables  in  the  system.  The  experimental  techniques  used  for  this
              purpose are based  on two-phase equilibria:  solubility  (solid/liquid),  paper  electrophoresis
              (paper/aqueous  solution),  solvent  extraction  (organic  solvent/water)  and  ion  exchange
              (resin/water).  The  equilibria  measured  have  been  shown  to  be  independent  of  the  metal
              concentration  in  the  range  10 .3  to  <  10 -13 M,  as long  as no  polynuclear  complexes  (i.e.
              with  several  metal  atoms per complex  molecule)  are  formed  (cf.  w
                (a)  Solubility
                The  sensitivity  of tracer  detection  makes  measurement  of  solubilities  relatively  simple.
              This  is  illustrated  by  the  first  radioactive  tracer  experiment  by  de  Heresy  and  Paneth  in
              1913  in which  the solubility of lead chromate was determined.  Chromate ions were added
              to  a  solution  of  PbCI 2  containing  a  known  amount  of  21~   precipitating  the  lead  as
              PbCrO 4. The precipitate contained 2030 "radioactive units", and had a weight of 11.35 mg.
              The specific activity was thus 2030/11.35  =  179 "units" mg -1. Shaking the precipitate with
              water  dissolved  2.14  units  per  10~  ml.  The  solubility  was  calculated  to  be  2.14/179  =
              0.012  mg 1-1 or 3.7 x  10 .8 M  Pb 2+.  If [Pb 2+ ]  =  [CrO42"] the solubility product would be
              Ks0  =  (3.7 x  10-8) 2  =  1.4 x  10 -15. The  modem  value  is 2x 10 -14.
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