Page 290 - Advanced Organic Chemistry Part A - Structure and Mechanisms, 5th ed (2007) - Carey _ Sundberg
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in which 
 is the transmission coefficient, which is usually taken to be 1, k is    271
                                                                             B
          Boltzmann’s constant, h is Planck’s constant, R is the gas constant, and T is the
          absolute temperature. If the activated complex is considered to be in equilibrium with  SECTION 3.2
          its component molecules, the attainment of the transition state (TS) can be treated as  Chemical Kinetics
          being analogous to a bimolecular reaction:
                                               TS
                                         ‡
                                        K =                                (3.15)
                                              A  B
          The position of this equilibrium is related to the free energy required for attainment
                                        ‡
          of the transition state. The symbol     is used to specify that it is a transition state or
          “activated complex” that is under discussion:

                                         ‡
                                       G =−RT ln K  ‡                      (3.16)
                           ‡
          The free energy  G is called as the free energy of activation. The rate of a reaction
          step is then given by
                                        
k T
                                          B
                                 Rate =       TS                           (3.17)
                                         h
                                        
k T    ‡
                                          B
                                 Rate =     e − G /RT   A  B               (3.18)
                                         h
          Comparison with the form of the expression for the rate of any single reaction step
                                     ‡
          reveals that the magnitude of  G is the factor that determines the magnitude of k at
                                                                             r
          any given temperature:
                                      Rate = k  A  B                       (3.19)
                                              r
              The temperature dependence of reaction rates permits evaluation of the enthalpy
          and entropy components of the free energy of activation. The terms in Equation (3.14)
          can be rearranged to examine the temperature dependence:

                                     
k T     − H /RT       S /R
                                               ‡
                                                       ‡
                                       B
                                k =        e        e                      (3.20)
                                 r
                                      h
                                                                      ‡
                             ‡
          The term  
k T/h e  S /R  varies only slightly with T compared to e − H /RT  because
                      B
          of the exponential nature of the latter. To a good approximation, then:
                                       k r    − H /RT
                                                 ‡
                                         = Ce                              (3.21)
                                       T
                                       k r  − H ‡
                                     ln  =        +C                       (3.22)
                                       T     RT
                                                                   ‡
          A plot of ln k /T  versus 1/T is a straight line, and its slope is − H /R. After  H ‡
                      r
                                     ‡
          is determined in this manner,  S is available from the relationship
                                          H ‡      hk
                                     ‡                r
                                    S =      +Rln                          (3.23)
                                          T        
k T
                                                     B
          which can be obtained by rearranging Equation (3.14).
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