Transition State Theory

Derivation of Transitional Partition Function q'T

                               

      To show

                                                                                                         (T1)

      Translational Energy

      We solve the Schrödinger equation for the energy of a molecule trapped in an infinite potential well. This situation is called “a particle in a box”. For a particle in a box of length a

                                                          V(X)

      The potential energy is zero everywhere except at the walls where it is infinite so that the particle cannot escape the box. Inside the box

                                                                                                                         (T2)

      The box is a square well potential where the potential is zero between x = 0 and x = a but infinite at x = 0 and x = a (A5p392). The solution is to the above equation is

                                                                                                     (T3)

            where                                                                                                              (T4)

                                                            

      We now use the boundary conditions

At x = 0 :   = 0, sin 0 = 0, and cos 0=1

     therefore  B = 0

      the wave equation is now

             = A sin kx

      At  x = a      = 0

      will be zero provided

      therefore ka = n

            where n is an integer, 1, 2, 3

                                                                                                                                         (T5)

      Substituting (T5) into (T4) we see that only certain energy states are allowed

                                                                                                                                 (T6)

      For particle in a a 3-D box of sides a, b, and c

                                                        

      Back to one dimension

                                                                  

      Therefore relative to the lowest energy level n = 1, the energy is

                                                                       (T7)

      Then

                                                                                                                     (T8)

                                                                                (T9)                     

Return to Transition State Theory