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1. The thermal interconversion of the axial and equatorial substitutents of the chair conformation of cyclohexane is extremely slow because the two forms are separated by a relatively high activation energy barrier, 10.5 kcal/mol (3672 cm−1). With CN substituents, the equatorial form is only 0.24 kcal/mol (84 cm−1) lower in energy than the axial form. Show how you would determine the ratio between the concentrations of the equatorial and axial forms using the Boltzmann distribution.

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Answer:

The ratio is  [tex]0.67 : 1[/tex]

Explanation:

From the question we are told that

 The activation energy separating the equatorial and the axial form is [tex]\Delta E = 0.24\ kcal/mol[/tex]

 Generally according to the Boltzmann distribution , the relationship between  concentration(in terms of number of molecule) of the equatorial form and  the concentration of the axial form is mathematically represented as

       [tex]N = N_o e^{-\frac{\Delta E }{RT} }[/tex]

Here [tex]N_o[/tex] is the number of molecule in equatorial form and  N is the number of the molecules in the axial form.

   R is the gas constant with value  [tex]R = 1.987 *10^{-3} \ kcal\cdot mol^{-1} \cdot k^{-1}[/tex]

     T is the temperature of the room with value  [tex]T = 25^oC = 298 \ K[/tex]

So

    [tex]\frac{N}{N_o} = e^{-\frac{0.24 }{1.987*10^{-3} * 298} }[/tex]

=> [tex]\frac{N}{N_o} = 0.67[/tex]

So the ratio of the concentration of equatorial form to the axial form is

      [tex]0.67 : 1[/tex]

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