Question 18.9: Consider the interconversion of the “boat” and “chair” confo...

Thermodynamics, Statistical Thermodynamics, & Kinetics

Consider the interconversion of the “boat” and “chair” conformations of cyclohexane: The reaction is first order in each direction, with an equilibrium constant of $10^{4}$ . The activation energy for the conversion of the chair conformer to the boat conformer is $42 kJ/mol$ . Assuming an Arrhenius preexponential factor of $10^{12}s^{-1}$ , what is the expected observed reaction rate constant at 298 K if one were to initiate this reaction starting with only the boat conformer?

0933

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Using the Arrhenius expression of Equation (18.82), $k_{B}$ is given by

$k=Ae^{-E_{a}/RT}$ (18.82)

k_{B}=Ae^{-E_{a}/RT}=10^{12} s^{-1}\exp\left[ \frac{-42,000  J   mol^{-1}}{\left( 8.314  J  mol^{-1} K^{-1}\right)\left( 298  K \right)} \right]

=4.34 ×10^{4}s^{-1}

Using the equilibrium constant, $k_{A}$ can be determined as follows:

k_{c}=10^{4}=\frac{k_{A}}{k_{B}}

k_{A}=10^{4}  k_{B}=10^{4}\left( 4.34 ×10^{4} s^{-1}\right)=4.34 ×10^{8} s^{-1}

Finally, the apparent rate constant is simply the sum of $k_{A}$ and $k_{B}$ :

k_{app}=k_{A}+k_{B}=4.34 ×10^{8}s^{-1}

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