The pre-exponential factor for the elementary reaction O + H2 → OH + H has been determined experimentally to be 8.04 × 10^11 cm³/mol·s at 500 K. Calculate the pre-exponential factor for this elementary chemical reaction at the same temperature using collision theory and compare your result with the

Question

The pre-exponential factor for the elementary reaction O + [latex]H_{2}[/latex] → OH + H has been determined experimentally to be 8.04 × [latex]10^{11}[/latex] cm³/mol·s at 500 K. Calculate the pre-exponential factor for this elementary chemical reaction at the same temperature using collision theory and compare your result with the given experimental value.

Accepted Answer

From Eqs. (17.6) and (17.19), the pre-exponential factor provided by kinetic theory is

A = B[latex]T^{n}[/latex], (17.6)

[latex]B=2 N_{A} \sigma_{A B}^{2}\left(\frac{2 \pi k T}{\mu} ight)^{1 / 2}[/latex], (17.19)

[latex]A=2 N_{A} \sigma_{A B}^{2}\left(\frac{2 \pi k T}{\mu} ight)^{1 / 2}[/latex].

From Appendix O, we may take the rigid-sphere diameter of [latex]H_{2}[/latex] to be 2.92 Å and that for atomic oxygen to be half of its value for [latex]O_{2}[/latex] or 1.72 Å. On this basis, we obtain, from Eq. (16.55),

[latex]\sigma_{A B}=\frac{1}{2}\left(\sigma_{A}+\sigma_{B} ight)[/latex] (16.55)

[latex]\sigma_{AB}=\frac{\sigma_{A}+ \sigma_{B}}{2}=\frac{2.92+1.72}{2}[/latex] = 2.32 Å.
From Eq. (16.6), the reduced mass for this binary reaction is
[latex]\mu=\frac{m_{1} m_{2}}{m_{1}+m_{2}}[/latex]. (16.6)

[latex]\mu=\frac{m_{A} m_{B}}{m_{A}+m_{B}}=\frac{(2)(16)}{2+16}\left(1.6605 imes 10^{-24} g ight)=2.952 imes 10^{-24} g[/latex].

Therefore, the pre-exponential factor from kinetic theory becomes

[latex]A=2\left(6.022 imes 10^{23} mol ^{-1} ight)\left(2.32 imes 10^{-8} cm ight)^{2}[/latex][latex] imes \sqrt{\frac{2 \pi\left(1.381 imes 10^{-16} g \cdot cm ^{2} / s ^{2} \cdot K ight)(500 K )}{2.952 imes 10^{-24} g }}[/latex]

= 2.48 × [latex]10^{14}[/latex] cm³/mol · s.

The calculated pre-exponential factor from kinetic theory is over 300 times greater than that provided by experiment. This huge discrepancy arises from the internal energy modes of molecular hydrogen, which are not accounted for in a purely kinetic model.

Question 17.1: The pre-exponential factor for the elementary reaction O + H...

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