Question 17.2: The pre-exponential factor for the elementary chemical react...
The pre-exponential factor for the elementary chemical reaction, O + H_{2} → 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 reaction at the same temperature using transition state theory and compare your result with the given experimental value.
From Eq. (17.31), the pre-exponential factor provided by transition state theory is
A=N_{A}\left(\frac{k T}{h}\right) \frac{\phi^{\ddagger}}{\phi_{ A } \phi_{ B }},
where the required partition functions per unit volume can be determined via Eq. (10.62):
\phi_{i}=\frac{Z_{t r}}{V} Z_{i n t}=\left(\frac{2 \pi m_{i} k T}{h^{2}}\right)^{3 / 2} Z_{i n t} , (10.62)
\phi_{i}=\left(\frac{2 \pi m_{i} k T}{h^{2}}\right)^{3 / 2} Z_{i n t} .Hence, for atomic oxygen, we obtain
\phi_{ A }=\phi_{t r}^{\circ} Z_{e l}=\left(\frac{2 \pi m_{ A } k T}{h^{2}}\right)^{3 / 2} Z_{e l},
so that, given the associated term symbol from Appendix J.1, we get Z_{el} = 9 and thus
\phi_{ A }=9\left[\frac{2 \pi(16)\left(1.66 \times 10^{-24} g \right)\left(1.38 \times 10^{-16} g \cdot cm ^{2} / s ^{2} \cdot K \right)(500 K )}{\left(6.626 \times 10^{-27} g \cdot cm ^{2} / s \right)^{2}}\right]^{3 / 2}
= 9(1.34 × 10^{26}) = 1.21 × 10^{27} cm^{−3}.
Similarly, for molecular hydrogen, we have, for the rigid-rotor/harmonic-oscillator model,
\phi_{ B }=\left(\frac{2 \pi m_{ B } k T}{h^{2}}\right)^{3 / 2} Z_{r o t} Z_{v i b} Z_{e l} .Accordingly, from Appendix K.1, the ground electronic state for molecular hydrogen mandates Z_{el} = 1. For the rotational mode of H_{2}, B_{e} = 60.853 cm^{−1} so that θ_{r} = (1.4387)(60.853) = 87.55 K. Similarly, for the vibrational mode of H_{2}, ω_{e} = 4401 cm^{−1} and thus θ_{v} = (1.4387)(4401) = 6332 K. Therefore, from Eqs. (9.25) and (9.47),
Z_{r o t}=\frac{T}{\sigma \theta_{r}}\left[1+\frac{1}{3}\left(\frac{\theta_{r}}{T}\right)+\frac{1}{15}\left(\frac{\theta_{r}}{T}\right)^{2}+\frac{4}{315}\left(\frac{\theta_{r}}{T}\right)^{3}+\cdots\right]. (9.25)
Z_{v i b}=\left(1-e^{-\theta_{v} / T}\right)^{-1} . (9.47)
Z_{r o t} Z_{v i b} Z_{e l}=\left(\frac{T}{2 \theta_{r}}\right)\left(1+\frac{\theta_{r}}{3 T}\right)\left(1-e^{-\theta_{v} / T}\right)^{-1} Z_{e l}=\frac{(500 K )}{2(87.55 K )}(1.058)(1)(1)=3.02.
On this basis, the molecular partition function per unit volume for H_{2} is
\phi_{ B }=3.02\left(\frac{2 \pi m_{ B } k T}{h^{2}}\right)^{3 / 2}=3.02\left(\frac{m_{ B }}{m_{ A }}\right)^{3 / 2} \phi_{t r}^{\circ}=3.02\left(\frac{2}{16}\right)^{3 / 2}\left(1.34 \times 10^{26}\right)
= 1.79 × 10^{25} cm^{−3}.
Now, for the activated complex, we assume a linear triatomic, i.e.,
O — H · · · · · H,
with only three vibrational modes, as the fourth such mode represents the weak bond that ruptures during chemical reaction. For simplicity in our analysis, we model the rotational mode of the activated complex using its analogous OH structure; hence, from Appendix K.1, B_{e} = 18.911 cm^{−1} and thus θ_{r} = 27.21 K. In a similar fashion, we model the vibrational modes based on H_{2}O so that, from Appendix K.3, we have a single stretch mode at ω_{e} \simeq 3700 cm^{−1} and two bending modes at ω_{e} \simeq 1600 cm^{−1}. Therefore, the characteristic vibrational temperatures for the activated complex are θ_{v1} = 5323 K and θ_{v2} = 2302 K, respectively. Similarly, from Appendix K.3, the ground electronic state for H_{2}O indicates that Z_{el} = 1. Based on this model for the activated complex, its partition function per unit volume becomes
\phi^{\ddagger}=\left(\frac{2 \pi m_{ X ^{\ddagger}} k T}{h^{2}}\right)^{3 / 2} Z_{r o t} Z_{v i b} Z_{e l}=Z_{r o t} Z_{v i b} Z_{e l}\left(\frac{m_{ X ^{\ddagger}}}{m_{ A }}\right)^{3 / 2} \phi_{t r}^{\circ}.
Evaluating the above internal partition functions, we have, from Eqs. (9.26) and (9.47),
Z_{r o t}=\frac{T}{\sigma \theta_{r}}. (9.26)
Z_{r o t} Z_{v i b} Z_{e l}=\left(\frac{T}{\theta_{r}}\right)\left(1-e^{-\theta_{v 1} / T}\right)^{-1}\left(1-e^{-\theta_{v 2} / T}\right)^{-2} Z_{e l}=\frac{(500 K )}{(27.21 K )}(1)(1.01)^{2}(1)=18.75.
Consequently, for the activated complex,
\phi^{\ddagger}=18.75\left(\frac{m_{X^{\ddagger}}}{m_{ A }}\right)^{3 / 2} \phi_{t r}^{\circ}=18.75\left(\frac{18}{16}\right)^{3 / 2}\left(1.34 \times 10^{26}\right)=3.00 \times 10^{27} cm ^{-3} .Having determined all relevant partition functions, we may finally calculate the pre-exponential factor, thus obtaining
A=N_{A}\left(\frac{k T}{h}\right) \frac{\phi^{\ddagger}}{\phi_{ A } \phi_{ B }}=\left(6.022 \times 10^{23} mol ^{-1}\right)\times \frac{\left(1.38 \times 10^{-16} erg / K \right)(500 K )\left(3.00 \times 10^{27} cm ^{-3}\right)}{\left(6.626 \times 10^{-27} erg \cdot s \right)\left(1.21 \times 10^{27} cm ^{-3}\right)\left(1.79 \times 10^{25} cm ^{-3}\right)},
so that our result from transition state theory becomes
A = 8.69 × 10^{11} cm³/mol · s.
Remarkably, the pre-exponential factor based on transition state theory for this elementary chemical reaction agrees with that based on experiment to within 8%, which is probably fortuitous given our rather simplified model for the activated complex. On the other hand, we must not forget in the midst of these long-winded calculations that transition state theory inherently accounts for all internal energy modes, thus providing predictions for chemical reaction rates that are undeniably much more realistic than those determined from collision theory.