At 700 K the rate expression for the decomposition of HI,
2HI\rightarrow H_{2}+I_{2}is given by (40) r=1.16\times 10^{-3}[HI]^{2} kmol/(m^{3}\cdot s)when [HI] is expressed in kmol/m^{3}. Compare this reaction rate expression with that predicted by the analog of equation (4.3.7), which corresponds to collisions of the type A–A. Given:
Molecular weight of HI = 127.9 , \sigma _{HI}=2.0\times 10^{-10}m
The measured activation energy is 186.1 kJ/mol.
On the basis of equation (4.3.4), the A–A analog of equation (4.3.7) is
r_{AA}=P_{s}2({n}'_{A})^{2}\sigma _{A}^{2}\sqrt{\frac{\pi k_{B}T}{m_{A}}}e^{-E_{c}/RT}
From equation (4.3.10), using R = 8.31 J/(mol×K),
E_{c}=186,100-\frac{8.31(700)}{2}=183,200 J/mol
In SI units, M_{A}=0.1279 kg/mol. If {n}'_{A} is measured in molecules/m^{3} and r_{AA} in molecules/m^{3}\cdot s, the expression for r_{AA} becomes
r_{AA}=P_{s}2({n}'_{A})^{2}(2.0\times 10^{-10})^{2}\times \left ( \sqrt{\frac{\pi (8.31)700}{0.1279}}e^{-183,200/[8.31(700)]} \right )
=6.35\times 10^{-31}P_{s}({n}'_{A})^{2}
To convert from number densities of molecules to reac-tant concentrations in kmol/m^{3}, one notes that
C_{A}=\frac{{n}'_{A}}{1000N_{0}}
where N_{0} is Avogadro’s number. A similar conversion is necessary to measure r_{AA} in kmol/(m^{3}\cdot s). In these units,
r_{AA}=6.35\times 10^{-31}(6.023\times 10^{23})\times 10P_{s}(C_{A})^{2}
=3.83\times 10^{-4}P_{s}(C_{A})^{2}
If the steric factor is comparable to unity, the rate calculated is within an order of magnitude of the experimental value of 1.16\times 10^{-3}[HI]^{2}. Authors of other textbooks have reported even better agreement between experimental values of this rate expression and those calculated from collision theory. Changes in the values used for the activation energy of the reaction and the molecular diameter are often sufficient to bring the calculated values into much closer agreement with the experimental values. Since measurements of different properties lead to significant differences in calculated values of hard-sphere molecular diameters, these quantities are not accurately known. Moreover, uncertainties of several percent in reaction activation energies are not at all unusual.
Consideration of a variety of other systems leads to the conclusion that very rarely can one employ the collision theory to predict rate constants that will be comparable in magnitude to experimental values. Although it is not adequate for predictions of reaction rate constants, it nonetheless provides a convenient physical picture of the reaction act and a useful interpretation of the concept of activation energy. The major shortcomings of the theory lie in its failure to relate the steric factor and the activation energy to molecular parameters from which a priori predictions can be made.