1. Introduction¶
1.1. Unimolecular Reactions¶
Unimolecular reactions are those that involve a single reactant or product molecule, the union of isomerization and dissociation/association reactions:
Gas-phase chemical reactions occur as the result of bimolecular collisions between two reactant molecules. This presents a problem when there is only one participating reactant molecule! The conclusion is that the above reactions cannot be elementary as written; another step must be involved.
For a unimolecular reaction to proceed, the reactant molecule \(\ce{A}\) must first be excited to an energy that exceeds the barrier for reaction. A molecule that is sufficiently excited to react is called an activated species and often labeled with an asterisk \(\ce{A}^\ast\). If we replace the stable species with the activated species in the reactions above, the reactions become elementary again:
There are a number of ways that an activated species \(\ce{A}^\ast\) can be produced:
Chemical activation. \(\ce{A}^\ast\) is produced as the adduct of an association reaction:
\[\ce{B + C <=> A}^\ast\]Thermal activation. \(\ce{A}^\ast\) is produced via transfer of energy from an otherwise inert species \(\ce{M}\) via bimolecular collision:
\[\ce{A + M <=> A}^\ast \ce{\mbox{} + M}\]Photoactivation. \(\ce{A}^\ast\) is produced as a result of absorption of a photon:
\[\ce{A} + h \nu \ce{-> A}^\ast\]
Once an activated molecule has been produced, multiple isomerization and dissociation reactions may become competitive with one another and with collisional stabilization (thermal deactivation); these combine to form a network of unimolecular reactions. The major pathway will depend on the relative rates of collision and reaction, which in turn is a function of both temperature and pressure. At high pressure the collision rate will be fast, and activated molecules will tend to be collisionally stabilized before reactive events can occur; this is called the high-pressure limit. At low pressures the collision rate will be slow, and activated molecules will tend to isomerize and dissociate, often traversing multiple reactive events before collisional stabilization can occur.
The onset of the pressure-dependent regime varies with both temperature and molecular size. The figure below shows the approximate pressure at which pressure-dependence becomes important as a function of temperature and molecular size. The parameter \(m \equiv N_\mathrm{vib} + \frac{1}{2} N_\mathrm{rot}\) represents a count of the internal degrees of freedom (vibrations and hindered rotors, respectively). The ranges of the x-axis and y-axis suggest that pressure dependence is in fact important over a wide regime of conditions of practical interest, particularly in high-temperature processes such as pyrolysis and combustion [Wong2003].
Plot of the switchover pressure – indicating the onset of pressure dependence – as a function of temperature and molecular size. The value \(m \equiv N_\mathrm{vib} + \frac{1}{2} N_\mathrm{rot}\) represents a count of the internal degrees of freedom. Over a wide variety of conditions of practical interest, even very large molecules exhibit significant pressure dependence. Figure adapted from Wong, Matheu and Green (2003).¶
B. M. Wong, D. M. Matheu, and W. H. Green. J. Phys. Chem. A 107, p. 6206-6211 (2003). doi:10.1021/jp034165g
1.2. Historical Context¶
The importance of bimolecular collisions in unimolecular reactions was first proposed by Lindemann in 1922 [Lindemann1922]. It was soon recognized by Hinshelwood and others that a rigorous treatment of these processes required consideration of molecular energy levels [Hinshelwood1926]. The RRKM expression for the microcanonical rate coefficient $k(E)$ was derived in the early 1950s [Rice1927] [Kassel1928] [Marcus1951]. In the late 1950s master equation models of chemical systems began appearing [Siegert1949] [Bartholomay1958] [Montroll1958] [Krieger1960] [Gans1960], including an early linear integral-differential equation formulation by Widom [Widom1959]. Analytical solutions for a variety of simple models soon followed [Keck1965] [Troe1967] [Troe1973], as did the first numerical approaches [Tardy1966]. Numerical methods – which are required for complex unimolecular reaction networks – became much more attractive in the 1970s with the appearance of new algorithms, including Gear’s method for solving stiff systems of ordinary differential equations [Gear1971] and efficient algorithms for calculating the density of states [Beyer1973] [Stein1973] [Astholz1979]. In the 1990s computing power had increased to the point where it was practical to solve them numerically by discretizing the integrals over energy.
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