As an example of a unimolecular/recombination fall-off reaction, consider methyl recombination. In the high-pressure limit, the appropriate description of the reaction is CH₃ + CH₃ C₂H₆ + M. In the low-pressure limit, a third-body collision is required to provide the energy necessary for the reaction to proceed, that is, the appropriate description is CH₃ + CH₃ + M C₂H₆. When such a reaction is at either limit, the (solely temperature-dependent) rate expressions discussed in the preceding paragraphs are applicable. However, when the pressure and temperature are such that the reaction is between the limits, the rate expressions are more complicated. To denote a reaction that is in this "fall-off" region, we write the reaction with the positive +M enclosed in parentheses, CH₃ + CH₃(+M) C₂H₆(+M).

There are several methods of representing the rate expressions in this fall-off region. The simplest one is due to Lindemann [10]. There are also now two other (and related) methods that provide a more accurate description of the fall-off region than does the simple Lindemann form. We provide three of these forms as options.

For the Lindemann form, Arrhenius rate parameters are required for both the high- and low-pressure limiting cases, and these are blended to produce a pressure-dependent rate expression. In Arrhenius form, the parameters are given for the high-pressure limit and the low-pressure limit as follows:

(4–14)

(4–15)

The rate constant at any pressure is then taken to be

(4–16)

where the reduced pressure is given by

(4–17)

and is the concentration of the mixture, possibly including enhanced third-body efficiencies. For this example, note that the units for are 1/sec, for are cm³/(mole sec), and for are 1/sec. If the in Equation 4–16 is unity, then this is the Lindemann form. The other descriptions involve more complex expressions for the function .

In the Troe form, [7] is given by

(4–18)

The constants in Equation 4–18 are

(4–19)

(4–20)

(4–21)

and

(4–22)

The four parameters , , , and must be specified as auxiliary input, as described in Pressure Dependent Reaction Parameters. It is often the case that the parameter is not used. For this reason, the use of either three or four parameters is allowed.

The approach taken at SRI International by Stewart, et al. [11] is in many ways similar to that taken by Troe, but the blending function is approximated differently. Here, is given by

(4–23)

where

(4–24)

In addition to the six Arrhenius parameters—three each for the low-pressure limit and high-pressure limit expressions: the user must supply the parameters , , and in the expression. The parameters and were not discussed by Stewart, et al., but we have included them as additional optional parameters to increase flexibility. If one wishes, and can be considered parameters that define a weak-collision efficiency factor, in the event that one wants to compute strong-collision rate parameters and correct them with such a factor.

As an example of a chemically activated bimolecular reaction, consider the reaction CH₃ + CH₃(+M) C₂H₅ + H(+M). This reaction, which is endothermic, occurs through the same chemically activated C₂H₆^* adduct as does the recombination reaction CH₃ + CH₃(+M) C₂H₆ (+M). As the pressure increases, deactivating collisions of C₂H₆^* with other molecules cause the rate coefficient for C₂H₆ formation to increase. At the same time, these deactivating collisions preclude the dissociation of C₂H₆^* into C₂H₅ + H, thus causing this rate coefficient to decrease with increasing pressure.

We assume the rate coefficient for a chemically activated bimolecular reaction to be described by the following equation:

(4–25)

where is analogous to the Lindemann form of Equation 4–16. Note that in Equation 4–25, is the pressure-independent factor, whereas in Equation 4–16 it is . The three choices for the function are exactly the same as for the unimolecular fall-off reactions, that is, the Lindemann (), Troe, or SRI forms.

Miller and Lutz [13] recently developed a generalized method for describing the pressure dependence of a reaction rate based on direct interpolation of reaction rates specified at individual pressures. In this formulation, the reaction rate is described in terms of the standard modified Arrhenius rate parameters. Different rate parameters are given for discrete pressures within the pressure range of interest. When the actual reaction rate is computed, the rate parameters will be determined through logarithmic interpolation of the specified rate constants, at the current pressure from the simulation. This approach provides a very straight-forward way for users to include rate data from more than one pressure regime.

For a given reaction, then, the user would supply rate parameters for a set of pressures. The set of pressure points for which rate parameters are specified must include at least two different pressures. If the rate expression at a given pressure cannot be described by a single set of Arrhenius parameters, more than one set may be provided. See Figure 3.2: Examples of PLOG data using multiples sets of Arrhenius parameters. In such cases, the reaction rates will be treated in a similar manner as for DUPLICATE reactions, meaning that Chemkin will use the sum of the sets of rates provided for the given pressure at that point.

During a simulation, if the current pressure is within 1% of one of the pressures for which rate constants were provided, then that set of rate parameters will be used directly. However, If the current pressure is in between the pressure points provided, then the rate will be obtained by a linear interpolation of as a function of (natural logarithms). For between and , is obtained using Equation 4–26.

(4–26)

If the rate of the reaction is desired for a pressure lower than any of those provided, the rate parameters provided for the lowest pressure are used. Likewise, if rate of the reaction is desired for a pressure higher than any of those provided, the rate parameters provided for the highest pressure are used.

This logarithmic interpolation method can be used as an alternative approach to describing any type of pressure dependence, including the multiple-well, multiple-channel reactions discussed in Multiple-well Multiple-channel Reactions Using Chebyshev Polynomials. It has the advantage of being conceptually straightforward to implement. However, the resolution or accuracy of the pressure dependence will depend on the number of pressure points included for each reaction.

An example of the multiple-well, multiple-channel chemically activated reaction is the reaction of the ethyl radical with oxygen, C₂H₅ + O₂. This chemically activated reaction occurs through three wells. The initial well corresponds to the chemically activated molecule CH₃CH₂OO formed by the radical addition process via a loose transition state. This activated molecule can further isomerize to the hydroperoxy radicals CH₂CH₂OOH and CH₂CHOOH leading to different products [14]. The Lindemann based methods described in the previous sections, that is, the Lindemann, Troe, and SRI forms, although accurate for representing the falloff behavior of single-well reactions, do not apply well to multiple-well reactions [15]. A method based on the Chebyshev expansions are proposed by Venkatesh, et al. [15] for approximating the pressure and temperature-dependent behavior of multiple-well reactions. The Chebyshev expansions provide accurate approximations over any given temperature and pressure domain for single- and multiple-well reactions. However, these approximates should not be used for extrapolative studies outside their defined domain.

Instead of using the modified Arrhenius form for the rate coefficient, the Chebyshev expansions approximate the logarithm of the rate coefficient directly as a truncated bivariate Chebyshev series in the reverse temperature and logarithm of the pressure. Since the Chebyshev polynomials are only defined in the interval of [-1, +1], the temperature and pressure boundaries for the approximation must be established first, that is,

and

The domain is then mapped onto a square bounded by using the transformations

(4–27)

and

(4–28)

The logarithm of the rate coefficient is approximated by the Chebyshev expansions as

(4–29)

where the Chebyshev polynomials of the first kind of degree are given by

(4–30)

with . The integers and denote the number of basis functions along the temperature and the pressure axis, respectively. The accuracy of the approximates will increase monotonically with and . The coefficients, , of the Chebyshev expansions are determined from a least-squares fit to a set of rate coefficient data points, , computed from a detailed theory such as the Rice-Ramsperger-Kassel-Marcus (RRKM) theory. To ensure the approximation is uniform over the desired domain, the computed rate coefficient data must be on the Gauss-Chebyshev grid. For example, if the computed rate coefficient data is on a Gauss-Chebyshev grid, its coordinates, and , must be the roots of a high-order Chebyshev polynomial of the first kind given by

(4–31)

(4–32)

where and . The integers and are the resolutions of the computed rate coefficient data in the temperature and the pressure direction, respectively. As a necessary condition, the number of basis functions in each direction of the Chebyshev expansions should be no greater than the data resolution of that direction, that is, and .

To use the Chebyshev expansions to represent the rate coefficient of a reaction, the user should provide the temperature and pressure limits of the expansion, , , (auxiliary keyword TCHEB) and , (auxiliary keyword PCHEB) the number of basis functions used in each direction, and , and the coefficients, , (auxiliary keyword CHEB). See Chebyshev Polynomial Rate Expressions and Equation 3–5 for more information about the use of these keywords.