An example of the multiple-well, multiple-channel chemically activated reaction is the reaction of the ethyl radical with oxygen, C₂ H₅ + O₂. This chemically activated reaction occurs through three wells. The initial well corresponds to the chemically activated molecule CH₃ CH₂ OO formed by the radical addition process via a loose transition state. This activated molecule can further isomerize to the hydroperoxy radicals CH₂ CH₂ OOH and CH₂ CHOOH leading to different products [21]. 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 [22]. A method based on the Chebyshev expansions are proposed by Venkatesh, et al.[22] 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,

(3–39)

and

(3–40)

Note that is the system pressure when the reaction uses (+M) as the 3rd body. When a species is used as the 3rd body, for example, (+N2) on the reaction line, the partial pressure of that species is used for calculation of the reaction rate constants.

The domain is then mapped onto a square bounded by ±1 using the transformations^[1]

(3–41)

and

(3–42)

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

(3–43)

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

(3–44)

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

(3–45)

(3–46)

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 Table 3.7: Alphabetical Listing of Gas-phase Reaction Auxiliary Keywords and Figure 3.5: Examples of Auxiliary Reaction Data of the Chemkin Input Manual for more information about the use of these keywords.