(13.29) it follows that: (13.30) Further progress can be made by applying the convolution theorem: (13.31) where and and and are transform pairs, and denotes convolution. Solution of 13.30 is Q q, G g possible by recognizing that (13.32) And (13.33) where if < and is unity otherwise, is the Heavyside step function. Convolving the ILTs E Ea in Eq. 13.32 and 13.33 gives: (13.34) The units should be input follows: in states per cm-1 in molecules cm-3 s-1 where is in units of cm-1 K-1, and in units of K. kB T0 The association ILT A similar expression can be obtained for the case where the Arrhenius expression is for the high pressure association rate coefficient. Under high pressure conditions the forward and the reverse rate coefficients are related by the equilibrium constant as indicated by Eq. 13.13: (13.35) If has a modified Arrhenius form then the equivalent expression to Eq. 13.30 is,
(13.29) it follows that: (13.30) Further progress can be made by applying the convolution theorem: (13.31) where Q and q, and G and g are transform pairs, and denotes convolution. Solution of 13.30 is possible by recognizing that (13.32) And (13.33) where if E < Ea and is unity otherwise, is the Heavyside step function. Convolving the ILTs in Eq. 13.32 and 13.33 gives: (13.34) The units should be input follows: in states per cm-1 in molecules cm-3 s-1 where kB is in units of cm-1 K-1, and T0 in units of K. The association ILT A similar expression can be obtained for the case where the Arrhenius expression is for the high pressure association rate coefficient. Under high pressure conditions the forward and the reverse rate coefficients are related by the equilibrium constant as indicated by Eq. 13.13: (13.35) If has a modified Arrhenius form then the equivalent expression to Eq. 13.30 is,
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