Reaction order estimate

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Reaction Order Estimate To observe the order of a reactionR→ (νP/νR)P, is to plot the conversion ξ registered after a timeτ, as a function of the initial concentrationC0. This estimate is of importance when probing the initial concentration effect to the conversion, as shown in the maleic anhydride synthesis. The rate law reads as follows: dC dt =−kC n which can be integrated to obtainC(t): { C1−n =C1−n 0 − (1−n)kt n ̸= 1 C =C0e−kt n = 1 Using the conversion definitionC =C0(1−ξ), then the functionξ(t) is obtained forn̸= 1, and it has general validity: ξ = 1− ( 1− (1−n)ktC n−1 0 )1/(1−n) Letting k = k0kn−1 1 such thatk0 = 1/τ, the observation timeτ is then independent from the reaction order. The objective term is thuskτ =kn−1 1 and the conversion relations become:    ξ = 1− ( 1− 1−n (k1C0)1−n )1/(1−n) n̸= 1 ξ = 1−e−1 n = 1 where k1C0 acts as a dimensionless variable, which is the independent vari- able with range(0.5, 4), chosen arbitrarily and for qualitative analysis. In practical situations, only the kinetic constantk and the observation time τ are known, and it is possible to recoverk1 = (kτ )1/(n−1), and from the Damköhler numberDa = kτC n−1 0 = (k1C0)n−1 is readily obtained: { ξ = 1− (1− (1−n)Da)1/(1−n) n̸= 1 ξ = 1−e−1 n = 1 However, sincen is unknown, we setk1 = 1 and the objectivek1C0 range becomes theC0 range andDa = Cn−1 0 . Theξ(C0) plots do not change their monotonic trends nor their derivative’s sign, which are the only properties of interest. We remark that this proceduredoes not extract the exact reaction order n, but it only gives a robust estimation procedure ifn lies above or below 1. We can derive a lower limit fork1C0, settingξ = 1 for n̸= 1: (k1C0)l = (1−n)1/(1−n) when kτ =kn−1 1 1 Figure 1: Dimensionless observation plot for a genericn-th order…