Kinetic Parameter Estimation for Catalytic H₂–D₂ Exchange on Pd

Abstract

Kinetic parameters have been estimated for the H₂–D₂ exchange reaction on a thin film Pd catalyst by fitting reaction data from T = 333 to 593 K over a range of inlet partial pressures, ${P}_{{H}_{2}}^{in}$ and ${P}_{{D}_{2}}^{in}$. A rigorous approach to estimating the 95% confidence regions of the kinetic parameters reveals some of the issues and complexities that are not routinely considered in the estimation of kinetic parameter uncertainty from catalytic data. Three different mechanistic models were used to assess the influence of subsurface hydrogen, H′: the traditional Langmuir–Hinshelwood (LH) mechanism, the Single Subsurface Hydrogen (1H′) mechanism, and the Dual Subsurface Hydrogen (2H′) mechanism. The fitting was performed by fixing the pre-exponential factors for all Arrhenius rate constants and equilibrium constants to their transition state theory values. The diffusion of H and D atoms from the surface into the subsurface was constrained to be endothermic (i.e. $\Delta {E}_{ss}$ > 0) and represented as an equilibrium process. Performance of the fitting routine was evaluated on a noiseless simulated dataset (created using $\Delta {E}_{ads}^{\ddagger }$ = 0, $\Delta {E}_{des}^{\ddagger }$ = 43, and $\Delta {E}_{ss}$ = 25 kJ/mol) and the same simulated dataset with the inclusion of 3% Gaussian noise. In both cases, the solver was able to return the chosen values of $\Delta {E}_{ads}^{\ddagger }$, $\Delta {E}_{des}^{\ddagger }$, and $\Delta {E}_{ss}$. Mapping of the behavior of the residual sum of squared errors, ${\chi }^{2}$, about its global minimum within 3D (${\epsilon }_{ads}$, ${\epsilon }_{des}$, ${\epsilon }_{ss}$) parameter space allowed quantification and visualization of the 95% confidence regions using 2D error ellipses for each pair of fitting parameters. For the experimental dataset on the Pd catalyst, fitting to the LH model predicted that H₂–D₂ exchange is adsorption rate limited, with $\Delta {E}_{ads}^{\ddagger }$ = 51.1 ± 0.6 kJ/mol with 95% confidence. On the other hand, fitting to both the 1H′ and 2H′ models led to predictions of $\Delta {E}_{ads}^{\ddagger }$ = 0, consistent with the current understanding that the barrier to H₂ dissociation on Pd is low. Thus, the results detailed herein provide supporting evidence for a non-LH mechanism for H₂–D₂ exchange on Pd while also illustrating the issues associated with quantification of uncertainty in kinetic parameter estimation.

Graphical Abstract