# Verify parameters variation in local sensitivity coefficients for microkinetic model

I’m performing a sensitivity analysis for a model of a set of elementary chemical reactions. I’m considering the variation in species concentration ($c_{k}$) with respect to the forward reaction rate constants ($k_{f}$). Let’s consider for the moment that there is only one reaction (for simplicity). So, I have to consider the following sensitivity coefficient, which must be thermodynamic consistent (equilibrium constant $k_{eq}$ held constant) in order to define system properties (rate limiting step, etc): $$\left ( \frac{\partial c_{k}}{\partial k_{f}} \right )_{k_{eq}}$$

The species concentration follows a governing equation in time “t” of the form: $$\frac{dc_{k}}{dt}=f\left ( c_{k}\left ( t,k_{f},k_{eq} \right ),k_{f},k_{eq} \right)$$

We can derive a differential equation for the sensitivity coefficients by taking partial derivatives with respect to the reaction constant $k_{f}$ (at $k_{eq}$ constant) on both sides of the above equation. This differential equation will have a jacobian term, which is no problem to calculate, and a term of the form: $$\left ( \frac{\partial f}{\partial k_{f}} \right )_{k_{eq}}$$

One can calculate the above term by forward finite difference evaluating the reaction constant at an incremental value $k_{f}^{1}=k_{f}+\Delta k_{f}$, at constant $k_{eq}$. Here stems my concern.

The reaction constant actually depends on other basic parameters, pre exponential factor ($A_{f}$) and heat of adsorption of the participating species ($Q_{k}$), $k_{f}=k_{f}\left ( A_{f},Q_{k} \right )$. Both parameters have a defined range and since the derivative is taken at $k_{eq}$ held constant, both parameters set will vary. I’d like to verify that $k_{f}^{1}$ can actually be obtained from the range values of $A_{f}$ and $Q_{k}$. One way to do this is actually calculate the values $A_{f}$ and $Q_{k}$ that yield $k_{f}^{1}$, but this is too expensive because the numbers of parameters is too high and there is no need to calculate these new parameters. Is there any strategy to verify that a given $k_{f}^{1}$ belongs to the range specified for its basic parameters $A_{f}$ and $Q_{k}$?