# Kinetic model of proton reduction

One of my colleagues is working on a homogeneous catalyst ($$\ce{X}$$) for proton reduction. The simplified proposed mechanism for it is the following.

$$\ce{X + 2 e^- ->[k_1] X^{2-}} \tag{1}$$

$$\ce{X^{2-} + H+ <=>[k_2,\,slow][k_{-2}] XH-} \tag{2}$$

$$\ce{XH- + H+ <=>[k_3][k_{-3}] X + H2} \tag{3}$$

However, this is not a true catalyst and it degrades over time:

$$\ce{X ->[k_\text{inact}] X_\text{inact}}$$

We monitor $$\ce{H2}$$ at different time intervals and we plot the turnover number (TON), defined as the total number of moles of $$\ce{H2}$$ produced per mole of initial catalyst (so if we produce 20 mmol of $$\ce{H2}$$ in an experiment where we started with 2 mmol of catalyst, the TON at that time is 10).

Is there a way to derive an equation for the TON vs time plot that only depends on total $$\ce{H2}$$ and time? (We get something that looks hyperbolic, like Michaelis Menten plots, but we don't know if it's hyperbolic, exponential, or something else).

My attempt:

Assuming steady-state of the intermediate, $$\ce{XH-}$$ (and I'm not sure we should!):

$$k_2[\ce{X^{2-}}][\ce{H+}] + k_{-3}[\ce{X}][\ce{H2}]= k_{-2}[\ce{XH-}][\ce{H+}] + k_{3}[\ce{XH-}][\ce{H+}]$$

$$\displaystyle [\ce{XH-}] = \frac{k_2[\ce{X^{2-}}][\ce{H+}] + k_{-3}[\ce{X}][\ce{H2}]}{(k_{-2}+k_3)[\ce{H+}]}$$

The rate law is given by:

$$\displaystyle \frac{d[\ce{H2}]}{dt} = k_3[\ce{XH-}][\ce{H+}] - k_{-3}[\ce{X}][\ce{H2}]$$

So:

$$\displaystyle \frac{d[\ce{H2}]}{dt} = k_3\left(\frac{k_2[\ce{X^{2-}}][\ce{H+}] + k_{-3}[\ce{X}][\ce{H2}]}{(k_{-2}+k_3)}\right) - k_{-3}[\ce{X}][\ce{H2}]$$

From here I am not sure how to proceed. I think it is safe to assume that $$[\ce{X}]$$ is constant in the catalytic cycle so we can say that $$[\ce{X}] = [\ce{X}]_0 e^{-k_\text{inact}t}$$. However, I'm not sure how to treat $$[\ce{X^{2-}}]$$, and I'm not exactly sure how to solve the integral for $$[\ce{H2}]$$ from there. Any suggestions are appreciated.

• Your equation ($ii$) is neither mass nor charge balanced. Would you please correct that first. – Mathew Mahindaratne Mar 6 '19 at 22:37
• Oops, I had it all in sequence and later decided to show it in steps for clarity and missed that. It is edited now. – ralk912 Mar 6 '19 at 22:39
• @ralk I would recommend maybe change your "C" for catalyst to something else. Some of the steps could be mistaken for unusual carbon compounds. – Tyberius Mar 6 '19 at 22:44
• Well maybe it is carbon :P – ralk912 Mar 6 '19 at 22:49
• Do you have any mechanistic prediction for how the catalyst inactivates? Specifically, which species is involved? For example, it might be more accurate to write your inactivation step as $k_{inact}[X^{2-}]$. That difference could be important in getting a good model for TON. More generally, I would take the approach of writing the rate laws as diff. eqs and fitting by simulation rather than making SS assumptions (that might not be valid) and trying to solve explicitly. Many computer programs are available to do this. – Andrew Mar 7 '19 at 20:19