# Coming up with a Mathematical Model to Predict how long a redox reaction can last

The description of my problem is as follows:

I will like to find out how can I obtain a mathematical equation which could give me a surface plot of the current output as a function of both concentration and time from an electrochemical cell which uses the following redox equation:

O2 + 2Pb -> 2PbO

I have the following information:

1. Mass of Pb - 10g
2. A fixed concentration of oxygen gas - 20.9% over volume of air in room

1. Unlimited supply of oxygen
2. The gas is supplied at constant room temperature and pressure (25 degree Celcius and 1 atm)

Tentatively, my plan was to use Farday's Law of Electrolysis (https://en.wikipedia.org/wiki/Faraday%27s_laws_of_electrolysis) to calculate the current which flows as a result of the transfer of electrons from the redox reaction. My hypothesis is:

1. The magnitude of the current generated by the following reaction will be proportional to the rate of oxygen consumption.
2. The magnitude of the current generated will decrease over time as the rate of reaction decrease (from the decrease in concentration of Pb).

My problem is that the plot I obtained shows me that the current vs time relationship is a reciprocal one under fixed concentration of oxygen gas. This does not agree with the experimental results I obtained which shows that current output is steady and starts decreasing after a long period of time (a few months at least). Can someone help point out my the flaw in my argument? What should I do instead to obtain the plot I described?

• It's a surface reaction - Pb isn't in solution, otherwise it could similar kinetics to what you've got. – Mithoron Aug 17 '17 at 14:02

$$D_{ox} \frac{\partial C_{ox}}{\partial x}= k_cC_{ox}-k_aC_{rd} \\ D_{rd} \frac{\partial C_{rd}}{\partial x}= -k_cC_{ox}+k_aC_{rd}$$
These lead to the Cottrell equation for the limiting flux (current density) $j=-nFC^o_{ox}\sqrt{D_{ox}/\pi t}$. To get what you want you will probably need to consult some specialist text books on electrochemistry.