# Relationship between surface area of electrode and reaction rate of hydrogen in fuel cells

I am looking at the effects of increasing the surface area of which the flowing hydrogen fuel is in contact with the anode in a fuel cell.

At the anode the hydrogen is catalytically split into protons and electrons. The catalyst used is most commonly platinum. So, for a fixed flow rate of hydrogen, increasing the contact surface area with the anode will increase the number of hydrogen atoms in contact with the catalyst, and the duration that they are in contact with it. Therefore the number of oxidization half cell reactions will increase. But what is the mathematical relationship that governs the number of reactions that will occur? Is it just a linear relationship, i.e doubling the surface area doubles the number of reactions, or is it more complex?

If it is impossible to get an exact mathematical relationship, what is a good approximation on how these variables will scale with each other? Also, how do chemist usually predict the number of reactions/reaction rate, and could this be applied to my example?

Any galvanic cell produces its equilibrium voltage at open-circuit. Let's say your cell is at +3 V before you allow any current to flow through it. As soon as you throw the switch, you allow that 3 V to push a current through the cell and some external load that you connect to it. There are several resistances within the cell that act to reduce the cell voltage from the initial 3 V. The electrolyte resistance, concentration dependent resistances at the electrodes and electron transfer resistances all contribute to the total voltage drop in the cell, and hence, effectively limit the amount of current it can generate. Typically, one of these resistances is much larger than any of the others depending on the exact configuration and operating conditions of the cell. It is the electron transfer resistance that you are concerned with in your question. The traditional way of expressing the voltage drop due to electron transfer resistance is the Tafel equation:

$$\eta _{a}=a+b\, log\, i$$

where $a$ and $b$ are empirical constants, $i$ is the current density and $\eta _{a}$ is what is called the activation overpotential. We see from this expression that the amount of voltage drop in the cell due to electron transfer increases linearly with the logarithm of the current density. It is desirable then to minimize $\eta _{a}$ for a battery or flow cell application so that we have the maximum amount of voltage available for pushing current through the load. A very good way to do this, as you pointed out, is to simply increase the electrode area. By doing so, we decrease the current density which decreases $\eta _{a}$.

So, the exact mathematical relationship you seek would be the empirical Tafel expression given above, or even better, the Butler-Volmer equation which is really just a more fundamental way of expressing the Tafel relationship. All you should need to do is locate some references on platinum polarization in aqueous solutions. There just happen to be tons of papers on this subject. Here is one to get you started:

Sheng W. et al, Journal of The Electrochemical Society, 157 11 B1529-B1536 2010

• This is a fantastic answer, Its the exact information I need! Thank you very much! I'm pretty certain I understand, but I might have some follow up questions in the morning once I fully wake up :) – Blue7 Jan 25 '15 at 2:49
• I have one question, I should probably know this, but it confuses me slightly: If the activation overpotential is proportional to the log of the current density, that means for a cell delivering a constant current the activation overpotential is proportional to log(1/A). Since power is current * voltage, that means for a cell delivering constant current the power the cell is capable of producing is also proportional to log(1/A). Is this correct? – Blue7 Jan 25 '15 at 19:09
• I guess what I'm asking is: Is the power of the cell proportional to the activation overpotential? – Blue7 Jan 25 '15 at 19:10
• The power output depends on both the internal resistance of the cell and the resistance of the load it is connected to. Play around with this widget to get a feel for how it works: hyperphysics.phy-astr.gsu.edu/hbase/electric/powtran.html . If you know the activation overpotential as a function of current density, you could calculate an estimate of the cell resistance over a range of current. Plug it into the widget and see what you come up with! – Qubit1028 Jan 25 '15 at 19:30
• Thankyou. One last question. This document, page 5, says the theoretical max voltage of a cell is $E_0 = 1.229V$. Does this mean the actual volatge of the cell is $E = E_0 - \eta_a - \eta_e - \eta_c$ where $\eta_e$ and $\eta_c$ are the electrolyte resistance and concentration dependent resistance? – Blue7 Jan 25 '15 at 21:02