What would happen to the cell potential if two half cells of a galvanic cell were at different temperatures?

I'm doing an experiment where I'm investigating the effect of temperature of the galvanic cell on the cell potential. But during the experiment, I noticed that the two half cells were at different temperatures. Could this be a limitation to the data that I collected? If so, how would it impact the cell potential that was measured?




Good experimental design means ensuring that you have repeatable and well defined experimental conditions - by not adequately controlling the temperature of the two half-cells you've introduced the variable $\Delta T\_\text{half-cell}$. Since you aren't varying $\Delta T\_\text{half-cell}$ in a systematic way and don't know if it's constant, you now have an undefined aberration convoluted in with whatever effect you wanted to measure. Either ensure that $\Delta T\_\text{half-cell}$ is small enough to ignore, change your experimental setup to eliminate the temperature difference and repeat your measurements, or preform a series of measurements where you alter $\Delta T\_\text{half-cell}$ in a systematic way. (This latter option is a lot of work since you'd have to vary $\Delta T\_\text{half-cell}$ at several different cell temperatures.)


With that out of the way, what effect would $\Delta T\_\text{half-cell}$ have on the overall cell potential? Lets start with the Nernst equation for the half-cell reaction $\ce{pP +ne^- -> qQ}$, which is:


$$
E\ce{(P|Q)} = E°\ce{(P|Q)} - \frac{RT\ce{(P|Q)}}{nF}\ln{\frac{([\ce{Q}]/c°)^q}{([\ce{P}]/c°)^p}}
$$


(from Chemical Structure an Reactivity 1e by Keeler and Worthers, p 842f). $E°$ is the cell potential at standard conditions, $R$ is the molar gas constant, $T\ce{(P|Q)}$ is the temperature of the $\ce{(P|Q)}$, $n$ is the number of electrons transferred, $F$ is the Faraday constant, and $[\ce{Q}]/c°$ etc. is the concentration of $\ce{Q}$ relative to the standard concentration.


For the other half-cell reaction $\ce{aA +ne^- -> bB}$ we can write:


$$
E\ce{(A|B)} = E°\ce{(A|B)} - \frac{RT\ce{(A|B)}}{nF}\ln{\frac{([\ce{B}]/c°)^b}{([\ce{A}]/c°)^a}}
$$


Remembering that the overall cell potential is $E = E\ce{(P|Q)} - E\ce{(A|B)}$, we arrive at the following:


$$
E = E° - \frac{R}{nF}\left[ T\ce{(P|Q)}\ln{\frac{([\ce{Q}]/c°)^q}{([\ce{P}]/c°)^p}} - T\ce{(A|B)}\ln{\frac{([\ce{B}]/c°)^b}{([\ce{A}]/c°)^a}}\right]
$$


So, is $\Delta T\_\text{half-cell}$ an important factor? It depends on the magnitude of $\Delta T\_\text{half-cell}$ relative to the temperature range studied, the precision with which you measure $E$, and whether it's a constant difference or changes with time and/or cell temperature. You'll have to figure this out on your own, but the equation above should give you some numbers to work with.


 1