I'm currently studying the temperature dependence of the following cell: $\ce{Cu_{(s)} + 2Ag^{+}_{(aq)} <=> 2Ag_{(s)} + Cu^{2+}_{(aq)}}$ ($E^⦵ = \pu{+0.46 V}$)

Experimentally, I recorded the initial temperature of the cell at $\pu{273 K}$, before increasing the temperature of the cell and recording its potential each time. This nicely followed the equation: $$E^⦵ = T\frac{\Delta S^⦵}{nF} - \frac{\Delta H^⦵}{nF}$$ Which apparently can be used to describe cell temperature dependence. It's worth noting that the activities of all species was 1.

The issue is that in retrospect, the entire experiment seemed kind of pointless, as it doesn't appear to actually tell me anything at all. Consider the following logic:

In a 'normal' equilibrium reaction, e.g. the Haber process, the reaction will proceed until it gradually reaches equilibrium, giving an equilibrium constant; let's call it $K_{1}$. Changing system temperature will shift the equilibrium until a new equilibrium is formed, with a new equilibrium constant $K_{2}$... And so on for each temperature change.

In a cell, this is not what happens. If we consider that $E^⦵$ is a 'guide' to equilibrium position, and assumed that the above was true, then changing temperature would've caused a new value of $E^⦵$ to equilibrate each time. However, we know it doesn't do this, as by definition equilibrium in a cell is when $E^⦵ = \pu{0 V}$. Therefore, this seems to me as though a cell will naturally reach equilibrium with the same equilibrium constant at any temperature.

Thus, apart from giving me rather accurate, experimentally derived values for $\Delta H^⦵$ and $\Delta S^⦵$ (assuming that these parameters aren't temperature dependent, I know), what does the experiment actually tell me? I mean, the cell is predisposed to run to equilibrium and thus have an $E^⦵$ value of $\pu{0 V}$ regardless of temperature, so where does the temperature dependence come into it?

Furthermore, this makes it confusing to study how the reaction quotient (and equilibrium constant) varied with temperature:

  • It's logical to extrapolate the graph and find the temperature at which equilibrium is apparently achieved; but as discussed, it doesn't work like that.
  • It's also logical to calculate both parameters at each temperature (which appears to fit nicely as my calculations show that $\log K$ decreases with temperature, vice versa for $\log Q$) and finding the point where $\log K = \log Q$ would surely give me the temperature at equilibrium; but again, as discussed it doesn't work like that, mainly because 'equilibrium' results in $E^⦵ > 0$

So is there any simple way to determine how the reaction quotient changed with temperature?


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