Why does Ecell change with concentration but not with coefficients

I understand that the voltage difference is an intensive property, which does not depend on the number of moles. So the reduction potentials don't change when you multiply coefficients by a common factor. For example,


$$\begin{align}
\ce{1/2 O2 + 2H+ + 2e- &-> H2O} & E\_\mathrm{red}^\circ &= \pu{1.23 V} \\
\ce{O2 + 4H+ + 4e- &-> 2 H2O} & E\_\mathrm{red}^\circ &= \pu{1.23 V}
\end{align}$$


However, since the number of moles does not matter, why does the concentration change the value of $E^\circ$ vs $E$ (under non-standard conditions).


$$ E\_\mathrm{cell} = E\_\mathrm{cell}^\circ - \frac{RT}{zF}\ln Q$$




This part of my answer is a bit silly; the electrodes in an electrochemical cell "know" when you change the concentration, but they don't "know" when you change coefficients of your chemical equation on your sheet of paper.


More seriously, intensive properties do change when you change concentration (an intensive property). However, they don't change if you increase the size of the cell (AAA, AA, C, and D batteries all have the same voltage, but AAA drains faster than D).


As a general principle, measurements don't change just because you change your mental model or the chemical equation used to describe a situation.


To make things more complicated, the molar enthalpy is also an intensive property, but the value does change when you multiply all coefficients by the same factor. If you relate this to measurements, however, nothing changes. This is because the changed coefficients appear in the calculation of, say, the amount of heat a certain amount of reactants give off. So even thought the molar enthalpy is different by a certain factor, the calculated heat (which you can measure) is not because the arbitrary factor cancels out when calculating the heat given off.


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