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Lets say we want the activity coefficient of $\ce{Cu^{2+}}$ ions in solution. Would we simply setup one half cell of a $\ce{Cu}$ electrode in a known concentration of $\ce{CuSO4}$ and then connect this to a Standard Hydrogen Electrode.
Using the standard electrode potential of the reaction $\ce{Cu^{2+} + 2e- -> Cu}$ and the Nernst equation could we determine the activity coefficient of $\ce{Cu^{2+}}$ ions ?
Is this all I would need to do ?
Sort of 'yes', sort of 'no'. Consider the following points:
Activity coefficients are notorious for changing with $\mathrm{p}\ce{H}$, ionic strength, and concentration (among other things). You might measure the the activity coefficient of $\ce{Cu^{2+}}$ in a $0.7~\mathrm{M}$ $\ce{CuSO4}$ solution and get one value, but try to measure the same thing in a low $\mathrm{p}\ce{H}$ and $1~\mathrm{\mu M}$ $\ce{CuCl2}$ solution and you may get a completely different value. In practice, people usually end up measuring activity coefficients across a range of conditions that are relevant to their specific use-case.
The Standard Hydrogen Electrode is a theoretical device; and actually constructing something similar is difficult. In practice, you would just want to use a commercially available double-junction silver-silver chloride or saturated calomel electrode. The potentials of those two are well-known and fairly stable.
See Chapter 2 of "Electrochemical Methods" 2nd Ed, by Bard and Faulkner for more information.
