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As my understanding, there should be an oxidation-reduction reaction:

$$ \ce{2 Fe^3+ + Cu → 2Fe^2+ + Cu^2+} $$

However, I always see the process using $\ce{FeCl3}$ to etch copper, but I never see people using $\ce{Fe2(SO4)3}$ to do this. Is it because such reaction does not occur, or is it because $\ce{FeCl3}$ is acidic in aqueous solution?

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In $\ce{FeCl3}$ etching of copper, the chloride ion is extremely important.

Chloride ions coordinate to $\ce{Fe^3+}$, $\ce{Cu+}$ and $\ce{Cu^2+}$.

See Copper Etching in Ferric Chloride Ind. Eng. Chem., 1959, 51 (3), pp 288–290 for the relative concentrations of the particular chloride complexes.

In contrast, if $\ce{Fe2(SO4)3}$ were used, there would only be aqueous ions.

At the copper surface, there is only a one-electron oxidation of $\ce{Cu}$ to $\ce{Cu+}$. As the above article explains, without coordination of chloride, the $\ce{Cu+}$ would be essentially insoluble.

The rate limiting step of the etching reaction is diffusion of ions from the surface of the copper. Coordination of chloride provides solubility and enables diffusion. This is the main reason that $\ce{FeCl3}$ is needed.

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