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If you replace the silver electrode by a carbon one, the cell would still work, as $\ce{Ag+}$ can still be reduced and deposit itself on carbon( that becomes a silver electrode). OTOH, if the zinc electrode is replaced by carbon one, there is no zinc to be oxidized to $\ce{Zn^2+}$ and the cell would not work, as $\ce{Zn^2+}$ cannot be oxidized either.
As any galvanic cell needs chemical processes providing electrons at their anodes and absorbing electrons at their cathodes. These processes may or may not chemically involve the material of the electrodes. Both inert electrodes can be used for electrolytes, that can be reduced at the cathode and oxidized at anode.
There is the commercial vanadium cell used for energy storage, using 2 redox half-reactions below. The interesting fact is vanadium has there four different oxidation numbers, $\ce{V^{II}/V^{III}}$ in one of compartments and $\ce{V^{IV}/V^{V}}$ in the other:
Charging
\begin{align} \ce{V^3+ + e- &->V^2+}\\ \ce{VO^2+ + H2O &-> VO2^+ + e- + 2 H+ }\\ \end{align}
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Discharging
\begin{align} \ce{V^2+ &-> V^3+ + e-}\\ \ce{VO2^+ + e- + 2 H+ &-> VO^2+ + H2O } \end{align}
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In the Hall–Héroult process of extracting aluminium from aluminium oxide, the electrolyte is a liquid mixture of cryolite $(\ce{NaAlF6})$ and aluminium oxide $(\ce{Al2O3}).$ The cathodes and the anodes are made of graphite.
During the electrolysis process, aluminium is deposited at the cathode and oxygen is liberated at the anode. Some of this oxygen reacts with the carbon in the graphite to form carbon-dioxide, thus slowly burning away the anodes. Thus, the anodes have to replaced periodically.
But if this is the case, why do they use graphite anodes at all? Why don't they use anodes of some other, less reactive material? I'm pretty sure there are many out there that would fit the bill.
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