Question

When a cell reaction reaches equilibrium, E = 0. E = 0. both E and E = 0. neither E nor E = 0.

Answer 1

Well, I think it must be Cell potential (E) and standard cell potential (E^o). At equilibrium, only cell potential becomes zero but not standard cell potential. Standard cell potential is the difference between standard electrode potentials of two electrodes (E^o_cathode - E^o_anode) but cell potential is given by E_cathode - E_anode. Note here that E^o_cathode and E_cathode are two different things. E^o_cathode or E^o_anode is a standard reduction potential value for a particular half cell reaction. Where E_cathode is given by,

E_cathode= E^o_cathode - RT/nF{ln([red.]/[oxid.])}

Where [red.] and [oxid.] represents concentration of reduced state and oxidised state respectively.

R is Gas Constant, T is temperature, n is no. of moles of electrons transfere in the reaction and F is Faraday constant.

Same for anodic process.

for example for a reaction like

Fe³⁺ (aq.) + 3e^- Fe(s)

Standard reduction potential (E^o) is - 0.04 volts.

At equilibrium, it is only Cell potential (E) that becomes zero but not E^o that is a standard value.

Also, according to Gibbs free Energy relation with cell potential E,

G = - nEF

Where 'n' is no. of moles of electron transfere in the reaction, F is faraday constant, G is gibbs free energy of the reaction and E is cell potential.

At equilibrium, G becomes zero therefore according to the relation, cell potential becomes zero.

Note here that there is also a relation between Standard Gibbs Free Energy (G^o)  and Standard cell potential (E^o) that is,

G^o = - nFE^o

But at equilibrium, Standard Gibbs free Energy have certain value ( not become zero) because Standard Gibbs Free energy is the energy change of the species only in their Standard state but not any other state. Therefore, E^o have a certain value at equilibrium.

Hope it should be clear what you are looking for.