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IUPAC's Gold Book defines the emf of a galvanic cell in the following way:

The limiting value of E(cell) for zero current flowing through the cell, all local charge transfer and chemical equilibria being established, was formerly called emf (electromotive force). The name electromotive force and the symbol emf are no longer recommended, since a potential difference is not a force.

Check: https://goldbook.iupac.org/E01934.html

I understand that zero current is required to find the emf since passing a current through the cell decreases its voltage. Moreover, we want to find the emf of the cell at some specific composition of the electrolytes, also ensuring a reversible reaction is taking place.

But what does it mean by "all local charge transfer and chemical equilibria being established"? Isn't the emf of the cell ZERO when the cell reaction attains equilibrium? What "equilibrium" does it refer to? And what does "local charge transfer" mean?

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We think of electrochemical processes happening immediately in one step, but that's not really true. In a galvanic cell, you have to have the dissolved ions move from the bulk to the surface, adsorb to the surface, and possibly undergo an activation step before they can reduce or oxidize. The last step is for the electron to actually move from the reduced to the oxidized species. This specific step is the "local charge transfer" reaction.

You are right that when the cell reaches equilibrium, the $E_{cell}$ would be zero. However, here we're interested in the equilibrium of each local process separately: eg. diffusion of ions from the bulk electrolyte to the surface will reach an equilibrium.

When you first submerge the electrode into the electrolyte, all of these steps are able to reach an equilibrium, and things like concentration gradients between the electrolyte bulk and electrolyte surface can effect the potential experienced by the electrode. In fact, some charge transfer even occurs initially, but because the half-cells are not connected, electrostatic forces build up and the concentrations are such that the chemical potential is quickly opposed and a steady-state is achieved.

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Electrochemical equilibria are just like chemical equilibria. The forward reactions and the reverse reactions happen at their own rates and when the two are equal, the system is in equilibrium. This is true regardless of the nature of the cell potential (electrolytic or galvanic).

If, you short circuit a galvanic cell externally, then, it will pass a current and go into a non-equilibrium condition until the current stops and the potential goes to zero. This however, does not change the fact that before the external short, there was an equilibrium inside the cell, whose potential could be measured.

The "local charge transfer" probably refers to any side reactions inside each half cell that may exist due to the initial reagents being prepared in a non-equilibrium concentration ratio.

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