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I am slightly confused about the concept of a salt bridge/porous disk. I understand that these are necessary in a galvanic cell since there is a charge buildup and the salt bridge and porous disks allow ions to flow in order to neutralize this charge and allow for the electricity to continue flowing.

What I am confused about is what ions are flowing. As an example I'm going to talk about a zinc-copper galvanic cell in which the zinc is oxidized at the anode and the copper is reduced at the cathode. What ions flow through the porous disk? Is it some of the $\ce{Zn^{2+}}$ ions and the anion from the copper solution, or is it separate ions? Is this the same in a salt bridge or is the salt used in the bridge the one that moves?

For example, if a salt bridge is made with $\ce{KCl},$ does the $\ce{K+}$ go to the cathode and the $\ce{Cl-}$ go to the anode or do the $\ce{Zn^{2+}}$ ions move through the salt bridge?

In other words, what ions flow through a salt bridge? Is it the ions from the separate solutions in the two half cells or the ions inside the salt bridge? What about a porous disk that has no ions inside of it?

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It depends on the geometry/construction of the bridge. In a typical glass-tube bridge, some small amount of the ions from the cell may eventually make their way all the way across the salt bridge, but over reasonable amounts of time, it's mostly the ions from the salt bridge that move into the half-cells (and the ions from the half-cells moving into the ends of the bridge). For a small filter paper strip being used as a bridge, the time it takes should be shorter, as there is much less solution in the salt bridge and the distance is likely shorter. Sometimes, glass salt bridges are filled with gels to limit diffusion further (at the expense of increasing the resistance of the bridge).

The goal of using a salt bridge and separating the half-cells is to prevent the solutions of each from directly mixing. If significant quantities of the ions from the half-cells are crossing the bridge, then they can react directly, bypassing the electrode connections. e.g. if we have a simple galvanic cell of $\ce{Zn∣Zn^2+∥Cu^2+∣Cu}$ and $\ce{Cu^2+}$ makes it across the separator to the zinc size, it can be directly reduced onto the zinc electrode, instead of by the electrons flowing through the electrode connections. Having a single beaker with both $\ce{Cu^2+}$ and $\ce{Zn^2+}$ would thus make a poor galvanic cell as most of the reagents would be wasted reacting directly, rather than doing work in the outside circuit.

With a simple porous separator, there are no other ions than the ones in the half-cells, so naturally, these are the ions that flow across, with the goal simply to prevent bulk mixing of the two solutions.

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The motor is being powered by a simple galvanic cell that uses a porous cup rather than a salt bridge to maintain the separation of copper and zinc ions. The separation of ions in crucial to the reaction as a mix between the two would cause an immediate reaction and would not require a lengthy movement of electrons across the wire which is what powers the battery. In this lab, the copper pennies are acting as the cathode where they in a Cu(II)SO4 solution making this the reduction part of the reaction. The opposite is true for the zinc washers, acting as the anode, which are placed in the ZnSO4 solution making this the oxidation part of the reaction. The anions are flowing towards the anode (zinc washers) to balance the out the positive ions (cations) that are being released by the anode and into the wire, into the motor, and into the cathode.

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