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I think the electrons can only flow from the anode to the cathode since oxidation always takes place in the anode and reduction in the cathode. The salt bridge exists to counteract the charge imbalance created by the one-way flow of the electrons. If the electrons could flow both ways, there would be no charge difference since the electron would be attracted to whichever side was positive and repelled by the negative. The salt bridge provides the charged ions to both solutions to maintain the positive and negative charge balance on both sides. This keeps both solutions neutral by preventing the repulsion of the electrons by the cathode which is building up negative charges due to the electrons it receives.
When batteries are being "charged" you are forcing the flow of the electrons to go the non spontaneous way, towards the anode, by adding a force and leading to a higher Delta G as time proceeds. To "use" this battery, the electrons are allowed to flow the spontaneous way, towards the cathode, without added force and leading to a lower Delta G as time proceeds.
in a galvanic cell, the electrons flow in the spontaneous direction, and this flow of electrons is how the battery works, and is generally the only direction of flow, (anode to cathode), the salt bridge allows for ionization and keeps the solutions neutral.
The salt bridge is just used to balance the charge as electrons flow to one side. For electrolytic cells, the reaction occurs in the nonspontaneous direction. For galvanic cells, the reaction occurs in the spontaneous direction.
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