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In a galvanic cell, electrons flow from the anode to the cathode. In the cathode solution, the electrons bond with the cations. However, this leaves cations in the anode that will gradually increase the charge of the anode solution. A "charge buildup" occurs when the anode solution is so positive that the electrons will no longer flow to the cathode, preventing electron transfer. The purpose of the salt bridge is for the cations of the anode solution to bond with the anions of the salt in order to neutralize the solution. By neutralizing the solution, electron flow will continue.
The electrons flow from the negatively charged anode to the positively charged cathode. So, overtime it makes sense that the electrodes would stop flowing because there would no longer be a difference in charge attracting electrons to either of the sides. Thats why the salt bridge is there, so that the charge difference remains and the reaction can continue to proceed.
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