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There is a charge difference between the 2 cells. Because there is a high concentration of electrons in the anode, it wants to reach equilibrium and move towards the low concentration of electrons in the cathode. It does this through the wire connecting the 2 cells, as the metals allows for the flow of electrons.
The electrons move in order to reach their lowest point of free energy, meaning that if there is a release of free energy with the redox reaction, it will be spontaneous. This is why dG and E can be connected through Faraday's constant and the number of moles of electrons. The electrons move through metal, as metals have the low electronegativity that allows for electrons to flow through without becoming associated with any one atom.
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