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Does flipping a reaction as written change the sign of the value of E or is it a completely different value?

For example if I have the value for the half reaction $\ce{Cl2 + 2e- -> 2Cl-}$ but need the value for the opposite reaction ($\ce{2Cl- -> Cl2 + 2e-}$), do I just put a negative sign and treat it like $\Delta H$ values?

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Yes, you are correct - just flip the sign. As quoted from the venerable UC Davis ChemWiki:

To determine oxidation electrodes, the reduction equation can simply be flipped and its potential changed from positive to negative (and vice versa).

They give the value for your (forward) reaction as +1.358 V, so for the reaction in question, you'll use -1.358 V.

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