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I have a question regarding the definition of the number of moles $n$ when using the equation- $$\Delta G=-nF\Delta E$$

I understand that $n$ is the number of moles transferred in the overall redox reaction, however, it seems to depend on how you write the equation.

If I write an equation as- $$\ce{Ag+ + e- -> Ag}$$ $$\ce{\frac{1}{2} Cl2 + e- -> Cl-}$$ Then, we have $n$=1.

But if I wanted one mole of chlorine and I write: $$\ce{Cl2 + 2e- -> 2Cl-}$$ then I get $n$=2. Which one is preferred and why?

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It does not depend on how you write the reaction. You were right in saying that, $n$ is the mole number transferred in the reaction. So, it is not the number of electrons you write arbitrarily in the equation that represents the reaction, it is rather proportional to the number of other substances, which, even being all arbitrary always satisfy that proportional relation.

Think of $n$ as the factor needed to relate one intensive property with one extensive property.

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As @user1420303 said, $\Delta G$ is an extensive quantity, and $E$ is an intensive quantity.

Firstly you should write $\Delta G$ as $\Delta G_m$ witch means that $\Delta G$ relating to $\pu{1 mol}$ extent of reaction. $$\ce{Cl2 + 2e- <=> 2Cl-}$$ $\Delta G_{m1} = -n_1FE_1, n_1=2$ $$\ce{2Cl2 + 4e- <=> 4Cl-}$$ $\Delta G_{m2}= 2 \Delta G_{m1}= -n_2FE_2$, where $n_2=4=2n_1$.

Hence, $E_1=E_2$

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