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Sodium hydroxide dissolved in water is a very strong base. I think this could mean it could de-oxidate the metal oxides during re-ox reaction. Basically, I ask whether the following happens: $$\ce{NaOH + Fe2O3 -> Fe + something}$$

I have read that it does reduce carbon dioxide producing sodium carbonate and water:

$$\ce{2 NaOH + CO2 -> Na2CO3 + H2O}$$

Would that work with metals? If yes, which ones?

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The short answer is no. Sodium Hydroxide is a soluble base (alkali) and Iron (III) oxide is a base too. An acid will react with metal oxides forming a salt and water. For example, if a sample of Iron(III) oxide was reacted with HCl, Iron chloride and water would form. It's possible to extract the pure Iron by electrolysis. Although, I am not sure that Iron Chloride is soluble so another acid may be more suitable or electrolysis could be performed using molten iron chloride.

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Your second reaction isn't a redox reaction. When carbon dioxide is dissolved in water, two equilibriums are forms:

\begin{eqnarray} \ce{H_2CO_3 + H_2O &<=> HCO_3^- + H_3O^+} \\ \ce{HCO_3^- + H_2O &<=> CO_3^{2-} + H_3O^+} \end{eqnarray}

Remember that water also is in equilibrium:

$$\ce{2H_2O <=> OH^- + H_3O^+}$$

Adding $\ce{NaOH}$ will — according to Le Chatelier's principle, the equilibriums are disturbed, and more carbonate ($\ce{CO_3^{2-}}$) is formed.

If you're interested in a nice redox reaction involving iron(III) oxide, aluminium is much better choice.

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