When methanol-water (2M) electrolysis were carried out for 10 hours at constant current; how can we calculate how much methanol and water consumed theoretically?
Theoretically, you can use the Farady's Law in order to determine the amount of reactant consumed. However, it assumes that your current efficiency is 100%.
If you know, approximately, how much of your charge is used in each reaction you can approximate the results.
The current efficiency is the yield based on the electrical charge passed during electrolysis. From Faraday's law of electrolysis the current efficiency equal the charge consumed in forming product (wnF/M) to the total charge (q). So, by this relations you can find the amount of consumed reactants.Note that the total charge is a function and depend on the total electrolysis time and the current.
As mentioned above, application of Faraday's law requires the value of current efficiency. Perhaps, the easiest way would be to measure the concentration of methanol in the quote of solution after electrolysis (say, IR spectroscopy?).
Of course via Faraday law, but the result will be not accurate in general due to the assumption that the current efficiency for your electrolysis is 100%.
Another problem in case of methanol, is that you do not know exactly the final product you get. for example if you doing oxidation, you will assume that methanol is oxidized to the final Product CO2 and then you will use 6e in your calculation. But this also will not be the case as the mechanism of methanol oxidation is not so simple and not all methanol will undergo complete oxidation
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