How Does Faraday's Law of Electrolysis Apply to Reversible Reactions?

[OmniReader]

According to Wikipedia Faraday's electrolysis law is

n_i = \frac{Q_{passed}}{Fz_i}

Where n_i is moles of species i, which has charge z_i, liberated at electrode. Q_passed is the charge that has passed, which is function of time t.

How is this law derived and are there any approximations taken - is it exact if Q_passed is an exact function of t? e.g. surely it only works if reaction is considered irreversible. So, how do we calculate n liberated precisely - e.g. along Butler-Volmer lines?

 

[epenguin]

According to Wikipedia Faraday's electrolysis law is

n_i = \frac{Q_{passed}}{Fz_i}

Where n_i is moles of species i, which has charge z_i, liberated at electrode. Q_passed is the charge that has passed, which is function of time t.

How is this law derived and are there any approximations taken - is it exact if Q_passed is an exact function of t? e.g. surely it only works if reaction is considered irreversible. So, how do we calculate n liberated precisely - e.g. along Butler-Volmer lines?

You can very easily derive that if you assume conversion of the metal to ion at one electrode and the opposite process at the other electrode is the only mechanism of charge transport - it is almost obvious and inescapable. It was not so obvious in Faraday's time because ions (and electrons) were unknown or just emerging as an idea, it was initially an experimental law I believe.

It doesn't matter in the above what function ΔQ is of t. You pretty often do have constant currents, i.e. ΔQ is proportional to t, but exactness of breakdown of this - as in battery exhaustion - is a secondary, not fundamental, question.

Whether it is reversible, e.g. if the process is happening in the opposite direction makes no difference as the net current flow and the net metal flow would both be the difference of the totals in each direction. But now you mention it I have never thought nor heard that I can remember of it being investigated experimentally nor theoretically, maybe I am having a blind spot. I'd say offhand that in typical electrolysis situations this must happen and be detectable.

 

[OmniReader]

Whether it is reversible, e.g. if the process is happening in the opposite direction makes no difference as the net current flow and the net metal flow would both be the difference of the totals in each direction. But now you mention it I have never thought nor heard that I can remember of it being investigated experimentally nor theoretically, maybe I am having a blind spot. I'd say offhand that in typical electrolysis situations this must happen and be detectable.

ok, so all we need to do is integrate the non-approximated Butler-Volmer equation for current over our time period, or other more exact current equation, and then we can get exact n_i produced at the electrodes, for each species as function of z_i?

along with Butler-Volmer we will probably need some ODEs for concentration at electrode surface as function of time for each species. this will calculate current as function of time using butler-volmer. then integrate from t_0 to t and we get the precise n_i liberated ... main question, does this retain all the precision of Butler-Volmer, is the Faraday law result for n_i as exact as the current function we integrate for Q?

you're right it is easy to derive but I wonder if I am missing something in the derivation. if not then it's often clear to others instead.

 

[epenguin]

The Faraday law is IMO obvious from the assumptions I listed.

I think it boils down to conservation of mass and charge and atomic structure of matter doesn't it?

I should recall that it was from Faraday's laws that the electron was first inferred.

So it is over and above the Butler-Volmer equation which I had never heard of or remembered till your post. Any detailed mechanistic equations such as those just have to conform to Faraday's overall.

 

[OmniReader]

a few of my doubts then - granted, integral of current dt will take care of all complications like reaction rates in different directions. so then please clarify three things

1) Faraday's law is exact, if Q_passed is known exactly

2) let's say equilibrium established at first electrode contains j species total and at second electrode contains k species total. all j species at first electrode and k species at the second electrode will have some value of n_i for that species. if n_i is positive then over whole electrolysis there has been a net increase in number of moles for that substance; if n_i is negative then there has been net decrease of number of moles. n_i in other words is change in n for that species, i.e. n(after electrolysis) - n(before electrolysis).

3) z_i is number of electrons transferred in reaction equation for 1 mol of species i, not charge i.e. stoichiometric coefficient on electrons in reaction equation, divided by coefficient on species i in reaction equation. I guess z_i is negative if species is being reduced in reaction equation, and positive if it is being oxidized?

please correct anything wrong here

 

[OmniReader]

any help? maybe my rephrasing will help.
first question is: is Faraday's law exact, if Q_passed is exactly known?
second question: I take it the value of n_i represents actually change in number of moles of this species 'i' in the system. should be positive if species increases in number of moles, negative if species decreases. for every species in the equilibrium at a particular electrode, n_i for that species at that electrode is number of moles change caused at that electrode by passing Q_passed charge. is this ok?
third question: z_i is no. of electrons transferred per mole of species 'i', so coefficient on electron / coefficient on 'i' in reaction equation, but is negative for species being oxidized and positive for species being reduced in equilibrium equation. so if e- is written on the left-hand side of redox equilibrium, then z_i is positive for reactants in the equilibrium and negative for products in the equilibrium as it is written then. have I understood correctly?

 

[epenguin]

Well frankly I feel nothing is served by laying down definitions that sound even more formalistic and hifalutin than the textbooks and getting hung up on them. Some of your paragraphs just made my head spin. Just use whatever definitions and conventions there are in your textbook. I note that neither your equation nor the laws as formulated in Wiki mention the direction of flow. It is quite clear that oxidation is removal of electrons.

1) Faraday's law is exact, if Q_passed is known exactly

As far as I know it is exact. All I can see it depending on (apart from the solid assumptions I have mentioned) is ion transport being the only conduction mechanism. As pure water without electrolytes has very low conductivity that seems a safe assumption. However it is equally true whether the Q passed is known exactly or inexactly, or not known at all.

However this question gives me the chance to point out something not often mentioned that I ever saw. The laws according to Wiki are: Faraday's 1st Law of Electrolysis - The mass of a substance altered at an electrode during electrolysis is directly proportional to the quantity of electricity transferred at that electrode. Quantity of electricity refers to the quantity of electrical charge, which measured in coulombs.

Faraday's 2nd Law of Electrolysis - For a given quantity of D.C electricity (electric charge), the mass of an elemental material altered at an electrode is directly proportional to the element's equivalent weight.
In a manner of speaking you could say that Faradays laws have nothing to do with electricity, and that the first law is superfluous.

To explain - the quantity of charge transferred is measured in Coulombs. For a long time the coulomb was defined electrochemically. If you look in old textbooks you will find the coulomb defined as the amount of charge that causes some ridiculous number of grams of silver to be deposited in an electrochemical cell.* If that that defines the quantity of electricity then the first law is contained in the second.

And together they could be restated something like “chemical equivalents in electrochemistry are the same as in ordinary chemistry”.

The historical fact (I do not know the history very well) is that the same Faraday who pioneered electrochemistry also pioneered electromagnetism. So no doubt he measured the currents in his electrochemical experiments with an electromagnetic device, and formulated the laws that way. He was deep into the unity of all these things. But someone else might have done just electrochemistry alone quite well and could have formulated the law chemically without ever seeing an ammeter, and we would probably be to this day teaching the subject slightly differently.

The silver deposition lends itself to fairly accurate measurement, that is why it became the standard. But at some point it proved possible to measure better, more precisely I suppose, by electromagnetic force of a current, and they made that the standard for the ampere and the coulomb that multiplied by a time. That is what it is now.

But don’t invest too much into it because next year they are going to change it into something more chemical again! http://en.wikipedia.org/wiki/New_SI_definitions

*then the ampere that per second or maybe at one time per hour

 

[OmniReader]

As far as I know it is exact. All I can see it depending on (apart from the solid assumptions I have mentioned) is ion transport being the only conduction mechanism. As pure water without electrolytes has very low conductivity that seems a safe assumption. However it is equally true whether the Q passed is known exactly or inexactly, or not known at all.

In other words, that our redox equilibrium equation covers all species capable of transferring current. Otherwise Q_passed would be inexact.

Well frankly I feel nothing is served by laying down definitions that sound even more formalistic and hifalutin than the textbooks and getting hung up on them. Some of your paragraphs just made my head spin. Just use whatever definitions and conventions there are in your textbook. I note that neither your equation nor the laws as formulated in Wiki mention the direction of flow. It is quite clear that oxidation is removal of electrons.

No conventions in my textbook, I'm learning this online ...
It's important to know that n_i is positive if change in number of moles of species 'i' is positive, and negative if the change in number of moles of species 'i' is negative. (Or if this is not true) if this is the case, then it can be deduced that (as Q_passed being positive favours the oxidizing direction, so species being oxidized will have negative n_i and those being reduced will have positive n_i), looking at the mathematical Faraday law this requires that species being reduced have positive z_i, whereas species being oxidized will have negative z_i.

I think the directions are important, otherwise you could get a calculation completely wrong! (e.g. opposite value of change in number of moles)

 

[epenguin]

In other words, that our redox equilibrium equation covers all species capable of transferring current. Otherwise Q_passed would be inexact.

I had meant to say and forgot. Your question was about Faraday's laws and i was perplexed you several times referred to some equilibrium. Faraday's Laws are not about equilibrium, they are about a transfer of mass and of charge. Then OK that's still comes in when you consider virtual or small transfers in the theory of redox potentials.

I cannot imagine your source has no definitions or conventions. At worst they will be implicit and easily discovered from context.

I advise just get on with real cases and all should become clear.

Sure, if there is more than one electrochemical process going on at the same time, which is fairly unusual, the current or charge transfer will be the algebraic sum of of the several. Just study cases of one thing happening at each electrode at first and it will be easier, the Ʃn_i stuff seems to me an unnecessary decoration.

Perhaps someone has to hand some Feynman observations about terminology and definitions, which might be summarised in the saying "Be a country boy".

 

[OmniReader]

As I said, I'm learning this online. I have no textbook nor practice problems (all those I can find online do not beg the question I want to answer). My "source" is Wikipedia which as you said has no definitions or conventions for n_i or z_i.

For real cases, yes, I can apply Faraday's laws. Plugging in numbers is obviously not difficult. but never have I seen a practice problem where n_i is negative. and for all problems I've seen, if the equilibrium established is A + e- -> B, it's always n_B they seem want to want, never n_A. So the application of Faraday's law for my more general purposes (how to find n_A?) is unclear and not covered by any practice problems.

 

[OmniReader]

As I said, I'm learning this online. I have no textbook nor practice problems (all those I can find online do not beg the question I want to answer). My "source" is Wikipedia which as you said has no definitions or conventions for n_i or z_i.

For real cases, yes, I can apply Faraday's laws. Plugging in numbers is obviously not difficult. but never have I seen a practice problem where n_i is negative. and for all problems I've seen, if the equilibrium established is A + e- -> B, it's always n_B they seem want to want, never n_A. So the application of Faraday's law for my more general purposes (how to find n_A?) is unclear and not covered by any practice problems.

 

[OmniReader]

Hello? do you know the answer? Sorry if I'm unnecessarily hurrying you, it's been a day or two.

I am wondering, we are always given cases like A + e- -> B and asked to find n_B being told that some current, always positive (e.g. 10 A) is passed for some period of time, possibly I(t). and n_B is always positive too. this is also with z_B being put in as positive. but if the current is positive this should support the reverse reaction (oxidation) over the forward (reduction) so surely, at the given electrode, if B is a product, n_B should be negative (being the change in number of moles of a product, when the reverse reaction > forward reaction). to get right n_B value I think z_B should be negative and z_A positive, then n_B will come out negative which it should if current is positive.

 

[epenguin]

Hello? do you know the answer? Sorry if I'm unnecessarily hurrying you, it's been a day or two.

I am wondering, we are always given cases like A + e- -> B and asked to find n_B being told that some current, always positive (e.g. 10 A) is passed for some period of time, possibly I(t). and n_B is always positive too. this is also with z_B being put in as positive. but if the current is positive this should support the reverse reaction (oxidation) over the forward (reduction) so surely, at the given electrode, if B is a product, n_B should be negative (being the change in number of moles of a product, when the reverse reaction > forward reaction). to get right n_B value I think z_B should be negative and z_A positive, then n_B will come out negative which it should if current is positive.

I invite someone else to come in on this as I'm short of time right now.

But press on, don't let it hold you back even if in back of mind.