Understanding the behaviour of exponential expressions

[Crispin]

Hi folks, hope somebody can help me understand this one please?

Gven an expression i = ia-ic = icorr(exp(n/Ba)-exp(-n/Bc)), we are told that if -n<<Bc then exp(-n/Bc) tends to 0 & the equation becomes i (is approx) = ia = icorr exp(n/Ba).

I find that exp(-n/Bc) tends towards 0 if I substitute decreasing numbers in place of -n. So that works fine.

But then the we are told that if n>>Ba the equation becomes i (is approx) = ic = icorr exp(-n/Bc). This must mean that the exp(n/Ba) term must have tended to 0 again, and be negated, hence why it is removed.

When I try substituting increasing value numbers in place of n for exp(n/Ba), I find the opposite trend, i.e. exp(5/2) = 12.1 exp(10/2) = 148.4 so with increasing n the trend is towards infinity not 0?

Any advice most appreciated

Crispin

 

[Crispin]

Hi folks, does this question not make sense, is that why I've not had a reply?

Please advise & if needs be I can check the notes & write the info exactly as written.

Thanks

Crispin

 

[Mark44]

Hi folks, hope somebody can help me understand this one please?

Gven an expression i = ia-ic = icorr(exp(n/Ba)-exp(-n/Bc)), we are told that if -n<<Bc then exp(-n/Bc) tends to 0 & the equation becomes i (is approx) = ia = icorr exp(n/Ba).

I find that exp(-n/Bc) tends towards 0 if I substitute decreasing numbers in place of -n. So that works fine.

But then the we are told that if n>>Ba the equation becomes i (is approx) = ic = icorr exp(-n/Bc). This must mean that the exp(n/Ba) term must have tended to 0 again, and be negated, hence why it is removed.

That doesn't make any sense to me. If n>>Ba, exp(n/Ba) is very large.

When I try substituting increasing value numbers in place of n for exp(n/Ba), I find the opposite trend, i.e. exp(5/2) = 12.1 exp(10/2) = 148.4 so with increasing n the trend is towards infinity not 0?

Sounds good to me.

 

[Crispin]

Thanks Mark, so I'm not going mad...

We are told that the sum of two reactions in equillibirum give the total current (i), and this is given by the equation: i = ia - ic = ix (exp(n/Ba) - exp(-n/Bc)).

As the "ic" term is negative, that's why it is subtracted, so as the the 2 negatives cancel out presumably.

Then the exact words are;
"When -n<<Bc (i.e. the overpotential is quite positive, and E > Ecorr), and exp(-n/Bc) tends to zero. In this case, almost no cathodic current flows & the equation becomes i (approx)= ia = ix exp(n/Ba)

When n>>Ba, the overpotential is negative & E < Ecorr. In this case almost no anodic current flows, and equation becomes i (approx)= ic = ix exp(-n/Bc)"

Is it something to do with the negative/positive signs that I may have missed?

Thanks

Crispin

 

[Mark44]

If ic is negative, then ia - ic will be larger than ia. (I'm assuming that ia >0.)

I don't know what all the terms represent, and particularly their signs, so I can't say anything very definite. But, if n > 0 and Ba > 0, and given that n >> Ba, it's a certainty that exp(n/Ba) is a large positive number.

 

[Crispin]

Thanks for your feedback Mark, its called the Butler-Volmer equation, and relates to the current density at the interface of an electrode in solution, where both anodic & cathodic reactions (electrochemical corrosion reactions), are contributing to the current being produced.

http://people.clarkson.edu/~ekatz/butler-volmer_equation.htm

If that doesn't shed any more light, then don't worry about it. Thanks for taking a look for me.

Crispin

 

[Mapes]

When n>>Ba, the overpotential is negative & E < Ecorr. In this case almost no anodic current flows, and equation becomes i (approx)= ic = ix exp(-n/Bc)"

Is it something to do with the negative/positive signs that I may have missed?

Yes, this should be -n>>Ba or n<<Ba (the text even states that n is negative). It's just a matter of seeing which current term dominates. It might be helpful to plot the current for typical values and letting n go from large negative to large positive. Then plot the two terms individually. You'll see how one of the terms will usually dominate.

Does this answer your question?

 

[Crispin]

Many thanks Mapes,

So it looks like a typo in the notes, it's a pdf file, and it definitely says "n>>Ba".

It should have read "n<<Ba", as then the term "exp(n/Ba)" tends to zero, and it can, therefore, be removed from the overal equation of i = ia-ic = icorr(exp(n/Ba)-exp(-n/Bc)), to just leave i = ic = icorr (exp(-n/Bc)

Thank you for your help, very much appreciated!

Kind Regards

Crispin