# Understanding the behaviour of exponential expressions

1. Nov 6, 2008

### 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?

Crispin

2. Nov 11, 2008

### 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

3. Nov 11, 2008

### Staff: Mentor

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

Sounds good to me.

4. Nov 11, 2008

### 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

5. Nov 11, 2008

### Staff: Mentor

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.

6. Nov 11, 2008

### 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 [Broken]

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

Crispin

Last edited by a moderator: May 3, 2017
7. Nov 15, 2008

### Mapes

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.

8. Nov 15, 2008

### Crispin

Many thanks Mapes,

So it looks like a typo in the notes, it's a pdf file, and it definately 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

Last edited: Nov 15, 2008