Nernst Equation

[shefv]

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I am confused with the derivation of the nernst equation.

My book gives this

1. If we start with G = Go + RT lnQ and sub in -nFE for G, where is the z (instead of n coming from in the equation?

2. What happens to Eo at the end?

3. I know that Q is Products/Reactants. I am confused with how that translates to ions outside/ions inside.

 

[lifetothefullest]

Frankly, that's a very confusing way to derive the Nernst equation. I also think that either the book has a typo in it or you miscopied it. So you know that delta G= delta G0 + RTlnQ. So delta G = -nFE and delta G0 = -nFE0. Sub those in and divide through by -nF. You get E = E0 - (RT/nF)lnQ. That's the general form of the Nernst equation. Note that n here refers to the number of moles of electrons that move around in the reaction (e.g. if LiCl to Li+ + Cl- is a one-electron process - you take an electron from Li and put it on Cl).

What you wrote is the equilibrium Nernst equation. At equilibrium, delta G = 0. Thus, E = 0 by delta G = -nFE. The Nernst becomes: 0 = E0 - (RT/nF)lnK (replacing Q with K to signify equilibrium conditions). That's where you get E0 = (RT/nF)lnK.

What I just discussed above can be used in any electrochemical cell.

Now, you're talking specifically about a membrane here. So you can imagine your "reaction" as an ion from inside the cell moving outside the cell. So ion(inside) ----> ion(outside). So your equilibrium constant for this "reaction" is ion(outside)/ion(inside). The charge that is moving around in the "reaction" is the charge on the ion, z (you can imagine it as electrons moving around if you like). So, just inserting those variables, you get E0 = (RT/zF)ln(ion outside/ion inside).

 

[shefv]

Frankly, that's a very confusing way to derive the Nernst equation. I also think that either the book has a typo in it or you miscopied it. So you know that delta G= delta G0 + RTlnQ. So delta G = -nFE and delta G0 = -nFE0. Sub those in and divide through by -nF. You get E = E0 - (RT/nF)lnQ. That's the general form of the Nernst equation. Note that n here refers to the number of moles of electrons that move around in the reaction (e.g. if LiCl to Li+ + Cl- is a one-electron process - you take an electron from Li and put it on Cl).

What you wrote is the equilibrium Nernst equation. At equilibrium, delta G = 0. Thus, E = 0 by delta G = -nFE. The Nernst becomes: 0 = E0 - (RT/nF)lnK (replacing Q with K to signify equilibrium conditions). That's where you get E0 = (RT/nF)lnK.

What I just discussed above can be used in any electrochemical cell.

Now, you're talking specifically about a membrane here. So you can imagine your "reaction" as an ion from inside the cell moving outside the cell. So ion(inside) ----> ion(outside). So your equilibrium constant for this "reaction" is ion(outside)/ion(inside). The charge that is moving around in the "reaction" is the charge on the ion, z (you can imagine it as electrons moving around if you like). So, just inserting those variables, you get E0 = (RT/zF)ln(ion outside/ion inside).

Thanks!

A related question - when looking at the Goldman equation - is that just a modification of the Nernst equation and can be applied to cell membrane potential?

How do we determine the outside versus inside in the numerator and denomenator for Na+, K+ and Cl-? i am trying to imagine it with a cell at resting membrane potential where Na+ wants to move in, K+ wants to move out and Cl- wants to remain outside the cell.

 

[lifetothefullest]

Yes, I believe the Goldman equation gives you the cell membrane potential when taking into account all the ions moving in and out of the membrane. You'd have to know the concentrations of the ions inside and outside of the membrane to solve that equation...

 

[shefv]

Why is it Na+ and K+ outside concentration but Cl- inside concentration in the numerator?

Isn't Na+ wanting to move into the cell, K+ moving out and Cl- remaining outside in a biological cell?

 

[lifetothefullest]

You have to take into account both outside and inside concentrations. In the Goldman equation, the concentrations that make up the argument of the natural log are summed. That is, you sum the stuff outside the cell and you divide it by the sum of the stuff inside. In the Nernst equation, you're doing the same thing but for individual ions. So for Nernst, it's E0 = (RT/nF)ln(Na+ outside/Na+ inside). Again, it's the same for Goldman but you're just summing all the ions up (K+ outside goes in the numerator and K+ on the inside goes in the denominator.

 

[shefv]

What about Cl-? That is inside/outside - why?