I have a question regarding the Nernst equation and energy in a system. I know this is also a chemistry question, but I was interested in the thermodynamics (the energy of a system), so I decided to post it in Physics SE. Please let me know if I should move it to Chemistry.
First, I want to confirm that I understand the Nernst equation correctly. Voltage is a measure of the energy/charge, so whatever the voltage is calculated for a half cell, is the amount of energy per charge in that half cell (even though voltage is arbitrary).
So for example, if there are exactly 1 coulomb worth of charge in the solution, if the half cell potential is calculated to be 1V, there is 1 joule per coulomb of charge, and therefore there is 1 J of energy in that half cell (again, yes this is arbitrary... I know there isn't actually 1 joule). I'm often wrong, so feel free to correct me if that's the case.
Okay, assuming the above is correct:
Let's use an example of a concentration cell - a voltaic battery that contains the same ions and electrodes in both half cells, and generates voltage due to differences in concentration of ions between the cells.
When I use the Nernst equation to calculate the energy (from the potentials) for two half cells at equilibrium, versus two half cells that aren't at equilibrium, why is the energy at equilibrium higher than when the energy isn't at equilibrium? Shouldn't the total energy at equilibrium be lower (as work is done by the system, lowering its total energy)?
Here's an example (for the sake of simplicity, I'll set the standard potential as 1V, and the constant that comes before the log(Q) as 1. I'll also assume that at 1M concentration, there is 1 coulomb of charge in solution):
Let's consider two situations: One at equilibrium, where the concentrations of both half cells is 0.5M (therefore there are 0.5 coulombs of charge in each half cell).
Plugging into the Nernst equation gives: 1 - (1)*log(1/0.5) = 0.699V for both half-cells.
0.699V * 0.5 C * 2 half cells = 0.699J of total energy.
The next situation is where the concentration of one half cell is 0.25, and the other is 0.75 (therefore 0.25 and 0.75 coulombs of charge each).
Half Cell 1: 1 - log(1/0.25) = 0.398V * 0.25C = 0.010 J
Half Cell 2: 1 - log(1/0.75) = 0.875V * 0.75C = 0.656 J
Total energy = 0.010J + 0.656J = 0.666J of total energy.
0.666J is smaller than 0.699J... but why? Shouldn't the energy at equilibrium be less than the energy when not in equilibrium? What am I doing wrong? Or where is this extra energy at equilibrium coming from?