Difference between energy and coulombic efficiency of perfect battery I have been trying to reconcile coulombic and energy efficiency for a perfect battery, but failed. Where have I gone wrong?
a coulomb is a unit of charge. Equal to lots of electrons.
a battery has a fixed amount of chemicals which react and need a fixed amount of electrons to complete reactions between empty and full.
therefore an ideal battery needs a fixed number of coulombs
an amp is a coulomb per second
so, charge at 100 amps for 2 hours is 200Ah which is 720000 coulombs
discharge at 50 amps for 4 hours is 200Ah which is 720000 coulombs.
Everybody happy. Ideal battery has coulombic efficiency of 100%. Coulombs is constant.
charge occurs at 13v which is 1300 watts which is 2600 Wh.
discharge occurs at 12v which is 600 watts which is 2400 Wh.
where has the 200Wh of energy gone .............
I think the answer lies in the maths and assumptions but where .....?
I'm guessing either a missing term or a misunderstanding of coulomb.
 A: An ideal battery has the capability of driving a certain amount of charge around a circuit that being 200 Ah or 720 kC in your example. This represents 12 $\times$ 720,000 = 8.64 MJ of energy stored in the battery.
That is about it for an ideal battery other than actually giving it an infinite capacity, a constant terminal potential difference independent of the current though it and its temperature which implies zero internal resistance.
Compared to a real battery quite a number of assumption have been made.
The chemical reaction is not 100% reversible so the capacity of the battery drops with time and so the battery has a finite lifetime usually stated as number of time that it can be recharged.
Both for charging and discharging of the battery there are lossy elements within the battery which can be lumped together as an "internal resistance" and so energy is converted into heat within the battery.
If a battery is discharged at a greater rate its capacity to supply electrical energy drops.
With time during the discharging process the potential difference across the terminals of the battery drops showing that non-ideal processes are occurring within battery.
