Fermi energy and batteries

[Moreno]

Someone told me that energy which a battery can deliver is approximately proportional to a difference in Fermi energy between anode and cathode. I guess it may not relate to metal-air fuel cells because Fermi energy is applicable more to a metals than gases. Fermi energy is also known as an electrochemical potential.

So, if we look at Fermi energy table for pure metals, then there is a substantial difference in Fermi energies between Aluminum (11.7 eV) and Sodium (3.24 eV). http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/fermi.html#c1

If we will make a battery with aluminum cathode and sodium anode (or contra) how much energy will it deliver? 

It is also interesting that though Aluminum and Sodium have different Fermi energies they both can serve as a high energy density anode in metal-air fuel cells.

 

Edited August 4, 2017 by Moreno

[Baron d'Holbach]

I think you are talking about "Fermi level" not "Fermi Energy". 

Isn't Fermi Level have to do with more with chemicals at the highest level but at zero ?

And Fermi Energy have to do with more with gases at the lowest level at zero?

Not related but related, I heard that only through Fermi laws we will achieve the "Speed of light technology and Warp Speed" .

 

[Moreno]

Why there exist a large discrepancy between electrochemical potentials (Fermi levels) and standard electrode potentials?

For example difference of Fermi energies between Sodium (3.24 eV) and Aluminum (11.7 eV) is larger than between Sodium and Copper (7 eV). http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/fermi.html#c1

However if we look at standard electrode potentials we will see that potential difference between Sodium (-2.71 V) and Copper (+0.33 V) is much larger than between Sodium (-2.7 V) and Aluminum (-1.6). https://en.wikipedia.org/wiki/Standard_electrode_potential_(data_page)

So, if we take in account difference between electrochemical potentials (Fermi levels) we would expect that reaction between Sodium and Aluminum will give us much more energy than reaction between Sodium and Copper. But if we look at standard electrode potentials we would expect otherwise. How can we explain it?