# Calculations on energies

Electrochemistry

## Gibbs energy, ΔG°

It is possible from the cells EMF to determine ΔG° for a reaction. In regards to electrochemistry and ΔG°, you have two relevant equations:

ΔG° = -z·F·E°
ΔG° = -R·T·ln(K)

 z = the number of electrons to be transferred F = Faraday's constant E° = standard electrode potential for the cell at standard conditions, i.e. E at 1 M and 25 °C R = the gas constant T = temperature in degrees Kelvin K = the equilibrium constant

Of these we need the formula

ΔG° = -z·F·E°

In some foreign collections of formulas and educational books, you can encounter this version for the formula:

ΔG° = -n·F·E°

where they prefer the use of n instead of z. Apparently there is no explanation for the use of n instead of z, when calculating ΔG°. Quite likely it is historical reasons.

## Gibbs energy outside the standard conditions, ΔG

Considering the way ΔG is calculated from chemical potentials (to be shown in details under chemical potentials when I ger around to that), you can extend the expression

ΔG° = -z·F·E°

to be a general expression for ΔG, i.e. it is also valid outside the standard conditions:

ΔG = -z·F·E

 z = the number of electrons to be transferred F = Faraday's constant E = the electrode potential for the cell

Therefore we can use the formula whether we work under standard conditions or not. You just have to remember to note whether it is under standard conditions or not by marking G and E with a ° under standard conditions.

## An example

If we take a rechargeable 1.5 V nickel/cadmium cell and try to calculate ΔG for this, it looks like this:

For a Ni/Cd cell, the redox reaction is:

Cd(s) + Ni2+(aq) Ni(s) + Cd2+(aq)

or the opposite direction, depending on whether the battery is producing a current or being recharged. E° for the Ni/Cd cell, according the the reaction, is 0.153 V (found by looking it up), so we are not working under standard conditions. From the reaction equation we see that z = 2 electrons, so we end up with

ΔG = -z·F·E

 z = 2 F = 9.65 · 104 C·mol-1 E = 1.5 V

ΔG = -2 · 9.65 · 104 C·mol-1 · 1.5 V
ΔG = -2.90 · 105 J/mol = -290 kJ/mol

Had it been the same cell under standard conditions, it would have looked like this:

ΔG° = -z·F·E°

 z = 2 F = 9.65 · 104 C·mol-1 E° = 0.153 V

ΔG° = -2 · 9.65 · 104 C·mol-1 · 0.153 V
ΔG = -2.95 · 104 J/mol = -29.5 kJ/mol