Using the following electrochemical cell and our understanding of the electrochemical potential, we would like to examine the dependence of the measured potential on the phase potentials at interface. The following electrochemical cell has 5 interfaces between different conducting phases. If we assume that each interface is at thermodynamic equilibrium, then an equilibrium exists between the compounds/ions/electrons on each side of the interface.
In this electrochemical cell, we can establish several interfacial equilibrium reactions. When writing these reactions, one must be careful to properly establish what the reactants and products are:
Interfacial Equilibria:
1. ePt 
eCu
2. ½H2 
H+
+ e
3. Cl-AgCl 
Cl-soln
Ag+AgCl 
Ag+soln
4. Ag+soln + eAg
AgAg
5. eCu' 
eAg
Because these interfacial reactions are all considered
to be at equilibrium for a potentiometric measurement, we can
establish relationships between the electrochemical potentials
of the reactant and products (Remember that 
)
remembering that Grxn = 0 for equilibrium reactions..
Electrochemical Potential relationships:
Adding all of these Electrochemical potential
relationships together:
(1)
Taking this expression and simplifying by eliminating
the terms that cancel each other leaves:
(2)
Now, expand the electrochemical potentials in
terms of the chemical potentials and the dependence on the phase
potential:
(3)
Once again, this equation can be simplified by
eliminating the terms that cancel each other out.
(4)
It is important to notice at this point that all
of the dependence on phase potential, with the exception of the
dependence on the phase potentials of the Cu and Cu' phases has
been eliminated from the equation.
Rearranging, and combining terms gives:
(5)
This equation can be simplified if we recognize
the following:
The activity of an electron in a metal phase is defined
as being equal to one. As a consequence,
and for any given type of metal (Cu in our example),
the chemical potential of electrons in the metal phase is equal
each other.
2. We can assign: 
Recognizing these relationships, we can simplify the chemical potential equation for this electrochemical cell to:
(6)
Solving for the phase potential difference gives:
(7)
For this electrochemical cell, the net redox reaction
is:
½H2 + AgCl Ag + H+
+ Cl-
and the Grxn is :
(8)
This expression is identical to the left-hand
side of the above equation, leaving us with:
(9)
Since one electron is transferred in the redox
reaction (n=1), this expression is equivalent to:
(10)
Where E, the potential that we measure is the
same as the phase potential difference between the phase potential
of the Cu phase and the Cu' phase [
].
If we now take equation (7) and expand the chemical
potential in terms of the standard state chemical potential and
the activity of the species we have:
(11)
Again, combining like terms leaves:
(12)
And simplifying by combing the logarithm terms:
(13)
Recognizing the relationship between various portions
of this equation allows one to find that this expression is simply
the Nernst equation:
1. The right-hand side of the equal sign is simply
equal to -FE.
2. (
) is just the
expression of Gorxn. Recognizing that Gorxn
is equal to -nFEo according to the Free Energy relationship
that we demonstrated above. For this particular cell reaction,
n = 1
These relationships allow one to rewrite equation
(13) as:
And we have shown that Thermodynamic representations for the equilibria that exist at each interface in a potentiometry measurement lead directly to the Nernst equation, as well as to the equation that relates the Free Energy change of an electrochemical cell reaction to phase potential differences.