Explanation of Misleading Nernst Slope by
Boltzmann Equation
K. L. Cheng
Department of
Chemistry, University of
MissouriKansas City, Kansas
City, Missouri
641
10
Received
March 12. 1998; accepted March
23, 1998
The common
misuse in the literature of the
terms Nernst slope, Nernst
factor, and Nernst potential
should be corrected. The
membrane electrode potential is
explained by the Boltzmann
distribution equation. The slope
of 59 mV per ionic unit is not
unique to the Nernst equation.
Both the Nernst equation and the
Boltzmann distribution equation
give the same 59mV slope based
on different mechanisms. The
slope indicates only the change
of D G regardless of
mechanisms. To avoid any
misleading inference to the
wrong equation mechanism, a
person's name should not be used
to denote a common lope. The
factors affecting the slope, the
electrode potential and it’s
measurements, the importance of
potential mechanism, and the
modified Boltzmann equation are
presented.
INTRODUCTION
There have
been widespread misconceptions
about the application of the
Nernst equation to explain the
mechanism of pH glass electrodes
and ion selective electrodes (ISE).
The terms Nernst slope (5,
12, 13). Nernst factor
(5, 9, 15). and Nernst
potential (5) have been commonly
misused in the literature. An
electrode potential may he
divided, at least, into two
types of electrode potential.
One is the cell potential
consisting of two halfcells,
cathode and anode, with redox
reactions such as a Zn in a
ZnCI2 solution; the other is the
capacitance potential based on
the adsorption of charges or
ions on a dielectric or
semiconductor such as a glass
without redox reactions.
Potential measurement may be
divided into two types of
potentiometry", namely, faradaic
potentiometry and nonfaradaic
potentiometry (1, 2). The former
is based on a device consisting
of two half cells with a salt
bridge for the potential
measurements and the latter is
based on a device with a
dielectric or semiconductor
adsorbed with charged particles
or ions against a reference
electrode; a conducting wire
instead of a salt bridge is
commonly used. A salt bridge can
also function as a conducting
wire for electron transfers;
however, a conductingwire
cannot substitute for the salt
bridge for ion transfers (see
fig. 1). The reasons for the
misuse of the Nernst equation
have been explained in the
literature (1, 2).
ELECTRODE
POTENTIAL MECHANISM
The electrode
mechanism based on the Nernst
equation. redox reaction. is
represented
E = E?/font>
+
2.3 RT/nF log([Ox]/[Red]).
(1)
The electrode
mechanism based on the double
and triple layers of a capacitor
is shown in Fig. 2.
BOLTZMANN
DISTRIBUTION EQUATION
The Boltzmann
distribution equation can be
written as follows:
Ni
/ N0
= exp{(Zi
eE) / kT}
(2)
The electrode
potential is measured with
respect to the bulk solution.
The following quantities are
found: z_{i} = 1
(for H^{+}); e = 1.6 x
10^{19}, electron
charge; k = 1.38 x 10^{23},
Boltzmann constant: T = 298 at
25 ?/font> C. The above equation
will become
E = 2.3
kT /?/font> (z_{i}
e log(N_{i} /N_{0}))
= ?/font> 0.059 log(N_{i}
/N_{0 }), (3)
where N_{i}
is ions or charge particles on
the electrode surface in charge
density and No is ion
concentration in bulk solution,
Here
2.3 RT / nF
= 2.3 z_{i}e / kT
= 59 mV/pC = D G.
(4)
The adsorption
of protons onto an insulator in
an acidic solution with a
Helmholtz double layer is
described as the Helmholtz
potential, which decreases about
60 mV per pH unit (3). In die
electron transfer expressions,
the expression exp((E_{tl}
 E_{F}) / kT)
based on the D
G change is called the Fermi
function (3), All the slopes are
59 mV based on
D
G for different mechanisms. This
causes some confusion. Now we
can see why we should not denote
a slope with a particular
person's name for the same
D
G that is not unique to any
mechanism. What is important is
the mechanism, not the slope. It
is clear that the slope does not
explain the mechanism, We may
suggest that the slope for the
pH glass electrode and ISE be
called the capacitance slope, as
it is related to the capacitance
potential (4).
MODIFIED
BOLTZMANN EQUATION
In pH and ISE
potential measurements, when
both cations and anions are
adsorbed at the same time onto
an electrode surface, such as a
pH glass electrode or an Ag_{2}S
electrode (9), we call the
capacitor a zwitterionic
capacitor. The original
Boltzmann equation is for an
electrode with a single type of
ion. Now for an electrode with a
mixture of positive and negative
charges, the Boltzmann equation
may be modified as follows:
where q is
charge density, C is
capacitance, and K is a
constant.
FACTORS
AFFECTING THE SLOPE
In dealing
with the pH glass electrode and
ISE, the following factors are
found to affect the electrode
potentials:
 Concentration;
 Temperature;
 Electrode surface
conditions;
 Number of charges of
ions (8);
 Stirring (6);
 Suspension (7);
 Zwitterionic nature, net
charge density;
 Anything changing ionic
adsorption;
 Isoelectric nature of
surface material;
 The Nernst equation
deals only with
concentration and
temperature;
 The Boltzmann equation
covers most factors.
These factors affect the
slope but do not change the
mechanism. At pH 68, the pH
glass electrode shows a
nonlinear curve, not like what
the Nernst equation predicts or
what most people expect. The
Nernst equation cannot he
applied to most anions, such as
OH^{} , which plays a
dominant role in the glass
electrode potential development
in basic media (11).
The resulting curve shown for
Ag^{+} by the Ag_{2}S
electrode was actually for both
the Ag^{+ }and the anion
ligands. it was misleading by
the Nernst equation and its
slope (9).^{}
The number 0.059 V for the
Nernst slope has been quoted by
most textbooks; however, a few
textbooks quoted 0.0591, 0.05915
(12), 0.05916 (13), and
0.0591594 (16), possibly for
better accuracy. The authors
forgot the meaning of
significant figures, As
discussed previously, the slope
is affected by many factors. To
have more significant figures
for the slope not only does not
mean much, but also misleads to
support the high accuracy of the
reaction mechanism. This number
with many decimals is just
another calculation number (14).
MECHANISM AND
THERMODYNAMICS
The mechanism is most
important for understanding the
electrode process and potential
origin. The thermodynamics shows
only the D G. the energy change,
not the mechanism. In the
textbooks of quantitative
analysis and electroanalytical
chemistry, the thermodynamics
has been overemphasized,
neglecting the mechanism. A
correct mechanism should be
based on careful experimental
data. A slope does not affect
the mechanism, but a mechanism
sometimes affects the slope. It
is improper to name a common
slope after a particular person,
suggesting incorrectly that the
slope is unique for that
mechanism and that when the
calculated slope matches the
experimental data, then the
mechanism must be correct. This
paper has proved that other
equations also give the same
59mV slope based on different
mechanisms.
CONCLUSIONS
The Nernst equation has been
misused in studies with membrane
electrodes, including the pH
glass electrode and ISE
electrodes. There is no such
thing as the Nernst slope,
Nernst factor, or Nernst
potential. First of all, the
Nernst equation is not related
to the capacitance potential.
Second, a common slope, 59 mV,
may be calculated from other
equations from different
mechanisms. It is also improper
to name the slope the Nernst
slope. the Boltzmann slope, or
the Fermi slope, and the
potential the Boltzmann
potential, the Helmholtz
potential, or the Nernst
potential. The potential origin
of membrane electrodes comes
from the adsorption of charged
ions or particles on the
electrode surface, following the
Boltzmann distribution,
capacitance law. E = q /C. and
Fruendlich isotherm. Contrary to
what's shown in the Nernst
equation, that the slope is
infinitely linear, E = q/C is
linear only in a limited range
depending on the nature of ions,
the electrode surface and
thickness, etc. Here, the
usefulness of a mechanism or
theory depends on how good it
can explain the electrode
potential phenomena. If it
cannot explain important facts,
then it should either be revised
or disposed of. It is a capital
mistake to theorize before one
has data. We have recently found
that the IUPAC conventional
redox mechanisms of calomel and
Ag/AgC] reference electrodes
have been erroneously postulated
on the basis of the Nernst
equation and Nernst slope (10).
This is another example of the
second Nernst hiatus (2).
ACKNOWLEDGMENT
We thank Jerry Y. C. Jean for
his assistance and the K. L,
Cheng Trust for financial
support.
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