Page 87 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN005M-206 June 15, 2001 20:25

Electrochemistry 165

For such a half reaction the free energy is given by the More rigorous methods for the calculation of aqueous ac-
relation tivity coefficients are available.
[red] The reaction of an electrochemical cell always involves
◦
G = G + RT ln , (29) a combination of two redox half reactions such that one
[ox]
species oxidizes a second species to give the respective
where− G indicatesthetendencyforthereactiontogoto
redox products. Thus, the overall cell reaction can be ex-
the right; R is the gas constant and in the units appropriate pressed by a balanced chemical equation
−1
for electrochemistry has a value of 8.317 J mol −1 K ; T
is the temperature of the system in K; and the logarithmic aox 1 + bred 2 cred 1 + dox 2 , K equil . (36)
terms in the bracketed expression represent the activities
However, electrochemical cells are most conveniently
(effective concentrations) of the electroactive pair at the
considered as two individual half reactions, whereby each
electrode surface. The free energy of this half reaction is
is written as a reduction in the form indicated by Eqs. (28)–
related to the electrode potential, E, by the expression
(32). When this is done and values of the appropriate
◦
◦
− G = nFE; − G = nFE . (30) quantities are inserted, a potential can be calculated for
each half-cell electrode system. Then that half-cell reac-
The quantity G is the free energy of the half reaction
◦
tion with the more positive potential will be the positive
when the activities of the reactant and product have values
terminal in a galvanic cell, and the electromotive force of
of unity and is directly proportional to the standard half-
that cell will be represented by the algebraic difference
cell potential for the reaction as written. It also is a measure
between the potential of the more positive half-cell and
of the equilibrium constant for the half reaction, assuming
the potential of the less positive half-cell,
the activity of electrons is unity,
E cell = E (more positive) − E (less positive) = E 1 − E 2 . (37)
◦
− G = RT ln K. (31)
Insertion of the appropriate forms of Eq. (32) into Eq. (37)
An extensive summary of E values is presented in the
◦
gives an overall expression for the cell potential,
compilation by Bard, Parsons, and Jordan (1985); the
most important are tabulated in Sawyer, Sobkowiak, and RT [ox 1 ] [red 2 ] b
a
◦ ◦ ln . (38)
E cell = E − E +
Roberts (1995). Standard potentials are thermodynamic 1 2 nF [ox 2 ] [red 1 ] c
d
quantities that usually are evaluated via caloriometry
The equilibrium constant for the chemical reaction ex-
for a cell reaction (e.g., 2 H 2 +·O 2 ·→ 2H 2 O; E ◦ =
cell
[E ◦ − E ◦ ]) and the relationship of Eq. (30). pressed by Eq. (36) is related to the difference of the stan-
+
O 2 H /H 2 dard half-cell potentials by the relation
When Eqs. (29) and (30) are combined, the resulting
Nernst expression relates the half-cell potential to the ef- ln K equil = (nF/RT)(E − E ). (39)
◦
◦
1
2
fective concentrations (activities) of the redox couple,
To apply potentiometric measurements to the determi-
RT [red ] RT [ox]
◦ ◦ nation of the concentration of electroactive species, a num-
E = E − ln = E + ln . (32)
nF [ox] nF [red]
ber of conditions have to be met. The basic measurement
The activity of a species is indicated as the symbol of system must include an indicator electrode, which is capa-
the species enclosed in a bracket. This quantity is equal ble of monitoring the activity of the species of interest, and
to the concentration of the species times a mean activity a reference electrode, which gives a constant, known half-
coefficient, cell potential to which the indicator electrode potential
can be referred. The voltage resulting from the combina-
[M a+ ] = a M a+ = γ ± C M . (33)
tion of these two electrodes must be measured in a man-
a+
Although there is no straightforward and convenient ner that minimizes the amount of current drawn by the
method for evaluating activity coefficients for individual measuring system. For low-impedance electrode systems,
ions, the Debye-Huckel relationship permits an evalua- a conventional potentiometer is satisfactory. However,
tion of the mean activity coefficient (γ ± ) for ions at low electrochemicalmeasurementswithhigh-impedanceelec-
trode systems, and in particular the glass-membrane elec-
concentrations (usually below 0.01 M),
trode, require the use of an exceedingly high-input-
√
µ impedance measuring instrument (usually an electrometer
log γ ± =−0.509z 2 √ , (34)
1 + µ amplifier with a current drain of less than 10 −12 A). Be-
cause of the logarithmic nature of the Nernst equation, the
where z is the charge on the ion and µ is the ionic strength
measuring instrumentation must have considerable sensi-
◦
1 2 tivity. For example, a one-electron half reaction of 25 C
µ = C i z . (35)
i
2 gives a voltage change of 59.1 mV for a 10-fold change