Fig. 1.12 Modified circuit including electronic leakage through the electrolyte
in the electrodes Eth even if there is no external current Iout flowing. This can be the
result of electronic leakage through the electrolyte that acts to short-circuit the cell.
This effect can be added to the previous equivalent circuit to give the circuit shown
in Fig. 1.12.
It is evident that, even with no external current, there is an internal current related
to the transport of the electronic species through the electrolyte Ie. Since the current
must be the same everywhere in the lower loop, there must be a current through
the electrolyte Ii with the same magnitude as the electronic current. There must be
charge flux balance so that there is no net charge buildup at the electrodes.
The current through the internal ionic impedance Zi generates a voltage drop,
reducing the output voltage Eout by the product Ii Zi, which is equal to IeZe.
Eout = Eth−IiZi (1.20)
In addition, the fact that both ionic and electronic species flow through the cell
means that this is a mechanism of self-discharge. This topic, discussed earlier in
Sect. 4.7, results in a decrease of the available charge capacity of the cell.
1.5.3 Transference Numbers of Individual Species
in an Electrochemical Cell
If more than one species can carry charge in an electrolyte, it is often of interest to
know something about the relative conductivities or impedances of different species.
The parameter used to describe the contributions of individual species to the transport
of charge when an electrical potential difference (voltage) is applied across an
electrolyte is the transference number. This is defined as the fraction of the total
current that passes through the system that is carried by a particular species.
In the simple case that electrons and one type of ion can move through the electrochemical
cell, we can define the transference number of ions as ti, and electrons
as te, where
ti = Ii/(Ii+Ie) (1.21)
and
te = Ie/(Ii +Ie) (1.22)