BNL-65847
Electromotive
force, emf (cells)
by
Mary D. Archer and Stephen W. Feldberg
Prepared by invitation for the forthcoming edition of the McGraw-Hill Encyclopedia of Science
and Technology
Submitted September 16, 1998
Electromotive
force, emf (cells)
The voltage or electric potential dilTerence across the terminals of a cell when no current is
drawn from it. Theernf ofacell
isthesum
of theekctricpotential
differences (PDs) produced
by a separation of charges (electrons or ions) that can occur at each phase boundary (or
interlace) in the cell. The magnitude of each PD depends on the chemical nature of the two
contacting phases. l%u~ at the interface between two difRerent metals, some electrons will have
moved from the metal with a higher free energy of electrons to the metal with a lower free
energy of electrons. The resultant charge separation will produce a PD (just as charge separation
produces a voltage across a capacitor) that, at equilibnu~ exactly opposes flu-ther electron flow.
Similarly, PDs can be produced when electrons partition across a metalJsolution interi%aceor
metallsolid intefiace, and when ions partition across a solutionlmembrane] solution interface.
The origin of emfi the Daniell cell as an example.
We can see how a cell ernf is composed of the sum of intefiacird PDs by considering the
historic Daniell cell shown in Fig. 1. This can be schematically represented by:
Cu’ I Zn I aq soln(l): ZnS04 ~aq soln(2): CUS04 ] Cu
a
c
b
Schem
d
where a solid line indicates a phase boundary and the dashed line a porous barrier permeable
to all the ions in the adjacent solutions. The barrier prevents physical mixing of the ZnS04 and
CuS04 solutions.
The cell emf is the open-circuit (i.e., zero current) potential difference measured between
the two Cu leads (any potential-measuring device must ultimately measure the potential
difference between two chemically identical phases).
1
We can write an expression for the cell
i
emf in terms of the electrochemical potentials of the species involved in the interracial equiibria.
The electrochemical potential ~~ of species j in phase a is defined by:
q
=
P,a‘ Zlw=
(2)
where F,= is the chemical potential of species j in phase a, z, is the charge (with sign) on the
species, F is Faraday’s constant (96497 C for Avogadro’s number of electrons), and @ is the
inner potential of phsse a. The term z~F@= in eq 2 is the electrical work required to move z~~
coulombs of charge into phase a from a vacuum at infinite dktance.
For a species in solution
eq 2 can be written:
5;
= p“~ + RTln aja + zjF@a
(3)
wherep,0“ is the standard chemical potential of species j in phase a, R is the gas constant
(8.3 144 Joules/moleldegree), T is the temperature in degrees Kel~
and aja is the activity of
species j in phase a. For a neutral species (i.e., when z~= O),
~;=P;=
p~
+ RThaja
For a filly ionized solution species at low concentrations,
(4)
a,= is well approximated by the
concentration Cj=; the activity of a pure phase (e.g., zinc and copper in the Daniell cell) is
defined as unity.
The value of ~j must be identical for any species j equilibrated between two different
phases. Thus for equilibrium of species j between phases a and #
(5)
More generally one can write for any interracial equilibrium between phases a and @
~ ~~j”””
I
2
= 0
(6)
where ~ is the stoichiometric number of species j (~ is positive for products and negative for
reactants) and ~~-e
is the electrochemical potential of species j in the phase in which it is
located. Thus, for the interracial equilibrium CU2++ 2e- @ Cu at interfaced in Scheme 1, eq
We now use these equations to quanti& the PDs across each of the interfaces in Scheme
1. We will assume that the PD across interfhce c is zero (a good approximation when the
separator is equally permeable to all the ions in the two adjacent phases). For interlaces a, b,
and d of Scheme 1 we can write:
(partitioning of electrons between Cu’ and Zn phases):
interface a
“Cul
Pe-
(7)
= ;:
(equilibrium: Zn’+ + 2e- @ Zn):
interface b
(8)
(equilibrium: Cu2+ + 2?- @ Cu):
interface d
(9)
Rearranging eqs 7–9 gives;
i;
-;::’
= Pe“Cu - ;.?
= ~ [ FZ –
7;.??)
-%;g-yz;;(l)
1
1[
(lo)
Since p~_cu = p~.cu’ and for an electron Ze- = -1, we can use eq 3 to write
>:->::’
= -F@cu
3
- tic”’)
(11)
The cell potential, Ed,
is simply the sum of all the equilibrium interracial PDs (remembering
that we’ve assumed that @’&Q) - #&(]) = O) :
Ed
= (~cm - gP@))
- @q
+ (p(’)
+ (p
- qp’)
. (@c” _ #cU’)
(12)
Combining eqs 3 and 11 – 12 (noting that the copper and zinc ions have a charge of :2+ and
that the activity of the metal phases is unity):
E &~ = (#c”- $&”’) = _ 1 -c”
&m=
(13)
where E=U is the potential difference between the two copper leads (Scheme 1) measured at
open circuit (i.e., when no current passes through the cell) and is therefore the emf of tlhe cell.
Dropping the superscripts denoting the solutions, eq 13 can be rewritten as:
Ed=
0
+
ECU2”ICM
~lnacu2.
[
-
1[
%
0
qzn
(14)
+ ~lnazn2+
1
where E& Z,lcu and Ez~2,1znare termed the standard electrode potentials (E’s) of the copper and
zinc half cells. Eq 14 is one example of a Nemst equatio~ which relates cell emf to the
activities of the cell constituents.
{see CHEMICAL
POTENTIAL;
EQUILIBRIUM;
ELECTROMOTIVE
Half cells, standard
electrode
CHEMICAL
FORCE
THERMODYNAMICS;
(EMF); THERMODYNAMIC
potentials
and reference
ELECTRODE
PRINCIPLES}
electrodes.
It is convenient to describe any electrochemical cell in terms of half cells. A half cell (consists
of an oxidant (Ox) and reductant (Red) such that: Ox + ne- *
4
Red; species Ox and IRed are
commonly referred to as a redox couple. The Daniell cell, for example, comprises the two half
cells: CU2++ 2e- @ Cu (redox couple is CU2+ICU)and Zn2+ + 2e- @ Zn (redox couple is
Zn2+~n) with the half-cell potentialsEc~z+lcuandfiznz+lz., @ven by
and
(16)
Such expressions are usefil only if the l?’ value for each half cell is known. Values of F can
be assigned to any given half cell by arbhrarily speci&ing that E“+F2, the l?’ for the standard
hydrogen electrode (SHE) half ce~ IY(aq,
ature dependence & ~.w/dT
a =
1) + e- @ ?&12(g,1 atm), is zero. The temper-
is also specified as zero. B values vs. the SHE for selected half
cells are given in Table 1. In principle, any E’ (and its temperature dependence) can be measured
dwectly vs. the SHE or another half cell whose electrode potential has been determined. Such
an electrode is termed a reference electruak The reference electrode is a half-cell designed so
that its potential is stable, reproducible, and that it neither contaminates nor is contaminated by
the medium in which it is immersed. Two convenient reference electrodes commonly used in
aqueous systems are the saturated calomel and the silverlsilver chloride electrodes (Table 1).
{ ELECTROCHEMISTRY
REFERENCE ELECTRODE ELECTRODE POTENTIAL}
F values and thermodynamics
The E’ values associated
thermodynamic tiormation
of the overall cell reaction.
with two half cells comprising a cell provide fimdamental
about the chemical reaction between the redox couples of the two
half cells. Returning to the example of the Daniell cell and combtig
eqs 13 and 14 we see that
E*Cuqcu – E;l,h
= ~
1
o all@)
[
PC”2+
–
p;u
+
P&
–
o So/n(l)
P&z*
1
(17)
The term on the lefthand side is the standard cell emf E~U (i.e., the value of E-U when all the
cell constituents are in their standard states of unit activity), and the bracketed term on the
righthand side is -AGo, where A@ is the standard Gibbs energy change of the cell rleaction
CU2++ Zn @ Cu + Zn2+. Hence we can write eq 17 as
EO
Cdl =–~
(18)
More generally, for the two half-cell reactions Oxl +
rile-
@
Redl and ~
+ n#
* Red~ E~u
is defined by
(19)
The cell reactio~ which involves the transfer of n,n2 electrons, is
n2 OxI
+
nl Red2 #
n2 Redl + nl 0X2
(20)
and eq 18 takes the general form
E;
where
n =
=-—
AGO .
nln2F
AGO
-—
nF
(21)
n,n2. If the cell is not in its standard state the relation becomes
E Cd =-~
nF
6
(22)
Using the general thermodynamic relation AG” = - K%
Kq we can also find the equilibrium
mnstant Kw of the cell reaction from the relation: 1
[Ox2]n’[RedJ% = exp
Kq =
[ClrJ”2[RedJ”
{seeCHEMICAL EQUILIB-
[1
nln2FE~u
AG 0
.—
= exp
RT
[ RT
(23)
CHEMICAL THERMODYNAMICS; THERMODYNAMIC
PRINCIPLES}
Membrane
potentials
as a source of emf.
When two ionic solutions of diflierent composition are separated by a membrane, a PD can
develop across the membrane, The complete cell requires electrical contacts with the solutions
on each side of the membrane, accomplished with reference electrodes:
re(lhs) I soln(lhs) I membrane ] soln(rhs) I re(rhs)
where re(lhs) and re(rhs) are the reference electrodes in the lefi and right solutions respectively.
The cell emf will be:
E cell = Er4rh,1 + q5s0h@hSJ
– qb’ohfi~ - Er4M,)
(24)
The PD across the membrane, @~@j - @“b@@> is a fimction of the membrane properties as
well as the composition of the adjacent solutions. The simplest example occurs when only one
1 Forthe specialcasewherenz= nz= n, thecell reaction(q 20) wouldbe writtenas Oxl + R@ * R@ + Oxz
andq 23 becomes
@21[Red,l
“q = [Ox,][Red2]
‘e+4=e4=’l
7
particular ion (e.g., hydrogen ioz W) is transported across the membrane. The transported ion
will tend to move from the side with high concentration to the side with low concentration;
however, in the process of doing so charge is separated across the membrane, producing a
potential dflerence which exactly counters the ion transport. The membrane PD will be
~ :Olrl(ws)
&WrW
where
a,
_
&WJLO
.
%1
z
r
J
~oln(rhs)
J
~
RT ~
p
CJ;owtis)
(25)
c +hs)
J
and c, are the activity and concentration of the transported ion. The cell emf is
K the concentration on one side, e.g., C;+’’”),iskept constant, E-u will reflect any variation
of the other concentratio~
soln(rhs)
c~
.
As long as the membrane transports only a single ionic species, the PD is thermodynamically
determined.
A soon as the membrane transports
more than one ionic species irreversible
thermodynamics come into play the membrane PD will exhibit a complicated dependence on the
concentrations and nobilities of the ions within the membrane as well as on their concentrations
in the adjacent solkons.
In certain circumstances, the response is still selective for one ion and
the membrane electrode has analytical utility. The glass electrode used for measurement of pm
in which a thin glass membrane responds selectively to the H+ io~ is an electrode of this type.
Potential dtierences called liquid junction potentials arise where two dtierent ionic wlutions
make contact through a permeable separator such as a glass frit.
thermodynamic
These interfere with
measurements and can be minimized by interposing a ‘salt bridge’, i.e., a
concentrated immobilized solution of KC1, or other salt in which the cation and anion have
nearly identical nobilities, between the two solutions. Numerous theories based on irreversible
thermodynamics have evolved to estimate the magnitude of liquid junction potentials.
8
{see ION SELECTIVE MEMBIUNES
The
nri-dllrd
- ..V emf
---=. . . . . .
.
hv
n Gnule
cdl
-= .“
“...=---
AND ELECTRODES; BIOPOTENTIALS}
fir
~ ~?
-.
Of ~~!!~ ~r ~~fi~~ (a
hatterv) -.
k IIW+
ac -a -v
IW
K- ------>,
-.-w
-
nnwer
~.. .,v.
source for a wide array of applications requ”tig DC electric power, ranging from powering wrist
watches to emergency power supplies. The emf of the saturated Weston cell
Hg H@304\ CdSO, (saturated aq SO~) ICd@g)
...
suu serves as a high-ievc i voitage reference for the Nationai institute of Standards and
a. .
.-. r-.
T-.c.-fifi.s-s-..,.
-RR.:
A..4W4 +LC9-AA
1 GL4111U1U&
> la..
llU WGVGI,
dU~GJJ1lWII
CU
1fJJ O ra-a
(U G -A..,
UU W QUL1=lUG1
GU L1lG lllUitL
Taatm-...l
. ,A14.m...P...-.&.
JJ1GVIC+G
VUlLCZ&2 1GLG1G11UG3
-.&-&.&
and .Zener diodes are used to produce reference voltages for many laboratory and field voltage
measurement devices.
There are also a few photoekctrochemical
cells with possible application in solar energy
conversio~ such as the Gratzel cell,
TiOz(s)l adsorbed RU2+dye I ~, 1- (non-aq soln)lC
where the cell emf is produced by absorption of visible light.
The emf of a cell can also be used as an indicator of chemical composition. Devices which
depend on the measurement of an open circuit-cell potential work well when the device is
sensitive to a single analyte but are notoriously sensitive to interferences (a familiar example is
the ‘alkdlne error’ which occurs when glass electrodes are used to measure very high pH
values).
For complex systems containing several species which can undergo electrochemical
reactio~ individual E values can be determined using electroanalytical techniques which involve
controlling the potential applied to a cell with measurement of the resultant current. Techniques
such as polarography and cyclic voltammetry,for example, involve changing the potential of an
indicator electrode and observing a wave or peak in the current at the redox potential of the
9
)
.
species in solutiors the height of the wave or peak indicates the concentration of the redox
species present in the solution.
{see ELECTROCHEMISTRY;
ELECTROCHEMICAL
ANALYSIS; BATTERY, FUEL CELLS}
Cell voltages
TECHNIQUES; POLAROGRAPHIC
.
when current passes.
E=fl values, which are thermodynamic quantities, do not give any tiormation
about the lkinetics
of the eiectrode reactions. ‘W-hencurrent passes t“mough a ceii, the cd voitage (i.e., the sum of
nn~.
ALUIG ru a
ceil
L-A....-d... 4-—: -..1.1 ..;11 x=-.
C-UGLWCGII UIC LGIIIUIIaJa)WUI UIIAGI UUIII
(LI fjJM~on
Of
MU
desitm)
and because
-.-.=.,
—----- —--
A... ..-c
LIE
GIIIA
the
---- kinetics
-—-------
k- . . . . . . -t .-.:.4 :..- 1----ucvmmc UI IGSISLIVGIUSQ
nf
-- the
---- ektrnde
------- ---
ramtimm
----------
..24.L:- .L-
WI IbUUI cm
are mnt
—.---- fast
-—.
enough to sustain thermodynamic (Nemstian) behavior at high rates of reaction (these kinetics
are very dependent on temperature and the nature of the electrode material). When a cell is
discharged through an external load, the cell voltage will be less than the efi, if the direction
of current flow is reversed (using an external power supply) the cell voltage will be greater than
the emf
In both cases, the greater the current, the greater the deviation of the cell voltage
from the emii “msome cases the eiectrode reactions are so siow that even the open-circuit ceil
voltage may not be a reliable measure of the emf
For example, a cell comprising the two half
cells, 2H+(aq) + 2e- * H2(g) and Oz(g) + 4H+(aq) + 4e- @ 2HZ0, both under standard
conditions, has a cell emf of 1.223V at 25 C (see Table 1). A highly sophisticated version of
this cell, the hydrogerdoxygen fiel cell, is designed to minimize internal cell resistance and has
electrodes tailored to maximize the rates of the electrode reactions; nevertheless, the voltage
under load is usually less than 0.8 V with most of the voltage loss occurring at the oxygen
electrode.
When an external power supply is used to electrolyte water the cell voltage required to
produce H2 and Oz at a reasonable rate is ~ 1.6 V and this again depends critically on the cell
design and the choice of electrodes.
10
6
.
#
Mary D. Archer and Stephen W. Feldberg
Acknowledgment.
SW
thanks the U.S. Department of Energy, Contract No. DE-AC02-
98CH10886, for support during the preparation of this article.
D. R. Crow, Principles
Bibliography.
R Denaro, Elemen@y
Electrochemistry,
and Application
of E!ectrochemis~,
4th Edition; A.
2nd cd.; D. J. G. Ives and G. J. J-
Reference
Electroa%s: Zheory and Practice, W. J. Moore, Physicul Chemistry 5th lMition; J. KogI@ IonSelective E1ectroa2s.
Table 1
Selected Standard Electrode Potentials at 25 C
Electrode
Electrode reaction
E“tv
Li+lLi
Zn2+lZn
H21H+lPt
Li+(aq) + e- * Li(s)
–3.045
-0.763
0
0.246
0.222
0.337
0.69
Zn2+(aq) + 2e- * Zn(s)
2H@q) + 2e- * H, (g)
Hg2C12(s)+ 2e- @ 2Hg(s) + 2CI- (sat. KC1)
AgCl(s) + e- @ Ag(s) + Cl- (a@
Cl- lHgzC121Hg
Cl- lAgCllAg
CU2+ICU
Fe(CN)~ -, Fe(CN)~ - lPt
O,p-rp
Cu2+(aq) + 2e- @ Cu(s)
Fe(CN)~ - (a@ + e- @ Fe(CN)~ - (a@
Oz(g) + 4H+(aq) + 4e- @ 2H20
F2(g) + 2H+ + 2e- @ 2HF(aq)
FZIF
1.223
3.06
{For more complete Tables see: Standhrd Electroak Potentials in Aqueous Solution, ed. A J
Bard, R. Parsons and J. Jordtq
Electro&
Potentials,
Encylcopedia
Marcel Dekker Inc., New York 1985; Tables of Stan&rd
G. Milazzo and S. Caroli, John Wiley & Sons, Chichester, 1978; Z?ze
of Chemical Electrode
Potentials,
Press, New York 1982. }
11
M. S. Antelmam and F. J. Harris, Plenum
-
Copper
)
Ieods
Clq
ZnS04
Im
lm
.-..
L
..d
Figure 1: The Daniell cell (reproduced from W. J. Moore, P@sica2
Chemistry, 5’”Edition, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1972; p.
524) - ~m$%.%<~~f
12