Download Voltaic cells for physicists: Two surface pumps and an internal

Voltaic cells for physicists: Two surface pumps and an internal resistance
Wayne M. Saslow
Department of Physics, Texas A&M University, College Station, Texas 77843-4242
~Received 12 March 1998; accepted 19 November 1998!
A proper discussion of voltaic cells—the basic elements of any battery—must invoke the fact that
chemical reactions occur at each of the two electrode–electrolyte interfaces. The corresponding
physical picture is that voltaic cells have two chemical-reaction-driven ‘‘ion pumps’’ and an internal
resistance within the electrolyte. ~These ion pumps can move electric charge; hence the historical
terminology of ‘‘electromotive force,’’ or emf.! A correct physical picture for the ‘‘seat of emf’’ of
a voltaic cell gives, not a ‘‘volume pump’’ ~represented by a linear rise in voltage, as in many
introductory physics textbooks!, but two ‘‘surface pumps’’ ~represented by two steplike jumps in
voltage!. Within this framework, some basic properties of voltaic cells are considered: their emf,
maximum current, internal resistance, and energy storage. We also discuss some diffusion effects
within voltaic cells, whose manifestations—but not origins—are well known when batteries are
charged and discharged. Because of these diffusion effects, treating the electrolyte as an internal
resistance is an oversimplification. © 1999 American Association of Physics Teachers.
I. INTRODUCTION
Voltaic cells convert chemical energy to electrical energy.
Therefore, a complete discussion of voltaic cells must discuss both chemistry and physics. However, introductory
chemistry textbooks typically discuss the chemical energies
associated with voltaic cells, but not their operation in electrical circuits; and introductory physics textbooks typically
discuss the use of voltaic cells in electrical circuits, but not
their chemical energies.
From a pedagogical viewpoint, this omission on the part
of physics texts misses at least two opportunities. First, without invoking chemistry one cannot explain many well-known
properties of voltaic cells, such as the size dependence of
their energy storage, as opposed to the size independence of
their emf. ~Voltaic cells can move electric charge; historically, this led to the terminology ‘‘electromotive force,’’ or
emf.! An operational definition of the emf of a voltaic cell, to
be justified in Sec. XI, is the open circuit voltage, as measured by a high-impedance voltmeter.
Second, without invoking chemical reactions localized at
the electrode–electrolyte interfaces, one cannot correctly describe the voltage changes on going around a circuit containing a voltaic cell. In many texts, figures of voltage profiles
around a circuit show a linear rise in voltage within the cell,
and the internal resistance is—somewhat mysteriously—
placed outside the cell. However, as is well known to electrochemists, there are two steplike voltage jumps, one at each
electrode–electrolyte interface, and the internal resistance is
associated with the electrolyte between the electrodes.1–3
The chemical reactions at the electrode–electrolyte interfaces provide the source of emf, or the ‘‘seat of emf,’’ as it
has been known historically. The location of the seat of emf
was the subject of dispute at the end of the 18th and the
beginning of the 19th century.
Note that a typical automobile battery consists of six 2 V
cells in series, to raise the emf to 12 V. Each of the 2 V cells
consists of a number of voltaic cells in parallel, to raise the
current-carrying capacity. Moreover, a 12 V battery for electronics use consists of eight 1.5 V voltaic cells in series; each
cell alone could be used to power an electronic watch. The
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purpose of the present work is to give a reasonably accurate
yet nontechnical description of what happens within a single
voltaic cell.
In Sec. II we discuss a specific but very simple voltaic
cell—the Zn–Cu cell–in order to have a concrete chemical
example. For this modest price of admission, we can resolve
the issue of the location of the ‘‘seat of emf’’ ~Sec. III! and
explain a large number of properties of ordinary voltaic cells
~Sec. IV!. Section V provides more detail about what happens at the electrodes, and discusses chemical terminology.
Section VI considers the electromotive-force series. Section
VII considers the energy stored by a voltaic cell. In Sec. VIII
we discuss charging of a capacitor by a voltaic cell, thereby
establishing that the open circuit voltage and the emf are
indeed the same. In Sec. IX we discuss voltaic cells on open
circuit. Section X discusses what happens when a voltaic cell
is used at high ~and low! discharge rates, where diffusion
processes in the bulk of the electrolyte cannot ~and can! keep
up with the demands at the electrode–electrolyte interfaces.
Section XI models the voltaic cell as having a fixed emf E
and a fixed internal resistance r. Section XII considers current flow in an actual circuit. Section XIII discusses the history of the voltaic cell, and how the location of the seat of
emf went from ~1! the ‘‘animal electricity’’ hypothesis of
Galvani to ~2! the ‘‘contact electricity’’ hypothesis of Volta
to ~3! the now-well-verified electrode–electrolyte interface
hypothesis first espoused by the more chemically inclined
scientists of the late eighteenth century.
II. THE Zn/ZnSO4 /CuSO4 /Cu CELL „THE DANIELL
CELL…
As a specific example of the emf due to a voltaic cell, we
discuss the Daniell cell, a voltaic cell based on Zn and Cu
electrodes and two electrolyte solutions, of ZnSO4 and
CuSO4. See Fig. 1. In real Daniell cells, the two solutions are
kept from mixing by a porous cup. Using chemical nomenclature, this cell is written as Zn/ZnSO4 /CuSO4 /Cu. 1–3 The
reactions correspond to a positive emf, which tends to drive
current to the right.4 We will use this to describe a set of
illuminating pedagogical experiments devised by Peter
Heller.5
© 1999 American Association of Physics Teachers
574
Fig. 1. Schematic of the Daniell cell during discharge. Zn21 goes into solution off the Zn electrode, and Cu21 comes out of solution onto the Cu
electrode. The electrolytes are H2O with ZnSO4 and CuSO4, which then
dissociate. See text for discussion of how mixing is avoided.
To establish the nature of the chemical reactions responsible for the emf of this cell, Heller modifies the Daniell cell.
Heller’s cell consists of a sandwich of a flat plate of zinc, a
sheet of blotter paper ~Z!, a second sheet of blotter paper ~C!,
and finally a flat plate of copper. However, both blotter C,
touching the Cu electrode, and blotter Z, touching the Zn
electrode, are moistened by ZnSO4 solution. When a circuit
containing this cell is closed, no significant current flows.
However, if a few drops of CuSO4 solution are placed on the
Cu side of blotter C, then current will flow, temporarily, an
indication that Cu21 ions in the electrolyte are flowing to the
Cu electrode. On preparing the cell by first applying, for a
short time, a reverse emf ~to drive Cu21 ions from the Cu
into blotter C!, current will also flow, for a time proportional
to the charging time, when the reverse emf is removed and
the circuit is closed.5
These two experiments provide clear evidence that the
chemical energy driving the reaction is due to the competition between Zn on the Zn electrode going into the electrolyte as Zn21, and Cu on the Zn electrode going into the
electrolyte as Cu21. The Zn wins, causing Cu21 from the
CuSO4 electrolyte to plate onto the Cu electrode. Without
Cu21 ions in the electrolyte, there would be no electroplating, and no emf. As discussed in Secs. VII and VIII, the emf
is a measure of the energy release per unit charge ~more
precisely, the Gibbs free energy release per unit charge! that
is transferred from one electrode to the electrolyte and then
to the second electrode.
If CuSO4 solution actually were directly in contact with
the Zn electrode, in the absence of current flow the Zn electrode would blacken due to ‘‘poisoning’’ by Cu plating onto
it.5 This is because the Zn tends to go into the CuSO4 solution, driving Cu21 out of solution at the Zn electrode. As
already indicated, Daniell cells perform this separation by
employing a porous cup. Specifically, a porous cup containing a ZnSO4 solution is placed in a larger container of
CuSO4 solution; then a piece of Zn is placed within the
ZnSO4 solution, and a piece of Cu is placed within the
CuSO4 solution. A poor man’s version of the Daniell cell
would employ a thick blotter Z moistened with ZnSO4
solution—thick enough to impede diffusion of Cu21 ions to
the Zn electrode—and a blotter C moistened with CuSO4
solution.5
III. ‘‘SEAT OF EMF’’ OF VOLTAIC CELLS
Although we have already discussed the location of the
seat of emf in the Introduction, the answer bears repeating, in
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Fig. 2. The chemical reactions at the Pb and PbO2 interfaces. Both reactions
produce PbSO4, with consumption of the ions constituting H2SO4 and production of H2O.
a bit more detail. In a voltaic cell, chemical reactions at each
electrode–electrolyte interface serve as ‘‘pumps’’ for electrons, with a ‘‘pump strength’’ that corresponds to the emf
~in V!, or energy per unit charge ~again, properly, the Gibbs
free energy per unit charge! that is released in the chemical
reaction. To within an atomic dimension, the ‘‘pump’’ is
localized at the electrode–electrolyte interface, with dimension d'12431028 cm. In Fig. 1, the two ‘‘pumps’’ are
represented by the hatched regions. Heller has suggested the
term electromotive pump, or emp, to describe any physical
effect that promotes the circulation of electric current around
a closed path.5 The emp can be specified quantitatively in
terms of volts.
When electrodes are placed into a solution, chemical reactions occur that add electrons to, or remove them from, the
electrode. This process stops when the potential difference
cancels out the effect of the chemical reaction. @This is implied by Eqs. ~20! and ~21!.# This transfer of electrons is
similar to what happens with contact electricity ~i.e., two
metals brought into contact!. However, it is more complex
because, rather than involving only electrons ~contact electricity!, here ions also must go into and/or out of solution.
An example of such a set of chemical reactions, as they
occur spontaneously, is given by Fig. 2. This represents a
lead-acid cell, the basis of the common automobile battery.
In this system, the electrolyte is H2SO4 in H2O ~a fully
charged battery is 35% sulfuric acid by weight!. The H2SO4
essentially completely ionizes to H1 and HSO2
4 . The electrodes are Pb and PbO2. Figure 2 shows that the reactions
spontaneously tend to deposit electrons on the Pb ~negative
electrode!, and to attract electrons to the PbO2 ~positive electrode!. The lead-acid cell is written as Pb/H2SO4 /PbO2.
IV. SOME PROPERTIES OF VOLTAIC CELLS
We are now in a position to discuss some basic properties
of voltaic cells.
~1! The chemical reactions driving voltaic cells cause electrons to be added to or removed from each electrode.
~They also cause ions to go into and/or out of the electrolyte, and neutrals to go onto or off of the electrodes.!
This provides the source of energy that drives an electric
current through a wire connecting the two electrodes.
~2! The chemical reactions at the electrode–electrolyte interfaces consume ionic fuel on both electrode and electrolyte. This explains why voltaic cells go dead; their reactants have been exhausted.
Wayne M. Saslow
575
~3! The more the ionic fuel on the electrodes and in the
electrolyte, the larger the energy storage by a voltaic
cell. This explains why large D cells last longer than
small AAA cells.
~4! The chemical reactions driving voltaic cells provide a
characteristic electromotive force, or emf ~energy per
unit charge transported around the circuit!. Hence cells
with the same chemical reactions ~e.g., AAA cells and D
cells! have the same emfs.
~5! The greater the electrode area, the greater the current a
voltaic cell can provide. This explains why larger D cells
provide more current than smaller AAA cells.
The first four of these items follow from our discussion of
energetics in Sec. VII. The fifth is implied by our discussion
of resistance in Sec. XI.
Physics courses—and physicists—gain credibility with
students by explaining such well-known facts about batteries, which typically are not discussed elsewhere. Moreover, a
student armed with the knowledge that chemical reactions
drive a voltaic cell might be able to recognize that someone
claiming to have a voltaic cell of 30 V is either lying, ignorant that his ‘‘cell’’ is really a battery of voltaic cells in
series, or is destined to receive the Nobel Prize in Chemistry
~or Physics! for discovering a new form of chemical bonding. Because the typical chemical bonding energy per electron is on the order of an eV, the typical voltaic cell has an
emf on the order of a volt. The maximum voltaic cell emf to
date is found in cells using Li, with emfs as large as 4.5 V.
The finite amount of energy stored in voltaic cells and
batteries of them ~in series and/or in parallel! can be seen in
the fact that they are rated in terms of the finite amount of
charge they can send around a circuit. This charge is measured in terms of amp-hours ~1 amp-hour equals 3600 C!. A
good car battery might be rated at 100 amp-hours, or 360 000
C. By way of comparison, note that a mole of electrons is
about 96 500 C ~a unit known as the faraday!.
V. ELECTRODES AND ELECTROLYTE IN MORE
DETAIL—SOME TERMINOLOGY
The reader may find it helpful to be reminded or informed
~as the case may be! of certain chemical and electrochemical
terminology. This material is not really necessary for the
sections that follow.
A. Half-cells
Electrochemists refer to each electrode and its associated
electrolyte as a ‘‘half-cell.’’ 1–3,6,7 Electrochemists distinguish between two types of electrolyte. Supporting, or
‘‘spectator,’’ electrolyte consists of nonreactant ions—which
can affect the electric field and can carry current in the bulk
of the cell, although they do not react at the electrodes. Nonsupporting electrolyte consists of the reactant ions.
B. Standard hydrogen electrode „SHE…
It has become conventional to determine the emf of each
half-cell relative to a standard electrode consisting of H2 gas
bubbled over platinum in water of pH7 ~the concentration of
H1 is 1027 moles per liter!. At atmospheric pressure and
temperature T525 °C, this is known as the standard hydrogen electrode, or SHE. The H2 gas serves as the reacting part
of the electrode, and the chemically unaffected platinum
serves as the part of the electrode that provides a source or
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sink of electrons. In Faraday’s very precise terminology ~see
Sec. V D, below!, where the extremities of the electrolyte
~not the electrodes! are the cathode and anode, the H2 gas is
the anode ~where the positively charged cations are released!
and the platinum is the associated electrode ~where electric
charge is released!. This reaction is given by
H2→2H112e 2 .
~1!
With the electron energy measured relative to infinity, the
emf for this reaction is about 4.5 V, a result obtained by
modern spectroscopic methods and by theoretical
calculation.8 That is, this reaction is exothermic, releasing
energy. Since the emf is given per unit charge, the energy
release in the above reaction is 2e times the emf.
C. Characterizing reactions and voltaic cells
Oxygen typically removes electrons when it forms a
chemical bond, and it releases electrons when that bond is
broken. Hence oxidation corresponds to the loss of electrons.
The term reduction corresponds to the addition of electrons;
its origin seems to lie in the fact that metallic oxides, when
heated, release their oxygen, thereby reducing in weight.
The term faradaic describes an electrode reaction that involves charge transfer of electrons to or from the electrode.
Equation ~1! is such a reaction. On the other hand, it is entirely possible to have ordinary chemical reactions occur at
an electrode, without charge transfer of electrons to or from
the electrode. Such reactions are called non-faradaic. They
consume energy, but they do not drive current. When an
automobile battery is left to sit on open circuit for six months
or so, it goes dead because of non-faradaic reactions. The
reaction that ‘‘poisons’’ the Zn electrode in a Zn/CuSO4 /Cu
cell ~no ZnSO4 solution, unlike the Daniell cell!, discussed
earlier, is non-faradaic.
Electrochemists—but not physical chemists—distinguish
between two types of voltaic cells. Those which discharge
spontaneously are called galvanic cells. Those that require an
applied voltage for current to flow, or galvanic cells to which
are applied a reverse voltage large enough to cause the cell to
charge ~i.e., reverse the current!, are called electrolytic cells.
D. Anode and cathode, anion and cation
Faraday defines anode and cathode in terms of current
flow to and from the cell; current enters the cell by the anode, and leaves by the cathode.9 Since electron flow is opposite current flow, the cell loses electrons at the anode ~oxidation!, and gains electrons at the cathode ~reduction!. ~As a
pnemonic, note that enter, anode, and oxidation all begin
with vowels; and leave, cathode, and reduction all begin
with consonants.! For the Zn/ZnSO4 /CuSO4 /Cu cell under
spontaneous discharge ~cf. Fig. 1! current enters the cell at
the Zn ~the anode! and leaves at the Cu ~the cathode!. For the
lead-acid cell under spontaneous discharge ~cf. Fig. 2!, current enters the cell at the Pb ~the anode! and leaves at the
PbO2 ~the cathode!.
According to Faraday’s definition of anions and cations,
anions go to ~or cations leave! the anode, and cations go to
~or anions leave! the cathode.9 For the Zn/ZnSO4 /CuSO4 /Cu
cell under spontaneous discharge, Zn21 leaves the Zn cathode, and Cu21 goes to the Cu anode. Hence positive ions
~Zn21 and Cu21! are cations; correspondingly, negative ions
are anions.
Wayne M. Saslow
576
The open circuit voltage—that is, the emf—of the
Zn/ZnSO4 /CuSO4 /Cu cell is about 1.1 V, with the Zn cathode negative and the Cu anode positive. If a back external
voltage is applied to the cell, of amount less than the emf, the
cell still discharges ~but at a lower rate!, so the meaning of
cathode and anode are the same as for spontaneous discharge. However, if this back external voltage exceeds the
emf, the cell now charges, and the meaning of cathode and
anode interchange. Now the Zn ~still negative! is the anode,
and the Cu ~still positive! is the cathode. This corresponds to
an electrolytic cell. Clearly, the chemical definitions of cathode and anode depend upon the net direction of the chemical
reactions, not upon their relative voltages.
At one point Faraday performed an experiment on moist
paper, supported on wax, and placed between the poles of a
static electricity machine.10 When the machine discharged, a
reaction took place in the paper. This showed that electrode
terminals were not necessary for electrochemical action. Faraday thus had reason to distinguish between the surfaces of
the electrodes and the extremities of the substance being decomposed. Specifically, for Faraday the anode was the extremity of the electrolyte surface in contact with the electrode, not the electrode itself; and similarly for the cathode.9
Our discussion of the standard hydrogen electrode in Sec.
V B shows the utility of Faraday’s definition, for electrons
actually go to the electrode ~Pt!, which is distinguished from
the bubbling H2 gas that serves as the anode.
Nevertheless, despite the example of the SHE, modern
electrochemists do not distinguish between the extremities of
the decomposing substance and the electrodes. Even Maxwell did not make this distinction.11 In what follows, neither
shall we.
E. On conflicting conventions for anode and cathode
Contemporary chemists follow Faraday’s original convention for anode and cathode, involving the relative direction
of the chemical reaction within the cell. However, contemporary physicists employ a convention where the meaning of
anode and cathode is determined by relative voltage.
If the two ends of a battery of voltaic cells are connected
to plates within a vacuum tube, then electrons can be emitted
from the more negative plate, called by 20th century physicists the cathode. However, this corresponds to a current entering that electrode, which by the chemical definition is considered to be the anode. We thus have a conflict in
terminology.
The convention for anode and cathode used by modern
physicists seems to have developed at about the same time
that physicists’ interests turned from the electricity within a
voltaic cell to the electricity outside a voltaic cell—
corresponding to the study of cathode rays, now called electrons. In a voltaic cell ~i.e., one that spontaneously discharges! the current enters the ‘‘interesting’’ region ~the
electrolyte! at the ~19th century physicist’s and the chemist’s
definition of! cathode. In a cathode ray tube, the current enters the ‘‘interesting’’ region ~the vacuum, which is outside
the electrolyte! at the ~20th century physicist’s definition of!
cathode. This switching of the region of interest may be responsible for the physicists’ inversion of meaning of cathode.
VI. THE ELECTROMOTIVE FORCE SERIES
The wires connected to the electrodes permit the electrons
released at one electrode to transfer to the other electrode.
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Fig. 3. ~a! Detailed view of a voltaic cell and ~b! schematic view of a voltaic
cell.
When the electrodes are the same, they pump with equal
strengths in opposite directions, so there is no net emf. Since
there is an emf associated with each electrode, the net emf E
is their difference,
E5E21[E2 2E1 52 ~ E1 2E2 ! [2E12 .
~2!
Here E2 .0 means that there is a tendency to pump charge
out of the electrode and into the wire ~negatively charged
electrons will thus tend to be drawn into the electrode from
the wire!. In writing ~2!, we assume that E21.0. See Fig.
3~a!. Typically, circuit diagrams show two plates, one longer
than the other; the longer plate corresponds to electrode 2.
See Fig. 3~b!. From a knowledge of E21 and E32 , we deduce
that
E135E1 2E3 5 ~ E1 2E2 ! 2 ~ E3 2E2 ! 5E122E32 .
~3!
Volta was the first to recognize this relationship. It means
that, given ten metals, a set of nine emf measurements relative to any of the metals will give the emf between any pair
of the metals. Someone not recognizing this ~and in the
1790s that included many competent scientists! would have
to measure all 1039/2545 possibilities. The set of values E
characterizing the chemical emfs of individual half-cells,
relative to the standard hydrogen electrode, is known as the
electromotive-force series or the oxidation-reduction potential series.12
When Zn goes into solution, the half-cell emf is 0.762 V
relative to the standard hydrogen electrode; in equilibrium,
the electrode side of the interface is 0.762 V lower than the
electrolyte side. When Cu goes into solution the half-cell
emf is 20.345 V relative to the standard hydrogen electrode;
in equilibrium, the electrode side of the interface is 0.345 V
higher than the electrolyte side.
Specifically, when a Daniell cell discharges, the electrode
reactions are
Zn→Zn2112e 2 ,
Cu2112e 2 →Cu.
~4!
Note that the Zn21 changes its environment, going from the
solid electrode to the electrolyte solution; hence the energy
~properly, free energy! involved here is not merely the energy to remove the electrons, which would involve only the
work function. By Volta’s relationship, Eq. ~2!, the emf associated with this is 0.7622~20.345!51.107 V. The Zn is
the negative electrode, so it would be the shorter electrode on
typical circuit diagrams, the Cu being the longer electrode. In
Fig. 3~b!, the Cu would be on the left, and the Zn would be
on the right.
At each electrode, the reaction stops when the emf for the
electrode reaction is cancelled by the voltage change brought
about by the charge separation.
Wayne M. Saslow
577
VII. ENERGETICS OF VOLTAIC CELLS
We now turn to a more detailed explanation of why ~2! is
true, and what causes a voltaic cell to run down. Consider a
voltaic cell with two electrodes, 1 and 2. The energy associated with electrode 1 takes the form
E 1 5N 1,i E 1,i ,
~5!
where N 1,i is one of the ions associated with the reaction, and
E 1,i is the total energy release at electrode 1, from all reactants, when a single ion of type i spontaneously reacts. Because the chemical balance requires specific integer relations
between the numbers of ions, it does not matter which ion is
chosen. Further, because of Faraday’s law of electrolysis, the
number of ions stored on electrode 1 is proportional to the
electric charge Q 1 that electrode 1 can provide to the external circuit. Hence we can rewrite ~5! by introducing the energy per charge E1 associated with the faradaic chemical reaction at electrode 1, and the charge Q 1 that can be driven
from the electrode to the circuit by that reaction:
E 1 5E1 Q 1 .
~6!
Similarly,
E 2 5E2 Q 2 .
~7!
Taking the voltaic cell to remain electrically neutral as a
whole, if charge d Q 1 flows from one side to the other, we
have d Q 2 52 d Q 1 . Thus, for a charge d Q 2 , the net energy
change is
d E5 d E 1 1 d E 2 5 ~ E2 2E1 ! d Q 2 .
~8!
By convention, E5E2 2E1 is positive, in which case we may
think of the charge Q of the cell as Q 2 . The cell will then
tend to discharge spontaneously, decreasing the value of Q 2 .
With d Q cell5 d Q 2 , we rewrite ~8! as
d E cell5Ed Q cell ,
E5E2 2E1 .
~10!
This relationship makes it clear that a voltaic cell has a finite
amount of energy, and that when Q cell50, the cell has discharged. Because E is simply a function of the chemistry,
different cells of the same type can store different energies
only by having more ionic fuel, or charge, Q cell . Each D cell
has a much larger ‘‘charge’’ than an AAA cell. However,
both have E51.5 V, since they employ the same type of
‘‘ionic’’ pump.
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Note that the electrode area is not relevant to the energy
storage. However, the greater the electrode area, the faster
the cell can charge and discharge.
VIII. CHARGING A CAPACITOR WITH A
VOLTAIC CELL
We have yet to establish that the voltage drop across the
plates of a battery equals what we have called the emf in
~10!. To do this, we consider how a voltaic cell charges a
capacitor. See Fig. 4. The capacitor energy is given by
E cap5
1 Q2
.
2 C
~11!
When the cell and the capacitor are connected, the cell
will discharge, and the capacitor will charge. If the cell has
(0)
an initial charge Q (0)
50, then by
cell , and the capacitor Q
(0)
charge conservation Q cell5Q cell2Q after the capacitor has
received a charge Q. In equilibrium, the total energy will be
a minimum. By ~10! and ~11! we thus have
05
F
d
d
Q2
0!
E~ Q ~cell
2Q ! 1
~ E cell1E cap! 5
dQ
dQ
2C
S
5 2E1
~9!
We would like to integrate this to find the energy available
to the voltaic cell. In so doing we must consider which type
of ion provides the limit on the chemical reaction. For example, consider the Zn/ZnSO4 /CuSO4 /Cu cell. If there is
more Zn on the electrode than Cu21 in the electrolyte, then
the amount of Cu21 in solution limits the amount of discharge that can occur; if there is more Cu21 than Zn, then the
amount of Zn on the electrode limits the amount of discharge
that can occur.
Thus, the amount of ‘‘charge’’ Q cell associated with a cell
is limited by its least abundant active component. ~For example, if the ‘‘charge’’ is limited by the number N Zn of Zn
atoms on the Zn electrode, then Q cell52eN Zn .! Taking E to
be constant—actually, it depends upon the state of the electrolyte and the electrodes, as well as the temperature and the
pressure—we can integrate ~9! to obtain
E cell5EQ cell .
Fig. 4. Charging of a capacitor by a voltaic cell.
D
G
Q
.
C
~12!
From ~12! we thus deduce that the voltage drop across the
capacitor, when there is no more charge flow, is
DV5
Q
5E.
C
~13!
Hence an electrometer measures the E of ~13!. Since the two
plates of the capacitor are connected to the plates of the
voltaic cell, and there is no voltage drop across the connecting wires, this establishes that the voltage across the electrodes of a voltaic cell indeed equals the emf.
Note that, if the initial ‘‘charge’’ is Q (0)
cell , then the final
(0)
5Q
2Q.
Hence
the
final
cell and capaci‘‘charge’’ is Q (1)
cell
cell
tor energy is
1!
1!
0!
E ~cell
1E ~cap
5E~ Q ~cell
2Q ! 1
Q2
2C
0!
0!
2CE2 1 21 CE2 5E ~cell
2 21 CE2 .
5E ~cell
~14!
2
Thus the cell discharges energy in the amount CE , half of
which goes to the capacitor, and half of which must go into
resistive losses. Until it has been discharged, in this model a
voltaic cell behaves as if it had a fixed potential E, so it acts
like a capacitor of infinite capacity. Once discharged, the
model takes the cell to behave as if it has zero emf. For
actual voltaic cells, the emf varies more gradually with the
state of ‘‘charge’’ of the cell.
Wayne M. Saslow
578
Fig. 6. Ion density profile for lead-acid cell: ~a! fast discharge and ~b!
relaxation after fast discharge.
Fig. 5. An open circuit containing a voltaic cell and the voltage profile
around the circuit.
The chemically inclined reader will note that we have considered the equilibrium condition for a system in a real environment, at a given temperature T and pressure P. In that
case, thermodynamic theory tells us to minimize the total
energy of the system ~voltaic cell, capacitor, and environment!. This is equivalent to minimizing the Gibbs free energy for the voltaic cell and capacitor alone. ~Since the capacitor is taken to have no volume or entropy, the Gibbs free
energy and the energy are the same for it.! Hence E really
should be interpreted as the change in Gibbs free energy per
unit charge.13 One of Heller’s experiments makes this point
clearly.5
IX. VOLTAIC CELLS ON OPEN CIRCUIT
It is well known that, for a parallel plate capacitor, the
larger the plate separation, the smaller the capacitance C.
Hence, if the two electrodes of an isolated voltaic cell are
thought of as the opposite plates of a capacitor, that capacitor
has a very small C. Similarly, a voltaic cell that is part of a
circuit with an open switch may be thought of as connected
to a capacitor of very small C. Small amounts of charge
~equal and opposite! go to the two sides of the switch, producing a voltage difference. By ~13! this corresponds to
DV5E, so that for a voltaic cell on open circuit, the terminal
voltage is the same as the emf. Hence, measuring the open
circuit voltage of the cell with a voltmeter ~as indicated in
the introduction! yields the emf. In Fig. 5 we show an open
circuit and the corresponding voltage for it as we go around
the circuit. To be specific, we have taken E2 51.4 V and E1
520.6 V ~so E52.0 V!; there is a voltage gain at each electrode. For later purposes, when we consider current flow, we
will take the internal resistance to have the value r50.1 V,
and the external resistance to have the value R50.4 V. Note
the jumps in voltage that occur at each of the electrodes, and
the change in voltage across the switch.
To produce DV 1 at electrode #1, charges 6Q (1)
electrode must
build up on the electrode and electrolyte sides of the interface. Treating that interface as a parallel plate capacitor of
plate separation d 1 and dielectric of dielectric constant k, we
have
DV 1 5E 1 d 1 ,
E 15
1!
Q ~electrode
A 1 ke 0
.
~15!
This corresponds to a capacitance per unit area
C 1 ke 0
5
,
A1
d1
~16!
which for k 55 and d510210 m takes on the value 0.5 F/m2,
a value characteristic of the typically large electrode capaci579
Am. J. Phys., Vol. 67, No. 7, July 1999
tance per unit area. It is a nontrivial matter to actually determine d 1 , known as the thickness of the dipole layer. Electrochemists also call this the electric double layer. In
practice, the capacitance per unit area is measured, and d 1 is
deduced. Since automobile batteries have plate areas of
about 5 m2, their capacitance is on the order of 1–2 F. This
large battery capacitance smoothes out voltage changes in
automobile electrical systems.
Each electrode of the voltaic cell possesses a dipole moment
~17!
p5Q electroded.
Because of this dipole moment p, each electrode of a voltaic
cell produces an electric field at a distance. We can rewrite
~17! as p5Q electroded5( ke 0 ADV). The values k 55, A
51024 m2, and DV51.5 V yield p57310214 C-m. At a
distance r, this produces an external electric field on the order of p/4p e 0 r 3 , which for r55 cm is 0.56 V/m. This field
will attract atmospheric charges ~ions! to the vicinity of the
electrode, causing the field to be ‘‘screened.’’ As a consequence, direct measurement of the dipole field of an electrode is very difficult.
For a freshly made plate placed in air, there is no double
layer; the double layer forms on immersion in electrolyte.
The charge buildup associated with this process has a timeconstant proportional to the interfacial resistance to current
flow and to the capacitance associated with the dipole layer.
X. SOME EFFECTS OF DIFFUSION
Most of us are familiar with the peculiar abilities of voltaic cells, under certain circumstances but not others, to appear to be discharged, only to recover after a short period of
time. Such effects are due to diffusion.
A. Starting a car: High current, short times
When a car battery is employed to start the electrical starting motor ~which then starts the primary gasoline-powered
motor!, the battery temporarily provides about 600 A. If the
spark plugs are wet, the car may not start after ‘‘cranking’’
on the starting motor for a minute or two, and then the battery appears to go dead. The reason the battery appears to go
dead is that 600 A is a large drain on the battery, and ionic
fuel is quickly consumed in both the electrodes and the electrolyte. This leads to a low ionic density in the vicinity of the
electrode–electrolyte interfaces. See Fig. 6~a!. Note that the
electrical conductivity s for each ion satisfies the Drude relationship,
s5
ne 2 t
,
m
~18!
where n is the ionic density, t is the ion collision time, and m
is the ion mass. Hence, when such ion depletion occurs, the
Wayne M. Saslow
579
Fig. 7. Ion density profile for lead-acid cell: ~a! fast charge and ~b! relaxation after fast charge.
electrical conductivity decreases, thus suppressing current
flow.
After getting out of the car, opening up the hood, drying
off the spark plugs, and then returning to the car, the battery
seems to have miraculously recovered, for it again is able to
provide power, and usually the car will start with the spark
plugs dried off. The reason for this recovery is that the ionic
fuel in the electrode and the electrolyte has had enough time
to diffuse to the electrode–electrolyte interfaces. See Fig.
6~b!. ~This is an example where the battery appears to be
discharged, whereas really it has ionic reserves in the electrode and electrolyte.!
B. Recharging a dead battery: High current, short
times
Sometimes one leaves on the car lights ~about a 6 A current! long enough to deplete the battery, so that it cannot
provide enough power to start the car. In that case, if jumper
cables are employed, thus placing a good battery in parallel
with a bad battery, the car will start. ~The energy is provided
by the good battery. Note that a battery charger provides
perhaps 10 A, nowhere near the 600 A needed to start a car.!
Now consider what happens if one next drives the car for
two or three minutes and then turns off the engine. At that
moment, the battery will be able to restart the car. However,
if one waits an hour or so, the battery will not be able to
restart the engine. The reason is that, while the car was being
driven, the generator partially recharged the battery, producing ions only in the vicinity of the electrode–electrolyte interfaces. See Fig. 7~a!. These ions do not diffuse immediately from the interfaces, and thus are available if one waits
only a few minutes to restart the car. However, if one waits
longer, the ions diffuse away from the electrode–electrolyte
interfaces, and then the background ion density at the
electrode–electrolyte interfaces is insufficient to provide the
current needed to start the car. See Fig. 7~b!. ~This is an
example where the battery appears to be charged, whereas
really it has no ionic reserves in the bulk of the electrode or
electrolyte.! If one drives the car for perhaps 20 minutes,
enough ions are produced that, even after they diffuse away
from the electrode–electrolyte interfaces, there are enough
ions near the interface to start the car.
Now consider a voltaic cell that has been sitting for long
enough that it has a nearly uniform charge density. When the
cell is connected to a circuit, chemical reactions occur at the
electrode–electrolyte interfaces, leading to large density gradients and ~therefore! large rates of diffusion near the interface. On the other hand, in the bulk of the cell, the densities
remain nearly uniform, current being carried only because
the ions respond to the local electric field. Clearly, a complete description of the electrolyte at short times is rather
complex.
Note that the characteristic diffusion time t across a cell of
width w is
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Am. J. Phys., Vol. 67, No. 7, July 1999
Fig. 8. Ion density profile for lead-acid cell: slow discharge leads to a
steadily decreasing, nearly uniform density.
t5
w2
,
D
~19!
where D is the smallest of the diffusion constant for the
relevant ions in the cell. For a car battery, this is on the order
of 20 minutes.14
C. Leaving on the car lights: Slow, steady discharge
Despite the complexity of the electrolyte response for
short times, the description simplifies at long times, provided
that the current is not too large. A simple experiment provides some clues as to what happens in this case.
After one has ~accidentally or deliberately! left the car
lights on overnight, so that the battery will not start the car,
one can turn off the lights and wait to see if the battery will
recover. A number of scientists—and not a few amateurs—
have performed this experiment; the battery does not recover. There is something to be learned from this otherwise
disappointing result.
What has happened is that, under the slow, steady 6 A
discharge of the headlights, the battery has been nearly uniformly depleted of ions. Although ions are only consumed at
the electrode–electrolyte interfaces, they are provided to the
interface by the mechanism of diffusion. Diffusion is a relatively slow process, but if the drain at the electrode–
electrolyte interfaces is small, then diffusion can keep up
with it. Hence, instead of the ions emptying at the electrode–
electrolyte interfaces without significant replacement ~as
happens for the 600 A drain when the car is started!, here the
ions are replaced by diffusion, and the ionic density falls
nearly uniformly in space. See Fig. 8. This has significant
consequences for voltaic cells, and especially for the leadacid cell.14
XI. A MODEL FOR A VOLTAIC CELL WITH
CURRENT FLOW
From the previous section it should be clear that a description of a voltaic cell in terms of only a few parameters ~rather
than a continuum of values for the ionic densities and ionic
fluxes throughout the cell! is an oversimplification. Nevertheless, let us make such a description, with the understanding that it must be taken with a grain of salt.
Let us consider each electrode separately, again identifying them as #1 and #2. For small currents, each electrode
satisfies an Ohm’s law ~this is a consequence of the fact that
the response must be linear for small currents!. Because electrode #2 will later be taken to dominate, let us consider it
first. We attribute to it a resistance r 2 , an emf E2 , and a
voltage difference DV 2 . The emf is considered to be positive
Wayne M. Saslow
580
when it drives current from the voltaic cell to the connecting
wire, whereas the voltage drop is considered negative in that
case. See Fig. 3~a!.
With this convention, a positive DV 2 corresponds to the
outer surface ~within the electrode! being positive relative to
the inner surface ~an atomic dimension away, within the
electrolyte!. Phenomenologically, we expect that Ohm’s law
will hold, in the form
E2 2DV 2
.
r2
~20!
Fig. 9. A closed circuit containing a voltaic cell and the voltage profile
around the circuit.
Positive DV 2 tends to drive current to the right. A measurement of DV 2 for I50 would yield E2 , and if I depends
linearly on DV 2 , a measurement of the slope of I vs DV 2
would yield r 2 . As for ordinary resistors, the electrode resistance varies inversely with area. This explains point 5 in Sec.
IV.
For electrode #1, positive emf drives current to the right,
so we take
present model, the largest current that a voltaic cell can provide. It cannot provide this current for very long, because the
ions in the vicinity of the electrodes will be rapidly depleted,
leading to an increase in r. The present discussion, with fixed
r, is valid only for discharges that are either so slow, or last
such a short time, that such depletion does not occur.
I5
I52
E1 2DV 1
.
r1
~21!
B. Voltaic cell in series with a resistor
Positive DV 1 tends to drive current to the left.
When no current flows, ~20! and ~21! yield
DV 2 5E2 ,
DV 1 5E1
~ no current flow! .
~22!
What happens is that, due to the chemical reaction, charge
flows to the inner surface, providing a driving force that
completely cancels out that of the emf. If the electrode voltages take any other value, current will flow through the system. When there is no complete circuit, no current flows.
If we consider the electrolyte as a simple ohmic medium
with resistance r electrolyte , across which there is a voltage
drop DV electrolyte , then
I5
DV electrolyte
.
r electrolyte
~23!
Like DV 1 , positive DV electrolyte tends to drive current to the
left.
Using ~20!–~23!, and the definition ~2!, for the cell as a
whole, including both electrodes, it follows that
I5
E2DV
,
r
r5r electrolyte1r 1 1r 2 ,
DV5DV 2 2DV 1 2DV electrolyte .
~24!
In ~24!, I is positive when it flows along the direction of
positive emf E, and DV is positive when it opposes the current. When the current is zero, DV5E, so that the larger
plate is at the higher voltage. A measurement of DV for
I50 would yield E and, if I depends linearly on DV, a
measurement of the slope of I vs DV would yield r.
XII. VOLTAIC CELLS IN SIMPLE CIRCUITS
A. ‘‘Shorted’’ voltaic cell
If the two electrodes of the cell are directly connected, or
‘‘shorted,’’ so DV50 in ~24!, then I5E/r. This is, for the
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Am. J. Phys., Vol. 67, No. 7, July 1999
When the switch is closed on the circuit in Fig. 5, a current
I5
E
r1R
~25!
flows through the circuit. For the set of values E2 51.4 V,
E1 520.6 V, r50.1 V, and R50.4 V, ~2! gives E5E2 2E1
52 V, and ~25! gives I54A, so Ir50.4 V and IR5DV
51.6 V. For simplicity, we treat the electrodes as having
zero resistance in ~20! and ~21! ~i.e., r 1 5r 2 50!, so even for
finite current flow, the voltage jump and the chemical emf
cancel out. See Fig. 9. Note the jumps in voltage that occur
at each of the electrodes, and the linear variations across the
internal resistances. ~Such linear variations of voltage across
the electrolyte apply at short times, but not at long times.14!
In this example both chemical emfs tend to drive current
from a to b through the interior of the cell. Hence both voltage jumps tend to drive current from b to a through the
interior of the cell.
Most current physics textbooks give voltage profiles
across a circuit with a voltaic cell where the voltaic cell is
shown to have two linear regions. This is physically incorrect. One linear region represents the effect of the chemical
pumping, and is placed between the electrodes. It implies
~incorrectly! that the voltaic cell is a volume pump. The
other linear region represents the effect of the internal resistance, and is placed outside the cell. This serves to distinguish between the two linear variations; if they were both
placed between the electrodes, then there would be only a
single linear variation. Correct descriptions, based upon the
idea of chemical pumps at the electrode–electrolyte interfaces, are given in Figs. 5 and 9. Many older physics textbooks give correct voltage profiles across a circuit with a
voltaic cell,15–17 although they may not describe a voltaic
cell as containing two surface pumps. Note that Refs. 15 and
16 give a three-dimensional schematic for the voltage profile,
whereas Ref. 17 gives a two-dimensional schematic, like that
of Fig. 9.
Wayne M. Saslow
581
XIII. DISCOVERY MUST INCLUDE
REPRODUCIBILITY; IT NEED NOT INCLUDE
UNDERSTANDING
The history of the voltaic cell and of the battery illustrates
that a great discovery or invention can be made even when
the discoverer or inventor does not understand how the phenomenon or device works.18
Fig. 10. ~a! A single Zn–Cu voltaic cell and ~b! Volta’s Pile, using Zn–Cu
cells, showing the extra, unnecessary metallic plates at the leads.
A. Galvani
In 1786 Galvani began a series of electrophysiology experiments on freshly prepared frog specimens: spinal cord B
connected to crural nerve C connected to leg muscle D. Connected to spinal cord B was grounded wire A, and leg muscle
D rested on metal frame E. Thus ABCDE were all connected
in order, but there was no circuit. The leg muscle would go
into convulsions when there was a nearby electrical discharge, either from an electrical device within his laboratory,
or from atmospheric discharge in the outdoors. ~The current
causing the convulsions is now known to be due to electrostatic induction.19!
On rare but noticeable occasions, the leg muscle would
convulse without apparent source of discharge. ‘‘Finally,
wearied by such fruitless waiting, I began to scrape and press
the hook @A# fastened to the backbone @spinal cord B# against
the iron railing @E, touching the leg muscle D#...I then noticed frequent contractions, none of which depended on
variations in the weather.’’ 18 That is, Galvani had connected
A to E, producing the circuit ABCDEA.
In this way, Galvani discovered that two dissimilar metals
~A and E! must be placed in a circuit with the frog specimen
~B,C,D! in order to cause the leg muscle to convulse, although he did not fully appreciate the significance of the
metals. He made ‘‘the assumption of a very fine nervous
fluid that during the phenomenon flowed into the muscles
from the nerves, similar to the electric current in the Leyden
flask @a two-plate capacitor#.’’ 18 Thus, his source of ‘‘animal
electricity’’ ~distinct from ordinary electricity! was the nervous system, which through the nerves ~assumed to be insulated on their outer surface! could bring animal electricity to
the inner part of the muscle, leaving the exterior of the
muscle oppositely charged.19 Spasms would occur when the
nerves and the muscle exterior were connected; here is
where, to Galvani, the role of the metals—as conductors of
electricity—entered.
Galvani’s explanation was incorrect, but he had made a
great discovery, which he published in 1791. Galvani is one
of the founders of the field of electrophysiology.19,20
B. Volta
The physicist Volta—already known for his discovery of
methane and for electrical studies that included the invention
of the electrophorus and the relation between charge and
voltage for capacitors—began to study Galvani’s effect. He
realized that the frogs’ legs were serving as extremely sensitive detectors of electricity, rather than as sources of electricity. By 1792 Volta had established that the two metals
~‘‘dry conductors’’! were necessary, but that the frog’s leg
was not. By using more conventional ~but sensitive! detectors of electricity, he found that the frog’s leg could be replaced by what he called a ‘‘moist conductor’’ ~i.e., an electrolyte, like salt water!. Volta went on to discover the
electrochemical series, which enabled him to determine that
an effective 1.55 V battery ~still in use today! can be made
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Am. J. Phys., Vol. 67, No. 7, July 1999
from silver ~Ag! and zinc ~Zn! electrodes. He also made the
observation that placing intermediate metals in the circuit
~e.g., between the silver and the zinc! had no effect on the
current produced.
In 1800, Volta invented what came to be known as the
voltaic pile: a battery of voltaic cells connected in series,
which made the effects that he had been studying much more
intense. See Fig. 10. One of his early piles had 32 Zn–Ag
voltaic cells in series, with about 50 V across the terminals
of the pile, a powerful battery indeed. Almost immediately,
more chemically inclined scientists utilized the voltaic pile to
decompose water, collecting hydrogen at one electrode and
oxygen at the other. Thus was born the subject of electrochemistry.
Utilizing his considerable experimental skill, in 1797
Volta also discovered another effect related to voltaic cells—
the contact potential between two dissimilar metals, which
causes a transfer of electric charge. The contact potential,
unlike the chemical emf, involves only electrons. It is a measure of the work function of a metal. ~Recall that the work
function is the minimum energy to remove an electron from
a metal. It is related to the minimum energy to remove an
electron from an isolated atom of that metal.!
Although Volta was aware that oxidation occurs at the
silver electrode, he focused only on the electric current aspect of his cells, considering the associated flow of chemicals to be a mere side effect, of no fundamental significance.
His view was that the contact potential at the metal–metal
interface was the power source that drove the electric current
in his cells, and that the electrolyte served only to bring the
other ends of the metals to the same electrical potential. In
other words, to Volta the voltaic cell was a perpetual motion
machine. As he writes explicitly on March 20, 1780:18
‘‘[The apparatus either] works without stopping, or its
charge is again automatically restored after each sparking. In other words it has an inexhaustible charge...[The
pile] consists of a plate of silver in contact with one of
zinc, and is connected with the next pair through a sufficient layer of moisture, which should better be salt water
than plain...’’
Volta performed his work some 50 years before conservation of energy was an established principle. He was wrong
about the location of the seat of emf, but his reputation,
justly deserved on the basis of his many contributions to the
science of electricity, kept the scientific community from seriously challenging his viewpoint for many years. Even after
Faraday’s work on electrolysis ~chemists refer to Faraday’s
law as the law of electrolysis! established that the electrical
and chemical effects are inextricable, many prominent scientists continued to accept the contact potential as the energy
source for the voltaic cell.
Wayne M. Saslow
582
C. The proof of Volta’s misunderstanding is found at
his battery’s leads
A simple experiment can demonstrate that Volta was
wrong about the source of energy in the voltaic cell. As 1915
Nobel Laureate in Physics W. L. Bragg writes, ‘‘The energy
for driving the current @in Galvani’s frog experiments# comes
from the slight chemical action between the metals and the
muscles or nerves they touch. In Volta’s Pile it comes from
an action of the liquid upon the metals themselves, and if
you look at illustrations of his pile you will see that there is
an extra silver plate at one end and zinc plate at the other
which are really unnecessary and do not help to increase the
strength of the pile.’’ 21 Figure 10~a! gives Volta’s basic unit,
a piece of silver ~Ag! beneath a piece of zinc ~Zn! beneath
wet pasteboard, and Fig. 10~b! shows how Volta connected
them to produce the pile. As Bragg implies, removal of the
extra plates at the extremities ~a Ag on the left, and a Zn on
the right! would have yielded no change in the emf, in contradiction to Volta’s view.
Ultimately, it was accepted that the energy source for the
voltaic cell was chemical in origin. However, there was no
full-fledged theory until chemistry itself had become more
developed, toward the end of the 19th century. A full discussion of voltaic cells, including what happens within the electrolyte and the electrodes, must consider both their chemical
and their physical aspects. Physicists and chemists are still
trying to develop a quantitatively accurate microscopic
theory of the voltaic cell. The most difficult part is the description of what happens at the electrode–electrolyte interfaces. As discussed in Sec. X, even the electrolyte is not as
simple as it might seem.
ACKNOWLEDGMENTS
I would like to thank H. Fisher, R. Haaser, P. Heller, M.
Hyman, J. Ross, M. Rowe, M. Shapiro, and M. Soriaga for
their comments, observations, and suggestions.
V. S. Bagotzky, Fundamentals of Electrochemistry ~Plenum, New York,
1993!.
2
A. J. Bard and L. R. Faulkner, Electrochemical Methods ~Wiley, New
York, 1980!.
3
J. Newman, Electrochemical Systems, 2nd ed. ~Prentice–Hall, Englewood
Cliffs, NJ, 1991!.
4
Chemists use the IUPAC convention ~International Union of Pure and
1
Applied Chemistry! where voltaic cells are drawn so that positive emf
corresponds to rightward current flow. Physicists employ a convention
where the larger electrode is used to indicate the direction of current flow.
This notation is more flexible, because it permits voltaic cells to be given
any orientation, as is needed for real circuits.
5
P. Heller ~unpublished!.
6
I. N. Levine, Physical Chemistry, 4th ed. ~McGraw–Hill, New York,
1995!. See Chap. 14.
7
R. A. Alberty and R. J. Silbey, Physical Chemistry, 2nd ed. ~Wiley, New
York, 1997!. See Chap. 7.
8
For a theoretical estimate, see H. Reiss and A. Heller, ‘‘The Absolute
Potential of the Standard Hydrogen Electrode—A New Estimate,’’ J.
Phys. Chem. 89, 4207–4213 ~1985!. For a measurement, see W. N.
Hansen and G. J. Hansen, ‘‘Absolute Half-Cell Potential—A Simple Direct Measurement,’’ Phys. Rev. A 36, 1396–1402 ~1987!. For a survey,
see S. Trasatti, ‘‘The Absolute Electrode Potential—The End of the
Story,’’ Electrochim. Acta 35, 269–271 ~1990!.
9
M. Faraday, reprinted in W. F. Magie, A Source Book in Physics ~Harvard
U.P., Cambridge, MA, 1963!. See pp. 492–495. Magie cites the original
material as M. Faraday, Philosophical Transactions of 1834, p. 77, and M.
Faraday, Experimental Researches in Electricity, Vol. I, p. 195.
10
L. P. Williams, Michael Faraday ~Basic Books, New York, 1965!. See p.
242.
11
J. C. Maxwell, A Treatise on Electricity and Magnetism ~Clarendon, Oxford, 1891! ~reprinted by Dover, New York, 1954!. See Sec. 237.
12
L. Pauling, General Chemistry, 3rd ed. ~Freeman, San Francisco, 1970!
~reprinted by Dover, New York, 1988!.
13
Chemists normally write the Gibbs free energy change DG, for the cell, of
emf E, in terms of the faraday F ~96 500 coulombs! and the number n of
faradays that pass through the cell under discharge. Thus DG52EnF.
14
W. M. Saslow, ‘‘What Happens When You Leave the Car Lights on Overnight: Violation of Local Electroneutrality in Slow, Steady Discharge of a
Lead-Acid Cell,’’ Phys. Rev. Lett. 76, 4849–4852 ~1996!.
15
O. Blackwood, General Physics ~Wiley, New York, 1943!. See figure on
p. 456.
16
E. M. Purcell, Electricity and Magnetism ~McGraw–Hill, New York,
1965!. See the two figures on p. 137. Note that these figures do not appear
in the second edition.
17
W. Scott, The Physics of Electricity and Magnetism, 2nd ed. ~Wiley, New
York, 1966!. See Fig. 5.5a ~p. 211! and Fig. 5.5b ~p. 212!.
18
The author’s primary resource on the early history of the voltaic cell is W.
Ostwald, Electrochemistry: History and Theory ~Veit & Co., Leipzig,
1896!. The Smithsonian Institution and the National Science Foundation
had this remarkable two-volume work translated into English, and published by the Amerind Publishing Company, New Delhi ~1980!.
19
H. Hoff, ‘‘Galvani and the Pre-Galvanian Electrophysiologists,’’ Annals
of Science 1, 157–172 ~1936!.
20
J. F. Fulton and H. Cushing, ‘‘A Bibliographical Study of Galvani and
Aldini Writings on Animal Electricity,’’ Annals of Science 1, 239–268
~1936!.
21
W. L. Bragg, Electricity ~Macmillan, New York, 1936!. See p. 50.
DO THE PROBLEMS
I have done my best to simplify this subject as much as possible ~but no further!, as will your
instructor. But finally it is up to you to wrestle with the ideas and struggle for total mastery of the
subject. Others cannot do the struggling for you, any more than they can teach you to swim if you
won’t enter the water. Here is the most important rule: do as many problems as you can!
R. Shankar, Basic Training in Mathematics—A Fitness Program for Science Students ~Plenum Press, New York, 1995!, p.
xi.
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Wayne M. Saslow
583