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Article
Electrolyte Additive Concentration for Maximum
Energy Storage in Lead-Acid Batteries
Andreas Paglietti
Department of Mechanical, Chemical and Materials Engineering, University of Cagliari, 09123 Cagliari, Italy;
[email protected]
Academic Editor: Joeri Van Mierlo
Received: 20 September 2016; Accepted: 16 November 2016; Published: 23 November 2016
Abstract: This paper presents a method to assess the effect of electrolyte additives on the energy
capacity of Pb-acid batteries. The method applies to additives of various kinds, including suspensions
and gels. The approach is based on thermodynamics and leads to the definition of a region of
admissible concentrations—the battery’s admissible range—where the battery can operate without
suffering irreversible changes. An experimental procedure to determine this range is presented.
The obtained results provide a way to assess the potential of electrolyte additives to improve
the energy capacity of Pb-acid batteries. They also provide a means to determine the additive
concentration that produces the maximum energy capacity increase of the battery. The paper closes
with an example of the application of the proposed approach to a practical case.
Keywords: Pb-acid batteries; electrolyte additives; battery energy capacity; electrolyte additive concentration
1. Introduction
Adding a chemical additive to the electrolyte of a Pb-acid battery may modify the specific energy
that the battery can store. This fact has been known since the invention of the battery and is currently a
research topic of great interest to the battery industry. This paper presents a general method to evaluate
the effect of electrolyte additives on the energy capacity of a Pb-acid battery and to determine the best
additive concentration to use. The electrolyte additive considered here is quite general. It can be a
chemical compound or a mixture of chemicals; a suspension or a gel of the kind used to immobilize the
electrolyte. The only restriction is that the additive—whatever it is—should be in chemical equilibrium
and have a low reactivity with respect to the other battery’s components.
Additives are also added to the battery electrolyte for a host of other reasons, such as to prolong
battery life, reduce electrode corrosion, improve conductivity, diminish gas evolution at the electrodes,
protect against overcharging or deep discharging, etc. Additives that are beneficial in some respects
may be detrimental in others. Thus, choice and concentration of the additive must always be assessed
against the side effects that it produces. This means, in particular, that an additive that enhances
the battery’s energy capacity may not be viable, at least at certain concentrations, owing to other
undesirable effects that it produces.
There are hundreds of papers, books and patents dealing with electrolyte additives and their
effects in Pb-acid batteries. A complete review of the literature would exceed the scope of the present
paper. Chapter 3 of Pavlov’s book [1] contains a comparatively short review of the main literature
on the subject up to about 2011. It concerns classic inorganic additives (phosphoric acid, boric acid,
citric acid, strontium sulphate, sodium sulphate), carbon suspensions, and organic polymer emulsions.
At the present time, the great potential of ionic liquids as electrolyte additives is being actively
studied [2] due to the capacity of these salts to widen the electrochemical window of water [3–5].
Furthermore, of great practical interest is the study of additives that produce gelled electrolytes, due to
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their applications in the field of electric locomotion [6–8]. An interesting study of the addition of an
additive to a gelled electrolyte was recently presented in [9].
The variety of available additives makes it impossible to give general rules on the best additive
and best concentration to use for a given purpose. Therefore, the present paper must, out of necessity,
be rather restricted in scope. For this reason, by ignoring other effects, this paper concentrates on the
influence of the additives on the energy storage capacity of the battery. The analysis presented provides
a general way to assess the effect of any electrolyte additive as far as this capacity is concerned. It also
shows how the additive concentration that maximizes this capacity can be determined from little basic
experimental data. Of course a positive assessment of an additive with respect to the energy capacity
of a battery does not rule out the need to ascertain whether or not, and to what extent, the additive
produces undesired side effects. However, in the search for the best additives to increase the energy
storage capacity of a battery, the results of the present paper may help to discard ineffective additives
quickly, thus greatly simplifying the selection process.
Central to the analysis of this paper is the observation that at any finite temperature the internal
energy of any finite volume system must be finite. This is a consequence of the energy conservation
principle or first law of thermodynamics. Under rather broad assumptions, which are met by most
natural systems, and by electrolyte solutions in particular, this observation, together with the second
law of thermodynamics, implies a limit to the specific free energy that the electrolyte can store and
supply isothermally. This point is discussed in Section 3. A similar analysis has been previously applied
in [10] to determine the maximum energy capacity of a living cell—a problem that is conceptually
analogous to the one considered here.
The present approach leads to the definition of the battery’s limit curve (Section 4). This curve
defines the limit concentration of electrolyte components beyond which the battery suffers irreversible
changes or damage, which may reduce the battery’s service life. In the case of a Pb-acid battery,
this damage materializes in the evolution of O2 at the positive electrode for excess of charge, or in
irreversible sulfation of the negative electrode for excess discharge. The said limit curve is instrumental
not only to find out the value of the maximum increase in the battery energy capacity that can be
achieved by the use of a given electrolyte additive, but also to determine the value of the additive
concentration that produces this maximum increase. It also leads to establishing the theoretical limits
of charge within which the battery can operate without suffering irreversible changes. A practical
example of the application of the obtained results is given in Section 5.
2. Free Energy of Battery Electrolytes with Additives
The free energy of a solution or a mixture is the sum of the free energies of its components. Thus,
if nH2 O , nH2 SO4 , and nj (j = 1, 2, . . . , k) denote the moles of water, the moles of sulfuric acid, and the
moles of additives, respectively, the Gibbs free energy of a Pb-acid battery electrolyte at pressure p and
absolute temperature T is given by:
k
G = G (nH2 O , nH2 SO4 , n1 , n2 , ..., nk , p, T ) = nH2 O µH2 O + nH2 SO4 µH2 SO4 +
∑
n j µj + C
(1)
j =1
Here µH2 O , µH2 SO4 , and µj are the partial molar Gibbs free energies or chemical potentials of water,
sulfuric acid, and additives, respectively, while C is an arbitrary constant. The chemical potential of
any component of a solution or mixture can always be expressed in the form:
µ = µo ( po , T ) + V ∆p + R T lna
(2)
In this equation, µo is the chemical potential of the considered component in a standard state at
pressure po and temperature T, while V is the partial molar volume of the same component, R is the
universal gas constant, ∆p stands for p − po and, finally, a is the activity or effective concentration of
the considered component.
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In what follows, the mole ratio:
n H2 O
xH2 O =
(3)
k
nH2 O + nH2 SO4 + ∑ n j
j =1
is taken as a measure of solvent concentration, while the concentrations of sulfuric acid and additives
are measured in molalities (moles per kg of H2 O) and denoted by bH2 SO4 and bj , respectively. That is:
bH2 SO4 =
and:
bj =
nH2 SO4
nH2 SO4
=
m H2 O
n H2 O MH2 O
nj
=
m H2 O
(4)
nj
n H2 O MH2 O
(5)
where MH2 O = 18.015 × 10−3 kg·mol−1 is the molar mass of water. In this notation, the activities of the
electrolyte components can be expressed as:
γH2 O nH2 O
aH2 O = γH2 O xH2 O =
(6)
k
nH2 O + nH2 SO4 + ∑ n j
j =1
aH2 SO4 = γH2 SO4 bH2 SO4 =
and:
a j = γj bj =
γH2 SO4 nH2 SO4
γH2 SO4 nH2 SO4
=
mH2 O
n H2 O MH2 O
γj n j
mH2 O
=
(7)
γj n j
(8)
n H2 O MH2 O
where γH2 O , γH2 SO4 , and γj stand for the appropriate activity coefficients, which, in general, depend
on nH2 O , nH2 SO4 , and nj , besides T and p. By expressing µH2 O , µH2 SO4 , and µj in Equation (2) and using
Equations (6)–(8), we can write Equation (1) as:
G
s
= nH2 O µoH2 O ( po , T ) + nH2 SO4 µoH2 SO4 ( po , T ) + ∑ n j µoj ( po , T ) + V∆p
j =1
+ R T [nH2 O ln
γH2 O nH2 O
k
nH2 O +nH2 SO4 + ∑ n j
+ nH2 SO4 ln
γH2 SO nH2 SO4
4
n H2 O MH2 O
k
γj n j
MH2 O
2O
+ ∑ n j ln n H
j =1
]+C
(9)
j =1
In writing this equation we have made use of the following equation:
k
V = nH2 O V H2 O + nH2 SO4 V H2 SO4 +
∑
nj V j
(10)
j =1
which relates the partial molar volumes V H2 O , V H2 SO4 , and V j of the electrolyte components to the
electrolyte volume, V.
Helmholtz free energy, Ψ, and Gibbs free energy are related to each other by the
well-known equation:
Ψ = G − p·V
(11)
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From this and from Equation (9), the Helmholtz free energy of the electrolyte is obtained:
Ψ
s
= nH2 O µoH2 O ( po , T ) + nH2 SO4 µoH2 SO4 ( po , T ) + ∑ n j µoj ( po , T ) − po V
j =1
+ R T [nH2 O ln
γH2 O nH2 O
γH2 SO nH2 SO4
4
n H2 O MH2 O
+ nH2 SO4 ln
k
nH2 O +nH2 SO4 + ∑ n j
k
γj n j
MH2 O
2O
+ ∑ n j ln n H
j =1
]+C
(12)
j =1
The above formulae are standard. However, as evident from Equation (7), the activity coefficient
γH2 SO4 introduced above refers to the overall concentration of sulfuric acid. This coefficient should
be distinguished from the mean activity coefficient of sulfuric acid ions, which may be denoted
±
by γH
and is normally considered in electrochemistry (although less frequently when dealing
2 SO4
±
with Pb-acid batteries). Using γH2 SO4 instead of γH
simplifies the formulae that follow, since the
2 SO4
details of dissociation of sulfuric acid into ions do not play any explicit role in the present approach.
The relationship between the two activity coefficients is:
γH2 SO4 = 4 bH2 SO4
3
2 ±
· γH2 SO4
(13)
±
This can be obtained from Equation (7) once aH2 SO4 is expressed as a function of γH
according
2 SO4
±
to the standard formulae for ionic solutes (see e.g., Section 7.4 in [11],). Both γH2 SO4 and γH2 SO4 depend
on bH2 SO4 and they are best determined from the experiment.
An important simplification of Equation (12) is obtained by introducing the following equation:
k
∑
n j ln
j =1
γj n j
n H2 O M H2 O
= nadd ln
γadd nadd
n H2 O M H2 O
(14)
a proof of which, in a slightly modified form, is given in [10]. In this equation we set:
k
nadd =
∑
nj
γj n j
(15)
j =1
and:
γadd
M H2 O
=
nadd
"
k
∏
j =1
n j #
1
neq
MH2 O
(16)
where the symbol Π indicates the product of a sequence, i.e.,:
k
∏ yi = y1 · y2 · ... · yk
(17)
i =1
By rewriting the right-hand side of Equation (12) as a sum of two parts and by using Equation (14),
the Helmholtz free energy of the electrolyte can, quite generally, be expressed as:
Ψ = Ψ0 + Ψ00
(18)
where functions Ψ0 and Ψ00 are given by:
Ψ0
= Ψ0 (nH2 O , nH2 SO4 , n1 , n2 , ..., nk , po , T )
s
= nH2 O µoH2 O ( po , T ) + nH2 SO4 µoH2 SO4 ( po , T ) + ∑ n j µoj ( po , T ) + C
j =1
(19)
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and:
Ψ00 = Ψ00 (nH2 O , nH2 SO4 ,
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γH2 O nH2 O
nH2 O +nH2 SO4 +nadd
i
nadd
+ nadd ln nγHadd
MH2 O −
2O
h
nadd , p, T ) = R T nH2 O ln
+nH2 SO4 ln
γH2 SO nH2 SO4
4
n H2 O MH2 O
po V
(20)
respectively. As discussed in the next section, Ψ00 is the part of Ψ that determines the admissible range
of the electrolyte. Thus, as far as the determination of this range is concerned, Equation (20) enables us
to substitute all of the electrolyte additives with just one single, fictitious additive of amount nadd and
activity coefficient γadd . Such an additive will be referred to as the equivalent additive.
Equation (20) is quite general. It applies to any combination of additives be they liquids, solid
suspensions, colloids, or any mixture thereof. Irrespective of the number and kind of additives, the
values of nadd and γadd can be determined experimentally by exploiting the fact that, as discussed in
the next section, there is a limit to the maximum amount of free energy that any finite system can store
in isothermal conditions. The details of the relevant experimental procedure are given in Section 5.
3. Free Energy Limit of the Electrolyte Solution
At any given finite temperature, the amount of non-thermal energy that a finite system can store
or supply is finite. This is an immediate consequence of the first law of thermodynamics. It implies a
limit to the maximum energy that the system can store. When considered in the light of the second
law of thermodynamics, the maximum energy limit entails a restriction on the states that a system
can reach without suffering irreversible changes in its constitutive properties. Under rather general
assumptions, such a restriction defines the region of all the states that the system can reach without
suffering irreversible changes in its properties. This region is the (thermodynamically) admissible
range of the system. Its boundaries are the system’s limit surface. The particular case of solutions, on
which we are concerned in this paper, is discussed in detail in [10]. A systematic introduction on the
subject, including general systems, is provided in [12].
From classical thermodynamics we know that, at a constant temperature, the amount of
non-thermal energy that a system can store or supply equals the change in the Helmholtz free energy
of the system. However, not all of the free energy of a system is subjected to thermodynamic limitation.
For instance, any purely mechanical part of the free energy of a system, e.g., the potential energy due
to the system’s weight, is not limited by thermodynamics. Therefore, when looking for the admissible
range of a system, the part of the system’s free energy that is not bounded by thermodynamics should
be ignored.
In the present case, the part of the free energy of the electrolyte that is not bounded by
thermodynamics is Ψ0 . This is apparent from Equation (19), as Ψ0 equals the sum of the free energies
of the electrolyte components in their standard state. Thus, Ψ0 depends on the amounts of these
components (nH2 O , nH2 SO4 , n1 , n2 , ..., nk ) independently of whether they are in a solution or separated
from each other. Since there is no thermodynamic limit to the amounts of material that can be
put together to form a system, there is no thermodynamic limit to the values that Ψ0 can assume.
The situation is totally different for Ψ00 . As Equations (6)–(8) and (20) should make evident, Ψ00 depends
on the concentration of the above components. Thus, it refers to the energy that these components have
as a result of their mutual interaction once they are mixed together. Any thermodynamic limitation
to the energy of the electrolyte solution must therefore be a limitation on Ψ00 , although the total free
energy of the solution is the sum of Ψ00 plus the part of energy, Ψ0 , that each component carries
irrespective of the presence of other components.
Actually, it can be verified that Ψ00 is only a small fraction of Ψ. The largest part of the total free
energy that a battery can store or supply is due to Ψ0 and it comes from the changes in nH2 O and nH2 SO4
that are produced by the chemical reactions taking place in the electrolyte. Small as it is, however,
Ψ00 determines the admissible range of the electrolyte. As a consequence, Ψ00 sets a limit to the total
free energy of the battery, Ψ, because it restricts the range of variation of nH2 O and nH2 SO4 . A similar
situation may also apply to solutions that contain chemically reacting components. For instance, in the
Batteries 2016, 2, 36
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case of a living cell, the part of the cytosol free energy that determines the admissible range of the cell
is only a fraction of the total free energy of the cytosol [10]. In that case, too, a small part of the total
free energy of the cytosol sets a limit to the amount of solution components, thus limiting the energy
that the living cell can store or release and, hence, its capacity to operate.
To make the following analysis independent of the amount of electrolyte, it is convenient to refer
to the molal concentration of Ψ00 per kg of solvent. This energy concentration is denoted by ψ00 and is
obtained by dividing both sides of Equation (20) by nH2 O MH2 O (i.e., by the weight in kilograms of the
water contained in the electrolyte):
ψ00 = ψ00 (nH2 O , nH2 SO4 ,
γH2 O nH2 O
RT
MH2 O ln nH2 O +nH2 SO4 +nadd
γH2 SO nH2 SO4
γadd nadd
nadd
4
n H2 O MH2 O + nH2 O ln n H2 O MH2 O
h
nadd , p, T ) =
+
nH2 SO4
nH2 O ln
i
−
po V
MH2 O
(21)
where V is the electrolyte volume per mole of solvent:
V=
V
n H2 O
(22)
In both the above equations, nH2 O is a variable because the number of moles of water in the
electrolyte varies as the battery charges or discharges.
In what follows, temperature is assumed to be constant. Moreover, the dependence of free energy
on p will be ignored, as is usually done in the absence of gaseous phases, and also when operating at a
00
constant pressure or nearly so. Thus, if ψmax is the value that ψ00 attains at the thermodynamic limit
mentioned above, the following relation:
ψ00 ≤ ψmax
00
(23)
applies to all states that the electrolyte can reach at the considered temperature. When taken together
with Equation (21), Equation (23) defines the admissible range of the electrolyte in the space of variables
nH2 O , nH2 SO4 , nadd . The limit surface of the electrolyte is the boundary to this range:
ψ00 = ψmax
00
(24)
It is, therefore, equipotential for ψ00 or Ψ00 (the same surface is not, however, equipotential for the
total free energy of the system, or for the part Ψ0 of it, as Equations (18) and (19) show).
Although V is variable, it suffers minor changes (less than about 0.3%) in normal battery operation.
As far as the present analysis is concerned, the term p◦ V/MH2 O that appears in Equation (21) can be
00
treated as a constant. As a consequence, its contribution to ψ00 and ψmax can, to a good approximation,
be neglected when applying Equations (23) and (24), because addition or subtraction of a constant
term to both sides of these relations is immaterial. Accordingly, when determining the admissible
range and limit surface of the electrolyte or the limit curve of the battery, we shall henceforth ignore
the term − p◦ V/MH2 O in the far right-hand side of Equation (21). With this proviso, the admissible
range of the electrolyte can be expressed as:
γH2 O nH2 O
nH2 SO4 γH2 SO4 nH2 SO4
RT
nadd
γadd nadd
00
ln
+
ln
+
ln
≤ ψmax
(25)
MH2 O
nH2 O + nH2 SO4 + nadd
n H2 O
n H2 O MH2 O
nH2 O n H2 O MH2 O
In the three-dimensional space (nH2 O , nH2 SO4 , nadd ) this relation defines the region of all the states
that the electrolyte can attain without suffering irreversible changes. The boundary of this region is the
limit surface of the electrolyte:
γH SO nH SO
γH2 O nH2 O
nH SO
n
γ
n
RT
00
+ 2 4 ln 2 4 2 4 + add ln add add
ln
= ψmax
(26)
MH2 O
nH2 O + nH2 SO4 + nadd
n H2 O
n H2 O MH2 O
nH2 O n H2 O MH2 O
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and it represents a surface in the three-dimensional space mentioned above.
4. Battery’s Admissible Range and Limit Curve
Not all states of the admissible range from Equation (25) can be accessed by the electrolyte
inside a battery. Under normal operating conditions, the battery does not exchange material with the
surroundings. In these conditions the total number of molecules of water and sulfuric acid inside the
battery remains constant. This is an immediate consequence of the well-known overall reaction that
controls the battery operation:
Pb (s) + PbO2 (s) + 2H2 SO4 (aq) 2PbSO4 (s) + 2H2 O (l)
(27)
The reaction proceeds from left to right during battery discharge. This produces two molecules
of water for every two molecules of sulfuric acid that are consumed. Charging the battery drives
the reaction in the opposite direction, thus producing two molecules of sulfuric acid for every two
molecules of water consumed. In both cases the sum of nH2 O and nH2 SO4 remains constant. At any
time of the battery’s charging or discharging process we have, therefore:
n H2 O + nH2 SO4 = n
(28)
where n is a constant. The value of this constant depends on battery preparation and can be determined
from the values of nH2 O and nH2 SO4 at any time of the battery’s life. In particular, let noH2 O and noH2 SO4 be
the values of nH2 O and nH2 SO4 of the electrolyte that is to be introduced into the battery. These coincide
with the values of nH2 O and nH2 SO4 of the electrolyte within the battery as the battery starts operating
once filled. The following equation must, therefore, hold true:
n = noH2 O + noH2 SO4
(29)
which fixes n.
Equation (28) can be used to eliminate variable nH2 O from Equations (25) and (26). This reduces
the number of independent variables appearing in these equations, thus further restricting the range
of states that the electrolyte can attain. More explicitly, by introducing Equation (28) into Equation (25)
we obtain the battery’s admissible range:
RT
MH2 O
ln
γH2 O (n−nH2 SO4 )
n+nadd
nH
SO4
2 SO4
+ n−n2H
ln
γH2 SO nH2 SO4
4
(n−nH2 SO4 ) MH2 O
+ n−nnadd
ln (n−nγadd nadd
)M
H SO
2
H2 SO4
4
H2 O
00
≤ ψmax
(30)
This is the region of plane (nH2 SO4 , nadd ) that contains all of the states that the electrolyte can
attain during normal battery operation without suffering irreversible changes. Its boundary is the
battery’s limit curve. It is obtained by taking the equality sign in Equation (30):
RT
MH2 O
ln
γH2 O (n−nH2 SO4 )
n+nadd
nH
SO4
2 SO4
+ n−n2H
ln
γH2 SO nH2 SO4
4
(n−nH2 SO4 ) MH2 O
ln (n−nγadd nadd
+ n−nnadd
H2 SO4
H2 SO4 ) MH2 O
00
= ψmax
(31)
This curve in the plane (nH2 SO4 , nadd ) delimits the region of all the states that the electrolyte can
reach reversibly as it operates inside a battery.
Equations (27)–(31) apply to Pb-acid batteries containing non-reacting electrolyte additives,
i.e., additives that do not react chemically between themselves or with other battery components.
Non-reacting additives are commonly used in commercial batteries. As previously stated, these are
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Equations (27)–(31) apply to Pb-acid batteries containing non-reacting electrolyte additives,
i.e.,
additives that do not react chemically between themselves or with other battery components.
Non-reacting
additives
commonly
in commercial
batteries.
previously
stated,
the only additives
aboutare
which
we areused
concerned
in this paper.
TheAs
same
equations
also these
apply,are
in
the
only
additives
about
which
we
are
concerned
in
this
paper.
The
same
equations
also
apply,
in
particular, in the absence of electrolyte additives, in which case nadd = 0.
00
particular,
in the
absence
additives,
in which
case of
nadd
= 0.
A typical
limit
curve of
ψ00electrolyte
= ψmax and,
thus, moves
the state
the
battery upward, i.e., along line
A
typical
limit
curve
ψ’’
=
ψ
"
,
as
obtained
from
Equation
(31),
represented
in Figure
1. The
max
AB of Figure 1. The battery’s admissible
range is the shadowed region is
within
the curve.
The amount
battery’s
admissible
range is remains
the shadowed
region
within
theand
curve.
The amount
of additive
in the
of additive
in the electrolyte
constant
during
charge
discharge,
because
the additive
is
electrolyte
remains Thus,
constant
during
charge and the
discharge,
because
additive
is the
chemically
chemically inactive.
charging
or discharging
battery within
thisthe
range
displaces
state of
inactive.
Thus,
charging
discharging
within
theIrreversible
state of thechanges
battery
the battery
up and
downor
a vertical
line, the
naddbattery
= constant,
inthis
the range
plane displaces
of Figure 1.
up
and
down
a
vertical
line,
n
add
=
constant,
in
the
plane
of
Figure
1.
Irreversible
changes
are
are produced in the electrolyte if the battery’s limit curve is exceeded. More specifically, charging
produced
the electrolyte
the thus,
battery’s
limit
specifically,
charging
the batteryinincreases
nH2 SO4 ifand,
moves
thecurve
state is
of exceeded.
the batteryMore
upward,
i.e., along
line ABthe
of
battery
Figure 1.increases nH 2SO4 and, thus, moves the state of the battery upward, i.e., along line AB of Figure 1.
Figure
Figure 1.
1. Typical
Typical limit
limit curve
curve of
of aa Pb-acid
Pb-acid battery
battery containing
containing aa non-reacting
non-reacting additive
additive in
in the
the electrolyte.
electrolyte.
The curve
curve is
is obtained
obtained from
from Equation
Equation (31).
(31). The
is
The
The electrolyte
electrolyte does
does not
not suffer
suffer irreversible
irreversible changes
changes if
if it
it is
kept
the battery’s
battery’s admissible
admissible range
range (shadowed
(shadowed in
in the
the figure).
figure).
kept within
within the
The process is reversible as long as the battery state remains within segment AB. However, if
The process is reversible as long as the battery state remains within segment AB. However, if
point A is exceeded, oxygen evolution takes place at the positive electrode, which makes the process
point A is exceeded, oxygen evolution takes place at the positive electrode, which makes the process
irreversible. A similar situation occurs during discharging. In this case, the discharging process
irreversible. A similar situation occurs during discharging. In this case, the discharging process
consumes sulfuric acid and the battery state moves downward along line AB. Point B on the
consumes sulfuric acid and the battery state moves downward along line AB. Point B on the battery’s
battery’s limit curve is the limit to reversible discharge. Beyond this point, the battery’s voltage
limit curve is the limit to reversible discharge. Beyond this point, the battery’s voltage becomes lower
becomes lower than the voltage needed to maintain the negative electrode reaction:
than the voltage needed to maintain the negative electrode reaction:
(32)
Pb + H2SO4 ⇌ PbSO4 + H2
Pb + H2 SO4 PbSO4 + H2
(32)
in chemical equilibrium. This makes the reaction proceed irreversibly to the right. The phenomenon
occurs
at a comparably
time
rate the
andreaction
is known
as sulfation.
It leadstotothe
theright.
formation
of insoluble
in chemical
equilibrium.low
This
makes
proceed
irreversibly
The phenomenon
PbSO
crystals
at the negative
concomitant
hydrogen.
and
occurs2 at
a comparably
low time electrode
rate and iswith
known
as sulfation.evolution
It leads toofthe
formationOxygen
of insoluble
hydrogen
evolution
at the
limits electrode
of the admissible
range are related
to the
windows
PbSO2 crystals
at the
negative
with concomitant
evolution
of electrochemical
hydrogen. Oxygen
and
of
water. The
readeratisthe
referred
tothe
theadmissible
relevant literature
details
on electrochemical
the chemical reactions
that
hydrogen
evolution
limits of
range arefor
related
to the
windows
of
regulate
thereader
electrochemical
stability
of water
in aqueous
electrolytes
(see e.g., reactions
([13–17]).that regulate
water.
The
is referred to
the relevant
literature
for details
on the chemical
The width of the
admissible
range
along aelectrolytes
vertical line(see
through
nadd is denoted as ΔnH 2SO 4 in
the electrochemical
stability
of water
in aqueous
e.g., ([13–17]).
The
width
of
the
admissible
range
along
a
vertical
line
through
n
denoted
as ∆n
in
add isper
2 SO4
Figure 1. This width represents the maximum amount of sulfuric acid that,
kilogram
ofHsolvent
Figure can
1. This
width
represents
the maximum
amount
sulfuric
acid that,
kilogram
of solvent
water,
react
reversibly
according
to Equation
(27). of
Thus,
the greater
thisper
width,
the greater
the
water,
can
react
reversibly
according
to
Equation
(27).
Thus,
the
greater
this
width,
the
greater
amount of energy that the battery can store and produce without the electrolyte deteriorating. The
the amount of energy that the battery can store and produce without the electrolyte deteriorating.
maximum value of ΔnH 2SO 4 is attained for nadd = n*add and∗ is denoted as ΔnH* 2SO 4 in∗ the above figure.
The maximum value of ∆nH2 SO4 is attained for nadd = nadd and is denoted as ∆nH2 SO4 in the above
Since
amount
of solvent
depends
the state
charge
battery,
it may
be convenient
figure.the
Since
the amount
ofwater
solvent
water on
depends
onofthe
state of
ofthe
charge
of the
battery,
it may be
to
define thetoadditive
concentration
with reference
a fixed state
of charge
battery.
This
will
convenient
define the
additive concentration
withtoreference
to a fixed
stateof
of the
charge
of the
battery.
be
taken
as
the
hypothetical
state
of
total
discharge,
which
the
battery
would
attain
once
all
sulfuric
This will be taken as the hypothetical state of total discharge, which the battery would attain once
Batteries 2016, 2, 36
9 of 14
all sulfuric acid in the electrolyte is consumed according to Equation (27). In this state the amount of
water in the electrolyte would be n H2 O = n, according to Equation (28). Therefore, when referring
∗
to this hypothetical state, the molal concentration of the electrolyte additive corresponding to nadd
is
given by:
∗
nadd
∗
badd
=
(33)
n MH2 O
This can be considered as the nominal molality of additive that is required to produce the
maximum capacity of energy storage in the battery.
Let ∆noH2 SO4 be the value of ∆nH2 SO4 when the battery electrolyte is free from additives
(see Figure 1). Since the energy that a battery can store or supply is proportional to the moles of
sulfuric acid which undergo Equation (27), the ratio:
ηmax =
∗
∆nH
− ∆noH2 SO4
2 SO4
(34)
∆noH2 SO4
represents the largest relative increase in the maximum capacity of energy storage that can be obtained
from a given electrolyte additive. Of course, ηmax depends on the additive being used, due to the
dependence on the additive of the battery limit curve.
5. Experimental Determination of the Limit Curve
00
To determine the limit curve of a battery, we need to know the values of ψmax and γadd which
are to be introduced into Equation (31). These values can be determined experimentally as follows:
We start by observing that Equation (31) holds true, in particular, when the electrolyte is free from
additives. In this case nadd = 0 and Equation (31) reduces to:
γH2 SO4 nH2 SO4
γH2 O (n − nH2 SO4 )
nH2 SO4
RT
00
ln
+
ln
= ψmax
MH2 O
n
n − nH2 SO4
(n − nH2 SO4 ) MH2 O
(35)
This equation applies at the limit of the battery’s admissible range. The quantity n appearing here
is given by Equation (29). It depends on battery preparation but not on the presence of electrolyte
additives. Thus, by operating on the battery deprived of electrolyte additive, we increase the battery’s
state of charge until we reach a limit point beyond which oxygen starts to develop at the positive
electrode in open circuit conditions (point A◦ of Figure 1). The appearance of this irreversible
phenomenon indicates that the state of the battery has reached the limit curve. We determine the value
00
of nH2 SO4 at this limit and we insert it into Equation (35). We can, thus, calculate ψmax .
As widely known, sulfuric acid concentration and battery voltage are related to each other
(see e.g., [18–21]). Therefore, rather than determining the limit value of nH2 SO4 , we may determine
the maximum open circuit voltage at which the battery maintains its charge without producing
oxygen at the positive electrode. This voltage is considerably higher than the standard voltage
(1.229 V) of water electrolysis [22], owing to the overvoltage that develops at the battery electrodes.
The extent of overvoltage depends on the property of the electrode surface and on the presence of
small amounts of various additives in the electrodes, introduced in their fabrication. As is apparent
from Equations (20) and (21), functions Ψ00 and ψ00 are independent of the free energy of the electrodes.
However, the overvoltage that the electrodes produce influences the battery’s admissible range and
00
limit curve, since it affects the limit value of nH2 SO4 and, hence, the value of ψmax . This makes the
battery’s admissible range and limit curve depend on the properties of the battery as a whole, and not
simply on the properties of its electrolyte.
00
The procedure to determine γadd is analogous to that followed to determine ψmax . In this case,
however, the battery electrolyte should contain a known amount of additive. Again, we charge the
battery up to the limit at which oxygen develops at the positive electrode in open circuit conditions.
We determine the corresponding value of nH2 SO4 and we insert it together with the considered value of
battery up to the limit at which oxygen develops at the positive electrode in open circuit conditions.
We determine the corresponding value of nH2SO4 and we insert it together with the considered value
of nadd into Equation (31). Since ψ "max has already been determined, the only unknown in this
equation is γadd, which can, thus, be determined. Due to the presence of transcendental terms, the
10 of 14
calculated graphically or numerically.
To exemplify, let us refer to a typical car battery at room temperature (T = 25 °C = 298.15 K). We
assume that, at the time of manufacturing,
the electrolyte within the battery contains 1 kg of water
00
nadd into Equation (31). Since ψmax has already been determined, the only unknown in this equation is
with, awhich
molarcan,
concentration
of sulfuric
acid
of bHo 2SOof
= 6 mol/kg. This means nHo 2O = 55.51
mol
γ
thus, be determined.
Due
to given
the presence
of γadd
is
4 transcendental terms, the value
add
o
best
graphically
Thus, nor=numerically.
55.51 + 6 = 61.51 mol, as follows from Equation (29). By leaving the
and calculated
nH 2SO 4 = 6 mol.
To exemplify, let us refer to a typical car battery at room temperature (T = 25 ◦ C = 298.15 K).
electrolyte free from additives, we charge the battery and find that bH 2SO4 = 7.25 mol/kg is the highest
We assume that, at the time of manufacturing, the electrolyte within the
battery contains 1 kg of water
o
sulfuric
acid
concentration
that
the
battery
can
maintain
in
open
circuit
conditions
without
with a molar concentration of sulfuric acid given of bH2 SO4 = 6 mol/kg. This
means
noH2 O = 55.51
mol
o
producing
at its
positive
electrode
concentration
corresponds
to a voltage
2.16leaving
V—or
and
nH2 SO4 oxygen
= 6 mol.
Thus,
n = 55.51
+ 6 (this
= 61.51
mol, as follows
from Equation
(29).ofBy
12.96
V for a six-cell
battery—according
to thethe
data
available
the literature
[19]).mol/kg
As observed,
the
electrolyte
free from
additives, we charge
battery
andfrom
find that
bH2 SO4 = 7.25
is the
battery
charge
and
discharge
occurs
at
constant
n
.
Therefore,
in
view
of
Equation
(28),
we find
that
highest sulfuric acid concentration that the battery can maintain in open circuit conditions
without
the above oxygen
value bHat
=positive
7.25 mol/kg
means
= 7.10 mol
and nH 2O = to
54.41
mol in
producing
electrode
(thisnHconcentration
corresponds
a voltage
of the
2.16battery
V—or
2SOits
4
2SO 4
Batteries
2016,
2, is
36 best
value of
γadd
12.96
V for aBy
six-cell
battery—according
to (35)
the data
available
the literature
[19]). As observed,
γ H O and
to this
electrolyte.
introducing
into Equation
the values
of from
γ H SO corresponding
2
2
4
battery charge and discharge occurs at constant n. Therefore, in view of Equation (28), we find that
value of bH 2SO4 as available from the literature and reported in the Appendix, and by recalling that R
the above value
bH2 SO4 = 7.25 mol/kg means nH2 SO4 = 7.10 mol and nH2 O = 54.41 mol in the battery
−1·mol−1 and M
−1 we calculate that, for the battery under
=
8.3143
J·K
=Equation
18.015 ×(35)
10−3the
kg·mol
electrolyte. By introducing Hinto
values, of
γH2 O and γH2 SO4 corresponding to this
2O
−1. literature and reported in the Appendix A, and by recalling that
value
of bH2 SO4 ψ
as"max
available
from
= −20.25
J·kgthe
consideration,
−1
R = 8.3143
J·K−1 ·mol
MHan
= 18.015 × 10−3 kg·mol−1 , we calculate that, for the battery under
To determine
γadd, and
we add
2 O arbitrary amount of the considered additive into the electrolyte of
00
−
1.
consideration,
J·kginstance.
the battery. Letψmax
nadd == −
5 20.25
mol, for
By operating the battery thus modified, we find that the
determine
, we addevolution
an arbitrary
amount
of the
considered
additive
the electrolyte
addoxygen
openTocircuit
limit γ
for
at the
positive
electrode
occurs
when into
the battery
charge
of
the battery.toLet
nadd = 5of
mol,
for instance.
By operating
themol.
battery
thus modified,
we find
nH SO = 6.74
nHthat
corresponds
an amount
sulfuric
acid of, say,
By
inserting
this
value
of
2
4
2SO 4
the open circuit limit for oxygen evolution at the positive electrode occurs when the battery charge
into Equation (35), we calculate that γadd = 0.64, as can be verified from the same equation once we set
corresponds to an amount of sulfuric acid of, say, nH
= 6.74 mol. By inserting this value of nH2 SO4
nadd = 5 mol, n = 61.51 mol, and ψ "max = −20.25 J·kg−12. SO4
into Equation (35), we calculate that γadd = 0.64, as can be verified from the same equation once we set
Finally, by inserting these values
of n , ψ " , and
γadd into Equation (35) and by making use of
00
nadd = 5 mol, n = 61.51 mol, and ψmax = −20.25max
J·kg−1 .
00
the expressions
of γ H 2O and
reported
in ,the
we obtain
analytic
expression
of
Finally, by inserting
theseγ Hvalues
of n, ψmax
andAppendix,
γadd into Equation
(35)the
and
by making
use of the
2SO 4
expressions
of γof
γH2 SO4 reported
thecurve
Appendix
A, weinobtain
of the
the limit curve
considered
battery.in
This
is plotted
Figurethe
2. analytic
From theexpression
same figure
we
H2the
O and
o the considered battery.* This curve is plotted in Figure 2. From the same figure we find
limit
curve
of
find that ΔnH 2SO 4 = 5.48 mol and ΔnH 2SO 4 = 6.14 mol. This implies ηmax = 0.12 according to Equation
∗
that ∆noH2 SO4 = 5.48 mol and ∆nH
= 6.14 mol. This implies ηmax = 0.12 according to Equation (34).
2 SOconsidered
4
(34). Thus, the electrolyte additive
in this example can produce up to a 12% increase in
Thus, the electrolyte additive considered in this example can produce up to a 12% increase in the
the battery’s energy storage capacity. The amount of additive needed to produce the maximum
battery’s energy storage capacity. The amount of additive needed to produce the maximum energy
energy storage capacity
n*add = 1.48 mol, as the figure shows. The corresponding nominal molality
∗ is
storage capacity is nadd
= 1.48
mol, as the figure shows. The corresponding nominal molality of
∗ b*= 1.34
=
1.34
mol/kg,
according
Equation
(33).
of additive
additive
is bis
mol/kg,
according
toto
Equation
(33).
add add
Figure 2.
2. Practical
Practical example
example of
of aa battery’s
battery’s limit
limit curve.
curve. The
The diagram
diagram refers
refers to
to 11 kg
kg of
of electrolyte
electrolyte solvent
solvent
Figure
(water). Thus,
Thus, the
the values
values reported
reported in
in the
the axes
axes can
can also
also be
be read
read as
asmolal
molalconcentrations.
concentrations.
(water).
Different additives may affect the battery differently. For instance, for the same battery considered
in the above example, an additive with γadd = 0.3 could produce a 25% increase in the energy storage
capacity of the battery. This can be checked easily from Equation (35), by constructing the limit
Batteries 2016, 2, 36
11 of 14
00
curve for γadd = 0.3 and the same values of n and ψmax given above. In this case the amount of
∗
additive producing the maximum energy storage capacity would be nadd
= 3.23 mol, which means
∗
badd = 2.91 mol/kg.
In the above analysis, we treated γadd as a constant, thus neglecting any possible dependence of
γadd on additive concentration. This may be acceptable, if the additive concentration is moderately
low (as is the case in many applications) or if we confine our attention to a sufficiently small portion
of the limit curve. Should more precision be required, the above procedure to determine γadd can be
repeated several times for as many different values of nadd as needed. The values of γadd thus obtained
could then be used to determine a function γadd (nadd ) which can be substituted for γadd into Equation
(35), if the approximation γadd = const. turns out to be inadequate.
Rather than charging the battery up to the oxygen evolution limit, represented by point A◦ in
00
Figure 1, we could—in principle—determine ψmax by discharging the additive-free battery down to
point B◦ of the same figure. This is the point of the battery’s limit surface at which sulfation of the
negative electrode start to take place. Once the concentration of sulfuric acid corresponding to this
lower limit is determined, the limit curve can be determined as described above. Both procedures
00
00
should provide the same value of ψmax , because both A◦ and B◦ belong to the same curve ψ00 = ψmax .
However, reference to the oxygen evolution limit appears to be more practical, since sulfation is a
rather slow phenomenon.
6. Conclusions
It is known that the energy capacity of a Pb-acid battery may be influenced by the presence of
additives in its electrolyte. The notion of equivalent additive, defined in this paper, helps to analyze
the effect of chemically inert additives and mixtures of such additives on the energy capacity of the
battery. This can be applied to determine an entire region of electrolyte concentrations, referred to
as the battery's admissible range, within which no irreversible change is produced in the battery
during charge or discharge. The boundary of this region is the battery’s limit curve. It corresponds
to the concentrations of sulfuric acid and, thus, to the range of open circuit voltages, that cannot
be exceeded without producing irreversible changes in the battery. The limit curve of a battery can
be constructed from a few experiments in which the battery is charged (or discharged) at different
additive concentrations. This provides useful information about the effectiveness of the additive to
increase the energy capacity of the battery, and the best additive concentration to adopt to this end.
The practical implications on the choice of the best additive to use are obvious. However, it should be
kept in mind that an additive can also produce undesired side effects, which are not considered in the
present work and require adequate scrutiny before any improvement of the energy capacity of the
battery resulting from the additive can be considered as practical.
Conflicts of Interest: The author declares no conflict of interest.
Appendix A. Activity Coefficients of Water and Sulfuric Acid in Solution
When expressed in molalities, the concentration of a solute does not depend on the presence of
other solutes in the same solution. Moreover, if the concentration of solutes is sufficiently small, the
activity coefficient of each solute is only marginally affected by the presence of other solutes in the same
solution. For this reason, due to the lack of better data concerning the influence of the addictives on the
activity coefficients of the electrolyte, the activity coefficient of sulfuric acid (γH2 SO4 ) is assumed to have
the same values γoH2 SO4 = γoH2 SO4 (bH2 SO4 ) that it assumes in the absence of additives. In other terms:
γH2 SO4 = γoH2 SO4 (bH2 SO4 ) = γH2 SO4 (nH2 O , nH2 SO4 )
(A1)
where the function γH2 SO4 (nH2 O , nH2 SO4 ) is obtained from γoH2 SO4 (bH2 SO4 ) after substitution Equation (4).
On the other hand, the activity coefficient of a solvent depends on its mole ratio. In the considered
case, the solvent is water and its mole ratio,xH2 O , depends on the amount of sulfuric acid and
( nH O , nH SO ) is obtained from  oHo SO (bH SO ) after substitution Equation (4).
where
where the
the function
function γHH22SO
SO44 ( nH22 O , nH22SO44 ) is obtained from γ H22SO44 (bH22SO44 ) after substitution Equation (4).
On
On the
the other
other hand,
hand, the
the activity
activity coefficient
coefficient of
of aa solvent
solvent depends
depends on
on its
its mole
mole ratio.
ratio. In
In the
the
x
considered
case,
the
solvent
is
water
and
its
mole
ratio,
,
depends
on
the
amount
of
sulfuric
considered case, the solvent is water and its mole ratio, xH 2O , depends on the amount of sulfuric acid
acid
H 2O
and
contained in the electrolyte, as is apparent from Equation (3). The analysis of
and additives
additives
ofofthe
the
Batteries
2016, 2, 36 contained in the electrolyte, as is apparent from Equation (3). The analysis 12
14
present
paper,
however,
concerns
unreactive
additives
at
comparatively
low
concentration.
Under
present paper, however, concerns unreactive additives at comparatively low concentration. Under
these
these conditions,
conditions, the
the influence
influence of
of additives
additives on
on solvent
solvent activity
activity coefficient,
coefficient, γHH22OO,, is
is expected
expected to
to be
be of
of
additives
contained
in
the
electrolyte,
as
is
apparent
from
Equation
(3).
The
analysis
of
the
present
secondary
secondary importance.
importance. For
For this
this reason,
reason, and
and in
in view
view of
of the
the already
already mentioned
mentioned paucity
paucity of
of
paper, however,
concerns
unreactive
additives
at comparatively
loware
concentration.
Under these

experimental
data,
the
values
of
of
an
electrolyte
with
additives
assumed
to
coincide
experimental data, the values of γHH22OO of an electrolyte with additives are assumed to coincide with
with
conditions, the influence of additives on solvent activity coefficient, γH2 O , is expected to be of secondary
o
o
o
o
o
those
without
introduced
is
γHo O ((xxHomentioned
O already
O ) . The quantity
those of
of the
the electrolyte
electrolyte
without
additives,
introduced here
here
is the
the
importance.
For this reason,
andadditives,
in view ofγHthe
paucityxxHHoof
data,
the
2 OO experimental
H22 O =
H22 O
H22 O ) . The quantity
2
values
of
γ
of
an
electrolyte
with
additives
are
assumed
to
coincide
with
those
of
the
electrolyte
= 0. That is:
solvent
ratio
H2 O
add = 0. That is:
solvent mole
mole
ratio when
when nnadd
o
o
without additives, γoH2 O = γoH2 O ( xH
). The quantity xH
introduced here is the solvent mole ratio
2O
nnH O 2 O
o
2 O
H
when nadd = 0. That is:
o
2
xxH O =
(A2)
2
(A2)
O2 SO 4
o H 2 O nnHH2 OOn+Hnn
2H
H 2 SO 4
xH
=
(A2)
2
2O
nH2 O + nH2 SO4
This
therefore:
This assumption
assumption about
about γγHH22OO means,
means,
This
assumption
about
means, therefore:
therefore:
H2 O
o
o

o
(x o )  
(n
,n
)
(A3)
(A3)
(A3)
O , nH 2 SO 4 )
=o γHH22O(Ox(oxHH22OO)) =
= γγHH22OO ( n(HHn22H
2 OO γ
γH2 OγHH=
O O ,Hn
H24 SO4 )
2
2 SO
H2 O
H2 O H2 O
2
o
o
The
by the
functions γoHo2SO
) and γHo2 O ((xoxHo2 O )) for
o 4 ((bbH 2(SO
o aa wide
The experimental
experimental values
values attained
attained
wide range
range
4 ) and
The
attained by
bythe
thefunctions
functions γ
b
)
and
γ
H 2H
SO SO
H 2SO
H
O
O( xfor
H
SO
4
4
2
2
4
HH22O
H2 O ) for a wide
2
4
of
are
in
papers
and
books.
For
aa critical
review
of
experimental
range
of concentrations
are reported
many
papers
and
For
a critical
review
of these
of concentrations
concentrations
are reported
reported
in many
manyin
papers
and
books.
Forbooks.
critical
review
of these
these
experimental
data
the
reader
is
referred
to
[23,24].
On
the
basis
of
these
data,
Equation
(A1)
is
assumed
to
experimental
data
the
reader
is
referred
to
[23,24].
On
the
basis
of
these
data,
Equation
(A1)
is
data the reader is referred to [23,24]. On the basis of these data, Equation (A1) is assumed
to be
be
approximated
by
the
expression:
assumed
to be approximated
by the expression:
approximated
by the expression:
(1.22 bH SO  6.5)
(1.22 b 2 4 − 6.5)
(1.22 bH2HSO
− 6.5) for b
2.12
≤~~ ∼
88 mol
//kg
2 SO
44
SO
γH2SO
= 2.12
for bbHH
8 mol/kg
γHH24SO
mol
kg
44 ≤
SO
SO4 = 2.12
H22SO
2
4
2
(A4)
(A4)
(A4)
4
represented
graphically
in
the
following
Figure
A1.
represented graphically
graphicallyin
inthe
thefollowing
followingFigure
FigureA1.
A1.
represented
Figure
A1.
showing
the
dependence
of
on sulfuric acid molality in
Figure A1.
A1. Plot
Plot of
of Equation
Equation (A4)
(A4) showing
showing the
the dependence
dependence of
of γγHH22SO
SO4 on sulfuric acid molality in
Figure
Plot
of
Equation
(A4)
H2 SO44 on sulfuric acid molality in
o
oo
J) of
of γ
aqueous
are taken
taken from
from Bullock
Bullock [24].
[24].
H 22SO
aqueous solutions.
solutions. Experimental
Experimental values
values ((J)
SO44 are
γ H
H 2SO 4
As
As for
for the
the solvent,
solvent, the
the following
following polynomial
polynomial approximation
approximation provides
provides aa good
good approximation
approximation to
to
activity
of
water
in
sulfuric
acid
aqueous
well-established
experimental
data
([23,24])
concerning
the
well-established experimental data ([23,24]) concerning the activity of water in sulfuric acid aqueous
0.4
≤1≤
solutions
<
0.4 
< xxxHH
1:: 1:
solutions for
for 0.4
2 O2 O
H2O
6
5
4
3
4+
4
xxH62 O+61699.80
 1699.80
xxH 2xOH32 O 3
γH2 O =
− 454.26
x H2 O
xHx HOO5 5 −2592.82
2592.82xxH HO2 O
2073.83
γHH2 O
= − 454.26
454.26
+ 2073.83
2073.83
H 2 O + 1699.80 x2H22 O − 2592.82 x H22 O
H 2O
2O
2
−919.55 xH222 O + 214.60 xH2 O − 20.61
−919.55
919.55 xxHH2 OO + 214.60
214.60 xxHH2 OO − 20.61
20.61
2
2
(A5)
(A5)
(A5)
This equation is plotted in Figure A2, together with the values of γoH2 O obtained from the data
reported in [24]. Equations (A4) and (A5) were used to determine the battery’s limit curve of the
example presented in Section 5.
ta the reader is referred
to [23,24].
On the basis of these data, Equation (A1) is assumed to be
Batteries 2016,
2, 36
proximated by the expression:
13 of 14
This equation(1.22
is bplotted
in Figure A2, together with the values of γ oH 2O obtained from the data
H 2 SO 4  6.5)
 H SO  2.12
for bH 2SO 4  ~ 8 mol / kg
(A4)
2
4
reported in [24]. Equations (A4) and (A5) were used to determine the battery’s limit curve of the
Batteries 2016, 2, 36
13 of 14
example
Section
presented graphically
in thepresented
followingin
Figure
A1.5.
Figure A1. Plot of Equation
(A4)
showingEquation
the dependence
of  H 2SO4 on sulfuric
acid molality
in
Figure
A2. Curve
aqueous
H 2 O for
Figure A2.
Curve of
of Equation (A5)
(A5) showing
showing γγ HH22OO as
as aa function
function of
of xxH
for sulphuric
sulphuric acid
acid aqueous
2O
oo
o
solutions.
Values
(J)
of
calculated
from
the
experimental
data
reviewed
in
Bullock
[24].
γ
are taken from Bullock [24].
aqueous solutions. Experimental
values (J) of γHH
solutions. Values
H2 O
O 4calculated from the experimental data reviewed in Bullock [24].
22SO
References
As for the solvent, the following polynomial approximation provides a good approximation to
Pavlov,
D. Lead-Acid
Science
Technology;
Elsevier:
Amsterdam,
The
1.
Pavlov,
Technology;
Elsevier:
Amsterdam,
The Netherlands,
Netherlands, 2011.
2011.
ll-established experimental
data
([23,24]) Batteries
concerning
theand
activity
of water
in sulfuric
acid aqueous
Zhao,
Y.;
Bostrom,
T.
Application
of
ionic
liquids
in
solar
cells
and
batteries:
A
review.
Curr.
Org.
Chem.
2015,
Zhao,
Y.;
Bostrom,
T.
Application
of
ionic
liquids
in
solar
cells
and
batteries:
A
review.
Curr.
Org. Chem.

1
utions for 0.4  xH 2.
:
2O
H
19, 556–566.
[CrossRef]
2015,
19, 556–566.
6
3 Compton, R.G. Effect of water
O’Mahony,
Silvester,
Aldous,
L.;
Hardacre,
C.;


454.26
x
 1699.80
x H 2 O 5 D.S.;
2592.82
x H 2 O 4L.;
 2073.83
x H 2 OC.;
3.
O’Mahony,
Silvester,
Aldous,
Hardacre,
Compton, R.G. Effect of water on the
H 2 O A.M.;
2O
(A5)
electrochemical window
and
potential
limits
of
room-temperature
ionic
liquids.
J. Chem.
Eng.
Data
2008,
53,
window and potential limits
of room-temperature
ionic
liquids.
J. Chem.
Eng.
Data
2008,
2
 919.55 x H O  214.60 x H O  20.61
2
2
2884–2891.
[CrossRef]
53,
2884–2891.
Rezaei, B.;
B.; Mallakpour,
Mallakpour, S.;
S.; Taki,
Taki, M.
M. Application
Application of
of ionic
ionic liquids
liquids as
as an
an electrolyte
electrolyte additive
additive on the
4.
Rezaei,
battery. J. Power Sources 2009, 187,
electrochemical behavior of lead acid battery.
187, 605–612.
605–612. [CrossRef]
5.
Rezaei, B.; Havakeshian,
ionic
liquids
as as
an an
electrolyte
additive
on the
Rezaei,
Havakeshian, E.;
E.;Hajipour,
Hajipour,A.R.
A.R.Influence
Influenceofofacidic
acidic
ionic
liquids
electrolyte
additive
on
electrochemical
and and
corrosion
behaviors
of lead-acid
battery.battery.
J. Solid J.State
Electrochem.
2011, 15,2011,
421–430.
the
electrochemical
corrosion
behaviors
of lead-acid
Solid
State Electrochem.
15,
[CrossRef]
421–430.
Tuphorn, H. Gelled-electrolyte batteries for electric vehicles.
6.
Tuphorn,
vehicles. J. Power Sources 1992, 40, 47–61. [CrossRef]
7.
Lambert,
D.W.H.;
Greenwood,
P.H.J.;
Reed,
M.C.
Advances in
in gelled-electrolyte
gelled-electrolyte technology
technology for
Lambert, D.W.H.; Greenwood, P.H.J.; Reed, M.C. Advances
valve-regulated lead-acid batteries. J. Power Sources 2002, 107, 173–179. [CrossRef]
Chen, M.Q.;
M.Q.; Chen,
Chen, H.Y.;
H.Y.; Shu,
Shu, D.; Li,
Li, A.J.; Finlow, D.E. Effects of preparation condition and particle size
8.
Chen,
silica gel
gel valve-regulated
valve-regulated lead-acid batteries performance. J.J. Power
Power Sources
Sources 2008,
2008, 181,
181,
distribution on fumed silica
161–171. [CrossRef]
9.
Tundorn, P.; Chailapakul, O.; Tantavichet, N. Polyaspartate as a gelled electrolyte additive to improve the
Tundorn,
performance ofofthethe
gel valve-regulated
lead-acid
batteriesbatteries
under 100%
depth
of discharge
anddischarge
partial-state-of
performance
gel valve-regulated
lead-acid
under
100%
depth of
and
charge conditions.
J. Solid
State Electrochem.
2016,
20, 801–811.
[CrossRef]
partial-state-of
charge
conditions.
J. Solid State
Electrochem.
2016,
20, 801–811.
10. Paglietti,
Paglietti, A.
A. Limits
Limits to
to anaerobic
anaerobic energy
energy and
and cytosolic
cytosolic concentration
concentration in
in the
the living
living cell.
cell. Phys.
Phys. Rev.
Rev. EE 2015,
2015, 92,
92.
10.
[CrossRef]
[PubMed]
doi:10.1103/physreve.92.052712.
11. Silbey,
Silbey, R.J.;
11.
R.J.; Alberty,
Alberty, R.A.;
R.A.; Bawendi,
Bawendi, M.G.
M.G. Physical
Physical Chemistry,
Chemistry, 4th
4th ed.;
ed.; Wiley:
Wiley:Hoboken,
Hoboken,NJ,
NJ,USA,
USA,2005.
2005.
12. Paglietti,
Paglietti, A.
A. Thermodynamic
Thermodynamic Limit
Limit to
to the
the Existence
Existence of
of Inanimate
and Living
Milan,
12.
Inanimate and
Living Systems;
Systems; Sepco-Acerten:
Sepco-Acerten: Milan,
Italy,
2014.
Italy, 2014.
13. Revie,
Revie, R.W.;
R.W.; Uhlig,
Uhlig, H.H.
H.H. Corrosion
Corrosionand
andCorrosion
CorrosionControl,
Control,4th
4thed.;
ed.;Wiley:
Wiley:Hoboken,
Hoboken,NJ,
NJ,USA,
USA,2008;
2008;Chapter
Chapter
13.
4.4.
14.
Jones,
D.A.
Principles
and
Prevention
of
Corrosion,
2nd
ed.;
Prentice
Hall:
Upper
Saddle
River,
NJ,
USA,
1996;
14. Jones, D.A. Principles and Prevention of Corrosion, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 1996;
Chapter 2.
2.
Chapter
15.
Crow,
D.R.
and Applications
Applications of
of Electrochemistry,
Electrochemistry, 2nd
2nd ed.;
ed.; Chapman
Chapman &
& Hall:
Hall: London,
London, UK,
UK, 1979;
1979;
15. Crow, D.R. Principles
Principles and
Chapter
8.
Chapter 8.
16. Swaddle,
Swaddle, T.W.
T.W. Inorganic
An Industrial
Industrial and
and Environmental
Environmental Perspective;
Perspective; Academic
Academic Press:
Press: San
San Diego,
Diego,
16.
Inorganic Chemistry:
Chemistry: An
CA, USA,
USA, 1997;
1997; Chapter
Chapter 15.
15.
CA,
17. Wessells, C.; Ruffo, R.; Huggins, R.A.; Cui, Y. Investigation of the electrochemical stability of aqueous
electrolytes for lithium battery applications. Electrochem. Solid State Lett. 2010, 13, A59–A61. [CrossRef]
18. Treptow, R.S. The lead-acid battery: Its voltage in theory and practice. J. Chem. Educ. 2002, 79, 334–338.
[CrossRef]
Batteries 2016, 2, 36
19.
20.
21.
22.
23.
24.
14 of 14
Berndt, D. Maintenance Free Batteries Based on Aqueous Electrolyte, 3rd ed.; Research Studies Press: Baldock,
UK, 2003.
Berndt, D. Electrochemical Energy Storage. In Battery Technology Handbook, 2nd ed.; Kiehne, H.A., Ed.;
Marcel Dekker: New York, NY, USA, 2003; Chapter 1, pp. 1–99.
Pinkwart, K.; Tübke, J. Thermodynamics and Mechanistics. In Handbook of Battery Materials, 2nd ed.;
Daniel, C., Besenhard, J.O., Eds.; Wiley-VCH: Weinheim, Germany, 2011; pp. 3–26.
Millet, P. Fundamentals of Water Electrolysis. In Hydrogen Production; Godula-Jopek, A., Ed.; Wiley-VCH:
Weinheim, Germany, 2015; pp. 33–62.
Robinson, R.A.; Stokes, R.H. Electrolyte Solutions, 2nd ed.; Butterworth: London, UK, 1959.
Bullock, K.R. The electromotive force of lead-acid cell and its half-cell potentials. J. Power Sources 1991, 35,
197–223. [CrossRef]
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