Modeling of Anodic Dissolution of Pb–Sb Alloys
in Chlorides Melts
Yu. P. Zaikov, P. A. Arkhipov, Yu. R. Khalimullina and A. P. Khramov
Institute of High-Temperature Electrochemistry, Ural Division, Russian Academy of
Sciences, Akademicheskaya str. 22, Yekaterinburg, 620219 Russia
e-mail: dir@ihte.uran.ru
INTRODUCTION
Anodic dissolution of lead–antimony alloys is mentioned in [1–4]. When
investigating the electrochemical separation of Pb–Sb alloys (10 wt % Pb) at
temperature from 973 to
1073 K in a KCl–NaCl eutectic melt containing 7 wt %
PbCl2, the authors of [1, 2] showed that, as the lead content in the anodic alloy
decreases from 10 to 0.03 wt %, the process has 100 % anodic current efficiency [2],
which decreases to 30–40 % if the Pb content is ≤0.03 wt % in the alloy [1]. During
the anodic dissolution of Pb–Sb alloys in the melt (mol %) 48PbCl2–35KCl–17NaCl
at T =773 K, the maximum polarization of the anode containing 0.7–46.0 wt % Pb at
the current density i= 0.5 A/cm2 is correspondingly 80–90 mV with respect to the lead
reference electrode [3, 4]. No other literature data on the dissolution of Pb–Sb alloys
are available.
In this work, we carried out measurement of the anodic polarization of lead–
antimony alloys at temperature 803 K in equimolar KCl–PbCl2 and modeling of
anodic dissolution of Pb–Sb alloys at this conditions were made.
EXPERIMENTAL
The main experimental data were obtained by the pulsed method using a
galvanostat, which formed calibrated (in amplitude) dc pulses from 10 mA to 10 A
with time intervals specified by a G5-56 pulse generator. The response of the cell
under study was recorded using an S9-8 oscilloscope. As additional detecting devices,
we used an APPA-109 multimeter and M-1104 pointer ammeter. The experiments
were performed in a quartz glass cell (Fig. 1), which was hermetically closed with
Teflon cap 2 with holes for electrodes and a thermocouple. Alundum crucible 4 was
placed onto special support 7made from refractory brick and arranged on the cell
bottom. Metallic lead 11, prepared electrolyte 9, reference electrode 10, working
electrode 6, and alundum case with thermocouple 3 were placed into the crucible. The
cell was sealed, evacuated, and filled with purified argon. Then the cell was placed
into a resistance furnace and heated to the specified temperature under an excess
pressure of argon.
The working electrode (anode) was a Pb–Sb alloy containing 30 mol % Sb.
Metallic lead placed on the bottom of the crucible played the role of an auxiliary
electrode. As the electrolyte for the auxiliary and working electrodes, we used a
mixture of potassium and lead chlorides. The measurements were performed with
respect to metallic lead 10, which was in contact with the melt of the same
composition. Electrolytes of the working electrode and reference electrode were
separated by a diaphragm made from the Gooch asbestos. The current lead to the
liquid metal electrodes was implemented using molybdenum rods 8 protected from
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contact with the melt by alundum tubes, the free end of which was closed with rubber
stoppers 1 in order to protect the cell’s seal.
1
A
r
2
8
3
4
5
9
10
6
11
7
Fig. 1. Layout of the electrolytic cell. (1) rubber stoppers, (2) teflon cap,
(3) thermocouple, (4) alundum crucible,(5) quartz test tube, (6) working electrode,
(7) support made from refractory brick, (8) current leads to electrodes, (9) electrolyte,
(10) reference electrode and (11) auxiliary electrode.
The setup was equipped with a system of automatic stabilization of
temperature, which was measured using Chromel–Alumel thermocouple 3.
Temperature oscillations during measurements were no larger than 2°C. To prepare
the electrolyte, we used reagents of analytical grade, which were additionally purified
and dehydrated by chlorination.
RESULTS AND DISCUSSION
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The anode processes on liquid metal alloys can be represented as follows. The
reduced form of metals is in the liquid alloy, while the oxidized form is on the
surface of the liquid metal electrode. In this case, the following electrode reactions
are possible:
Pb
Pb–Sb
– 2e ↔ Pb2+,
(1)
Sb
Pb–Sb
– 3e ↔ Sb2+.
(2)
Let us introduce the following notation for the concentrations: the lead and
antimony contents in the alloy bulk are
surface,
o
С Pb
and
o
С Sb
, respectively; on the alloy
s
s
o
o
С Pb
and С Sb ; ions in the electrolyte bulk, СPb2 and С Sb3 ; and near the
electrode, СPb2  and С Sb3 . Under equilibrium conditions, we have
s
s
o
s
С Pb
= С Pb ,
o
s
s
o
o
s
СPb
2  , С Sb = С Sb and С 3 = С 3 .
2 = С
Pb
Sb
Sb
The thermodynamic relation of the equilibrium potential (Ee) for this electrode
can be expressed through the lead and antimony electrodes:
Ep  E
o
Pb2  / Pb
Ep  E
o
o
С Pb
2  f
R T
Pb2 

ln
o
o
2 F
С Pb
  Pb
o
Sb3 / Sb
(3)
o
o
R  T С Sb3  f Sb3

ln
o
o
3 F
С Sb
  Sb
(4)
o
o
where EPb2 / Pb , ESb3 / Sb are the standard potentials of lead and antimony;
o
o
 Pb
,  Sb
are the activity ratios of lead and antimony in the alloy;
f Pbo 2 , f Sbo 3 are the activity ratios of the lead and antimony ions in the melt;
T is the temperature, K; F= 96 495 cal/mol; R= 8.31 J/(mol·K).
On switching on the dc electric current, a diffusion layer is formed both from
the electrolyte side  Pb2  ,  Sb3 and from the alloy side  Pb Sb . Under these
conditions, the concentrations of the reduced and oxidized forms of lead and antimony
near the electrode surface vary. For lead, the content of the reduced form decreases
s
o
o
s
С Pb
 C Pb
, and that of the oxidized form increases СPb2  CPb2 . For antimony,
both characteristics increase С Sb  C Sb and СSb3  CSb3 .
To analyze the obtained experimental data, let us calculate the theoretical
polarization curve of the anode made from the Pb–Sb alloy (30 mol % Sb).
Taking into account the fact that the activity ratios of lead ions vary
insignificantly in the concentration range under consideration [5], the steady-state
electrode potential (E) is expressed according to the equations
s
o
s
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o
EE

Pb
s
N Pb
2
R T

ln
,
s
2F
N Pb
(5)

Sb
s
N Sb
R T
3

ln
s
3 F
N Sb
(6)
EE


where E Pb , E Sb are the conventional standard potentials of
antimony [6]:
f 2
f 3
R T
R T

ln Pb , E Sb
 Eo 
ln Sb
2F
 Pb
3 F
 Sb

E Pb
 Eo 
lead and
(7)
s
s
where N Pb
are the ion fractions of their particles near the surface of
2 , N
Sb3
s
s
the Pb–Sb alloy; N Pb , N Sb are their molar reactions, respectively.
Hence, after some transformations, we obtain
s
 N Pb
 s
 N 2
 Pb
1
2



s
 N Sb
3


 Ns
 Sb
1
3

 
  exp  F E Pb

 ESb
 R  T



(8)
Taking into account that
N
C
(9)
V
where N is the ion (molar) fraction; V is the molar volume, cm3/mol; and C is the
molar volume concentration, mol/cm3; and making the notation 1 for Pb2+, 2 for Sb3+,
3 for Pb, and 4 for Sb, we write the expression
C

C

s
3
s
1
1
2
1
3
1
6
  С   Vall 
 F

 
  





exp
E

E
3
4
 R  T


 

  C   Vel 
s
2
s
4


(10)
where Vel , Vall are the molar volumes of the electrolyte and the alloy.
The flows of the mass transfer under conditions of steady-state diffusion can
be expressed by the equations
j1 
D1
C
s
1
 C10

1
D
j 2  2 C2s  C20 
2
D
j4  id C4s  C40 
4
1-68
(11)
(12)
(13)
where j1 and j2 are the fluxes of the lead and antimony ions from the surface of the
liquid metal electrode into the electrolyte bulk; D1 and D2 are the diffusivities of the
Pb and Sb ions in the electrolyte; 1 , 2 are the thicknesses of the Nernst diffusion
layer for the Pb and Sb particles from the electrolyte side; j4 is the interdiffusion flux
of metallic lead and antimony; Did is the interdiffusion coefficient; and  4 is the
thickness of the Nernst diffusion layer from the alloy side.
s
s
Since N 3  N 4  1 , we can write
C3s  C 4s 
1
Vall
(14)
0
0
The molar fraction of antimony particles in the alloy bulk is N 4  C 4  Vall .
Taking into account the notions of mass transfer through the Nernst diffusion layer,
we have
С 40  Vall 
j2  j4
j 2  j 4  j1
(15)
The rate of removal (ia) of the oxidized forms of lead and antimony from the
electrode surface follows the equation
ià  j1  2 F  j 2  3F
(16)
Let us rewrite eq. (5) taking into account (9):

E  E Pb

R  T  C1s Vel 

ln 

2  F  C3s Vall 
(17)
As a result of the above conclusions, we have the set (10)–(17) consisting of
eight equations with nine unknowns.
Let us express j1 from formula (15):
j1 
 1

j2  j4







j

j

j

j


1
2
4
2
4
 С 0 V

С40  Vall
 4 all

(18)
From (11) and (12), we derive the expression for the concentration of the lead
and antimony ions at the surface of the Nernst diffusion layer:
С1s  j1 
С2s  j2 
1
 1
 
 C10 , C1s   j 2  j 4    0
 1  1  C10
D1
 C 4  Vall
 D1
2
D2
 C20
(19)
(20)
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Combining expressions (13) and (14), we have
С 4s  j 4 
С3s 
4
Did
 C 40
(21)

1
1
 С4s 
 j 4  4  C40
Vall
Vall
Did
(22)
After a series of transformations of eq. (10), we write


0 
 j2  2  C 2 
D2



4
0 
 j4  D  C 4 
id


1
3



1
 j4  4  C40


Vall
Did





 1
 
 1  1  С10 
  j2  j4    0
 C4  Vall  D1


1
2
1
6
V 
 F


  all   exp 
 E Pb
 E Sb 
 R T

 Vel 


(23)
Equation (23) gives the relation between two unknown quantities: f2 = f(j4).
Let us substitute D1 = 2 10–5 cm2/s [5–7], D2 = 10–5 cm2/s, Did = 3 10–5 cm2/s, C10 =
0.013 mol/cm3, С 20 = 2.44 10–8 mol/cm3, С 40 =0.016 mol/cm3, Vall = 18.8÷19.6
cm3/mol, Vel = 37.8÷51.4 cm3/mol, F = 96 500 cal/mol, R = 8.31 J/(mol· K), T = 803


 E Sb
K, E Pb
= 0.279 V into this expression. Using the step-by-step method and
varying the kinetic parameters
1  2
,
and
D1 D 2
4
, we can calculate the flux of
Did
antimony ions from the surface of the liquid metal electrode into the electrolyte bulk
at a specified interdiffusion flux.
The current density was calculated by the equality derived by the substitution
of the expression for j1 into formula (16):

 1

i  2  F   j 2  j 4    0
 1  j 2  3  F

 C 4  Vall

(24)
The equation for the steady-state potential is obtained by the substitution of the
corresponding expressions into (17):

E  E Pb


 1
 1
0




j

j


1


С
 2

4
1
 C 0 V
 D
Vel 
R T 
1
4
all



ln

4
1
2 F 
Vall 
0

j


C


4
4
Vall
Did


(25)
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
The value of E Pb is 0.012 V at T = 803 K [7]. Then we took a definite
number of pairs of values of j2 and j4, and for each of them, we found the current
density by equality (24) and the potential by (25).
Figure 2 represents the experimental data and theoretical polarization curve of
the Pb–Sb alloy (30 mol % Sb) at T = 803 K. It is evident from the polarization curves
that, in the studied range i = 0.001–3.7 A/cm2, the theoretical and experimental points
fit one line in the error limits ± 0.002 V. The calculations show that the parameters
1

and 2 remain constant, namely, 430 and 1500 s/cm, respectively. Taking into
D1
D2
account that the diffusivities of lead ions in chloride melts are 2 10–5 cm2/s [8, 9], the
diffusion layer is 0.0086 cm. If we take the diffusivity of antimony ions to be 10–5
cm2/s, the thickness of the diffusion layer is 0.015 cm. The parameter
4
varies
Did
from 2000 to 13 000 s/cm in the course of the numerical selection. Taking into
account the fact that, as the antimony content increases, its diffusivity in the alloy
decreases [10], the interdiffusion coefficient also decreases. At low densities of the
polarizing current, this coefficient is
3 10–5 cm2/s [10]. The thickness of the
diffusion layer is 0.06 cm. If we assume that this characteristic varies insignificantly
as the anodic current density increases, then, during the dissolution of the Pb–Sb
alloys, the interdiffusion coefficient decreases by an order of magnitude.
Figure 3 represents the dependence of the calculated concentrations of metallic
lead and its ions near the metal–salt phase interface as a function of the current
density. As i increases, the calculated Pb content on the surface layer of the alloy
decreases, while the Pb2+ concentration in the electrolyte near the electrode increases.
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Fig. 2. Anodic polarization curve of the
Pb–Sb alloy (30 mol % Sb) at T = 803
K. Symbols correspond to the
experiment, and the curve corresponds
to the calculation.
Fig. 3. Calculated lead concentration
versus current density.
i: (1) 0.04, (2) 0.14, (3) 0.25, (4) 0.46, (5)
0.74, (6) 0.90, (7) 1.24 and (8) 2.09
A/cm2.
According to the phase diagram of the Pb–Sb system [11], the liquids’ point at
T = 803 K corresponds to 30 mol % Pb. Calculations show that this lead concentration
occurs at i = 0.7 A/cm2. This circumstance indicates that a thin solid layer is formed
on the surface of the liquid metal. The diffusion rate of metals in solids is
considerably lower than in liquids. Consequently, a deficit of the electronegative
component is formed in the reaction region, while the antimony concentration at the
metal–electrolyte interface increases, which can lead to the dissolution of Sb by
reaction (2). According to the above reasoning, dissolution of the Pb–Sb alloys in the
50PbCl2–50KCl melt in a temperature 803 K proceeds under conditions of diffusion
kinetics with a limiting stage of mass transfer from the alloy side.
CONCLUSIONS
The theoretical anodic polarization curve for the Pb–Sb alloy (30 mol % Sb) is
calculated, and the dependence of the surface concentrations of lead and its ions on
the anodic current density is shown. It is established that the limiting stage is the mass
transfer in the Pb–Sb liquid metal system.
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