CC BY
0META^yPria
UDC|669.431.6:66.087.2:519.87
Candidate of chemical sciences Smirnova I. V., Bozhanova E. S. (DonSTU, Alchevsk, Ukraine)
MATHEMATICAL MODEL OF ELECTROREDUCTION OF METALS ON AMALGAM
ELECTRODES OF LIMITED VOLUME
A mathematical model of electrode processes in systems with amalgam electrodes of limited volume is considered. The influence of the concentration values of the supporting electrolyte solution (background) on the peak height of dissolution is researched.. The mechanism of the influence of the double electric layer on the kinetics of dissolution of metal amalgam is offered. Data obtained by mathematical model allow to calculate basic characteristics of amalgam extraction of valuable metals from metal-containing wastes.
Key words: mathematical model, amalgam electrodes of limited volume, double electric layer, metal-containing wastes.
A mathematical modeling of electrode processes in systems with amalgam electrodes of limited volume is performed in the research [1].
The scheme of amalgam electroextraction of a metal from low concentrated metal systems (wastes), dissolved in water, can be represented by:
t:
e.
> •
,n+
le)
where Me — volumetric body (metal
71 Jt £
ions in the bulk solution); — surface body — metal (ion of metal) on the surface of
the mercury foam; — kinetic constant of
surface body formation ; — kinetic
71 jff £
constant of separation from the surface
of the mercury foam; 31 — kinetic constant of metal reduction from the surface body M k
; vz — kinetic constant of metal tarnish-MaB k*
ing by surface body ; 1 — kinetic constant of metal reduction by free surface of
k*
electrode; 7 — kinetic constant of metal tarnishing by free surface of electrode;
fr* ir
Hereinafter ^ = H'a
This process is described by diffusion, surface and kinetic equations in the bulk solution and on the surface of the amalgam mercury-foam electrode and in its volume.
Dissolution of metal from amalgam is the basic stage of electroextraction of metals from aqueous multicomponent metal solution. Velocity of this stage (and of the whole process) depends first of all on scanning velocity of potential and kinetic constant of electroex-traction. Provided that, characteristic of metal nature is peak potential.
Theoretical correlation of peak potential and kinetic constant of electrochemical dissolution stage of metal dissolution from amalgam is suggested in the work [2]:
V =
RT fiZF
ln (ffl)
+ -
RT fiZF
■ ln
l fiZF
V ka
RT
(1)
. R
where ' ? — peak potential; — absolute
T Li
gas constant; — system temperature; ^ —
coefficient of charge transfer; ^ — body of
©Smirnova I. V., 2014 ©Bozhanova E. S, 2014
META^yPria
electrons taking part in electrode process; '' — Faraday constant; — scanning veloc-
r
ity of potential; J' — thickness of electrode mercury foam.
Peak potential increases linearly with the logarithm of scanning velocity of potential. According to this correlation, the greater kinetic constant (i.e. to a lower activation energy of the process) corresponds to a smaller peak potential (more negative), and vice versa.
In the study of the influence of supporting electrolyte concentration value (background) in solution on the height of dissolution peak [3] it was discovered that change in concentration of the background causes also the potential shift of the peak. Moreover, increase of background concentration results in decrease of peak potential, accelerating the process of metal dissolution.
Figure 1 shows the correlation of peaks potentials and natural logarithm of scanning velocity ln(^) for ions Zn2+, Cd2+ and Pb2+. Linear correlations confirm formula (1). From the slope of these curves the value ( can be estimated.
Such regularity at first sight seems paradoxical: background cations under the influence of the negative field of electrode concentrate at its surface, forming double electric layer (DEL), and must prevent metal ions yielding from amalgam.
Figure 1 — Correlation of peak potential and ln(«) for ions Zn2+ (1), Cd2+ (2) and Pb2+ (3).
In addition, the rise of coefficient of
charge transfer @ under increasing of background concentration is discovered.
In the description of DEL it is divided into two parts: a dense layer comprising solvated counterions (CI), almost closely drawn to the electrode surface, and diffuse layer, formed by ions, located at a greater distance from the electrode, than CI. In highly concentrated solutions (more than 0.1 M), diffusion layer is practically absent: all ions DEL are concentrated in dense part [4]. In this case, the electrochemical kinetics of metal ions yielding from amalgam into solution is affected only by CI of the dense layer.
Metal in amalgam can be represented in the form of ions Men+, which, like mercury ions, are immersed into electron gas environment. The process of metal dissolution from amalgam into solution is stepwise. At first ion diffuses in the volume of mercury to its boundary (Figure 2, a) and takes marginal position (Figure 2b). Thus, it loses some of the kinetic energy and acquires a certain potential (surface) energy [5], which increases its time on the amalgam surface. In this position, the separation of the ion of metal from mercury is prevented by the activation barrier of ionization, which is reduced with approaching of the electrode potential to the potential of metal dissolution in the process of inversion. Removal of ion Men+ from the surface of the amalgam occurs in the direction of reaction coordinate of its ionization. At decreasing of activation barrier ion Men+ may take more remote location from mercury atoms penetrating into the volume of solution. Such penetration leads to a positive micropotential causing local rearrangement of the dense layer CI: background cations replace anions (Figure 2, b, c). To implement such rearrangement it is necessary that the time of metal ion remaining on the surface of electrode in the process of diffusion drift was much more than the time of rearrangement of the CI content in the field of micropotential. Thus penetrated into a dense layer anions create in turn negative potential y (Figure 2, c), which reduces the activation barrier of separation of metal ion from the surface of the mercury electrode. Detachment occurs at a distance 5, proportional to the radius of the solvated ion (Figure 2, c).
META^yPria
a 6 b
Figure 2 — Stages of the dense part DEL rearrangement in the process of metal dissolution from amalgam
The timing parameters can be estimated as follows. Period between the acts of diffusion drift of metal in the volume of amalgam is
iWV
estimated as
sHg _
:6Db(Hg=T°eXp T
[5],
where D
Pb(Hg)
ts=T°exp
t) ■
where Ws — activation en-
ergy for the act of metal diffusion transition from the surface position to volumetric. The
correlation of ts and tv is:
L=exp
Ws-Wv
kT
Diminution of energies Ws -Wv is defined by the surface tension coefficient of mercury
<JHg : Ws -Wv=2k rpb«Hg, where rPb — ion radius Men+, and 2k rpb — area of the hemi-
sphere. Thus: —=exp
T v
^2k rpb«H\ ^ kT
—diffusion coefficient of
metal in amalgam, Wv — activation energy of drift act in the volume, k— Boltzmann constant, r0 — average period of the fluctuations in the position of temporary equilibrium, and %g —average distance between the positions of temporary equilibrium, defined by interatomic distance in amalgam. Value %g for
mercury is approximately 3-10_8cm according to the evaluation by the formula:
*h§=3N^H§H [5] where ^Hg and PHg —
molar mass and mercury density respectively, Na — Avogadro's constant.
The time of metal ion remaining on the surface of mercury is estimated as
Time of rearrangement rE of CI content in the field of micropotential can be estimated from the velocity of ion migration u in electric field with E intensity, connected by the relation: u=bE, where b — ion mobility [4]. E intensity of Coulomb field 2e of ion Men+ (e — elementary charge) is defined by the 2e e
formula: E=-2=-2, where s0 —
4n:s0sr 2xs0sr
electrical constant, s — relative dielectric constant of water, r — distance from Cou-
be
lomb field. Thus: u(r) =-2. Differential
2nssr
of ion displacement dr in time dt is equal to:
be
dr=u(r)dt=-2dt. Time of displacement
2nssr
at distance from rb to rf (required time rE) is defined by integral:
r o 2 o
TE = j nS]S dr = 3Ss0S(rf3" rb3) . Evaluation
rb
for the situation when background cation leaves the dense part of DEL passing the distance equal to the thickness of the dense layer h (wherein rb = h, rf = 2h ) leads to the ex-
META^yPria
pression: rE =1 ^bO ^' Value of h is usually
considered to be equal to the sum of the diameter of the solvent molecules (diameter of water molecule is equal to 0.276 nm [6]) and ion radius [7].
It is possible to describe the dependence of peak potential of metal dissolution from potential y numerically, using the correlations outlined in the work [2]. Within the frames of two proved approximations a kinetic equation for the concentration of CRed of reduced form of metal Red in the volume of amalgam was derived:
dCRed dç
_ k c
col 1
Red
(2)
dC,
Red _
k
dç col
Red
exp(rRT [ç-^(ç,Cf )])
. (3)
The second derivative of equation (3) can be written as:
dCRed_ k
dçÇ ci
dCRed rfë'iï dV
L dç +CRed'rRT1 Ç
. (4)
Since the analytical signal is proportional to
dCRed
the derivative
dç
peak potential çp can
d2C
be found on condition that-=0 . Substi-
dç2
tution in (4) the value of the derivative (3) and equating the result to zero determines the required formula for peak potential:
ç _pHri?®kMÇp'Q )
RT . f ln
1-
a-(çp,Cf dç /
Çp y
where: k — kinetic constant of ionization, rn — rate of potential inversion, l — thickness of the mercury-foam electrode (MFE). Adopted approximations: Red concentration is constant across the width of mercury-foam electrode; analytical signal is generated only by ionization reaction. In view of potential impact y(<,Cf), which may generally depend on < as well as on concentration of background electrolyte Cf, the expression (2) becomes [4]:
PZ^XBT k) l> PZF
To test the experimental data the task vof the certain type of a function y(<,Cf) is
needed. In the classical works on electrochemistry a theoretical derivation of the equation of the double layer is outlined and approximate formulas for potential y are given. Unless taking into account weak dependence of y on the electrode potential, then in the case of 1-1-valent of background electrolyte with a high concentration for potential, created by anions, the following approximate formula is valid:
W~Wo - Fln (Cf ):
(6)
where yo — invariable. In this case, the functional dependence of peak potential on the concentration of the background electrolyte is as follows:
çp(cf )_ jZF in(rRT4)—-F ln(cf)
. (7)
Thus, the proposed mechanism explains the influence of DEL on kinetics of dissolution of metal from amalgam, and mathematical model of dependence of the peak potential on the concentration of the background after its experimental confirmation will allow to calculate main characteristics of amalgam extraction of valuable metals from metal-containing wastes and evaluate ecological-economic efficiency of the use of metal-containing wastes as secondary raw materials.
References
1. Smirnova I. V. Modeling of electrode processes in electrochemical methods of analysis: Monograph. / I.V. Smirnova, V. V. Shelepenko, Z. P. Popovich. — Alchevsk : DonSTU 2010. — 145 p.
ISSN 2077-1738. Збгрник наукових праць ДонДТУ. 2014. № 2 (43)
МЕ ТАЛУРГ1Я
2. Shelepenko V. V. Approximation formula of symmetric anode analytical signal in stripping volt-ammetry with a mercury-foam electrode / V. V. Shelepenko, I. V. Smirnova, Z. P. Popovich // Ukr. chem. magazine. — 2010. — № 11. — P.17-22.
3. Identification of kinetics parameters of mass transfer processes in liquid multicomponent heterophase systems: Research report (final) / Donbass State Techn. University. — 173 GB; Number GR 0109U002510; Inv. № 0210U005439. — Alchevsk, 2010. — 254p.
4. Stromberg A. G., Semchenko D. P. Physical Chemistry. — Moscow: Higher School, 2001.—527p.
5. Bogoslovskiy S. V. Physical properties of gases and liquids: Teaching aid / SPbSUAI. St. Petersburg., 2001. — 73 p.
6. Chemical Encyclopedia: in 5 v: V. 5: Tryptophan - Yatrochemistry / Editorial Board: Zefirov N. S. (chief ed.) and others. — M. Great Russian Encyclopedia, 1998. — 783 p.
7. Salem R. R. Double layer theory. — M. : FIZMATLIT, 2003. — 104 p.
Recommended to printing by Doctor of Engineering Science, Prof Novokhatskiy A., Doctor of Engineering Science, Prof. of NMA Ukraine Boitchenko B.
Received on 18.06.14.
к.х.н. Смирнова И. В., Божанова Е. С. (ДонГТУ, г. Алчевск, Украина) МАТЕМАТИЧЕСКАЯ МОДЕЛЬ ЭЛЕКТРОВОССТАНОВЛЕНИЯ МЕТАЛЛОВ НА АМАЛЬГАМНЫХ ЭЛЕКТРОДАХ ОГРАНИЧЕННОГО ОБЪЕМА
Рассмотрена математическая модель электродных процессов в системах с амальгамными электродами ограниченного объема. Исследовано влияние величины концентрации раствора индифферентного электролита (фона) на высоту пика растворения. Предложен механизм влияния двойного электрического слоя на кинетику растворения метала из амальгамы. Полученные с помощью математической модели данные позволят рассчитать основные характеристики амальгамного извлечения ценных металлов из металлосодержащих отходов.
Ключевые слова: математическая модель, амальгамные электроды ограниченного объема, двойной электрический слой, металлосодержащие отходы.
к.х.н. Смирнова I. В., Божанова О. С. (ДонДТУ, м. Алчевськ, Украгна)
МАТЕМАТИЧНА МОДЕЛЬ ЕЛЕКТРОВ1ДНОВЛЕННЯ МЕТАЛ1В НА АМАЛЬГАМНИХ ЕЛЕКТРОДАХ ОБМЕЖЕНОГО ОБ'€МУ
Розглянута математична модель електродних процеав в системах з амальгамними елект-родами обмеженого об'ему. До^джено вплив величини концентрацИ' розчину тдиферентного електролту (фону) на висоту тку розчинення. Запропонований мехатзм впливу подвтного еле-ктричного шару на ктетику розчинення металу з амальгами. Отримат за допомогою матема-тичног моделi дан дозволять розрахувати основн характеристики амальгамного вилучення цтних металiв з металовмiсних вiдходiв.
Ключовi слова: математична модель, амальгамн електроди обмеженого об'ему, подвтний електричний шар, металовмiснi вiдходи.
