The limit solubility relationships based on Frenkel’s heterophase fluctuations theory
For a weak one-component solution, in which the solvent and dissolved substances are not chemically bound, a simple analytical approximation was found for temperature dependence of limiting solubility of the dissolve substance. The derivation of the approximation is based on the Frenkel theory of he...
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Shapovalov, R.V. Osmayev, O.A. 2016-10-14T10:34:40Z 2016-10-14T10:34:40Z 2012 The limit solubility relationships based on Frenkel’s heterophase fluctuations theory / R.V. Shapovalov, O.A. Osmayev // Вопросы атомной науки и техники. — 2012. — № 1. — С. 273-276. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 05.10.Gg, 02.60.Ed, 64.60.A– https://nasplib.isofts.kiev.ua/handle/123456789/107156 For a weak one-component solution, in which the solvent and dissolved substances are not chemically bound, a simple analytical approximation was found for temperature dependence of limiting solubility of the dissolve substance. The derivation of the approximation is based on the Frenkel theory of heterophase fluctuations. It was shown how the parameters of this analytical approximation can be related to experimental data. Для слабого однокомпонентного раствора, в котором растворитель и примесь не образуют химического соединения, получено аналитическое приближение, позволяющее найти предельное количество растворенного вещества в зависимости от температуры. Вывод приближения основан на теории гетерофазных флуктуаций Френкеля. Показано, как параметры аналитического выражения могут быть связаны с экспериментальными данными. Для слабкого однокомпонентного розчину, у якому розчинник та домішка не створюють хімічної сполуки, отримано аналітичне наближення, яке дозволяє знайти граничну кількість розчиненої речовини в залежності від температури. Наближення знайдено на базі теорії гетерофазних флуктуацій Френкєля. Показано, як параметри аналітичного співвідношення можуть бути пов’язані з даними експериментів. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases The limit solubility relationships based on Frenkel’s heterophase fluctuations theory Cоотношение для предельной растворимости, полученное на основе теории гетерофазных флуктуаций Френкеля Співвідношення для граничної розчинністі, отримане на основі теорії гетерофазних флуктуацій Френкеля Article published earlier |
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The limit solubility relationships based on Frenkel’s heterophase fluctuations theory |
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The limit solubility relationships based on Frenkel’s heterophase fluctuations theory Shapovalov, R.V. Osmayev, O.A. Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
| title_short |
The limit solubility relationships based on Frenkel’s heterophase fluctuations theory |
| title_full |
The limit solubility relationships based on Frenkel’s heterophase fluctuations theory |
| title_fullStr |
The limit solubility relationships based on Frenkel’s heterophase fluctuations theory |
| title_full_unstemmed |
The limit solubility relationships based on Frenkel’s heterophase fluctuations theory |
| title_sort |
limit solubility relationships based on frenkel’s heterophase fluctuations theory |
| author |
Shapovalov, R.V. Osmayev, O.A. |
| author_facet |
Shapovalov, R.V. Osmayev, O.A. |
| topic |
Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
| topic_facet |
Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
| publishDate |
2012 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Cоотношение для предельной растворимости, полученное на основе теории гетерофазных флуктуаций Френкеля Співвідношення для граничної розчинністі, отримане на основі теорії гетерофазних флуктуацій Френкеля |
| description |
For a weak one-component solution, in which the solvent and dissolved substances are not chemically bound, a simple analytical approximation was found for temperature dependence of limiting solubility of the dissolve substance. The derivation of the approximation is based on the Frenkel theory of heterophase fluctuations. It was shown how the parameters of this analytical approximation can be related to experimental data.
Для слабого однокомпонентного раствора, в котором растворитель и примесь не образуют химического соединения, получено аналитическое приближение, позволяющее найти предельное количество растворенного вещества в зависимости от температуры. Вывод приближения основан на теории гетерофазных флуктуаций Френкеля. Показано, как параметры аналитического выражения могут быть связаны с экспериментальными данными.
Для слабкого однокомпонентного розчину, у якому розчинник та домішка не створюють хімічної сполуки, отримано аналітичне наближення, яке дозволяє знайти граничну кількість розчиненої речовини в залежності від температури. Наближення знайдено на базі теорії гетерофазних флуктуацій Френкєля. Показано, як параметри аналітичного співвідношення можуть бути пов’язані з даними експериментів.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/107156 |
| citation_txt |
The limit solubility relationships based on Frenkel’s heterophase fluctuations theory / R.V. Shapovalov, O.A. Osmayev // Вопросы атомной науки и техники. — 2012. — № 1. — С. 273-276. — Бібліогр.: 5 назв. — англ. |
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2025-11-26T11:41:53Z |
| last_indexed |
2025-11-26T11:41:53Z |
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| fulltext |
THE LIMIT SOLUBILITY RELATIONSHIPS BASED ON
FRENKEL’S HETEROPHASE FLUCTUATIONS THEORY
R.V. Shapovalov 1 ∗and O.A. Osmayev 2,1
1National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
2Ukrainian State Academy of Railway Transport, 61050, Kharkov, Ukraine
(Received October 30, 2011)
For a weak one-component solution, in which the solvent and dissolved substances are not chemically bound, a simple
analytical approximation was found for temperature dependence of limiting solubility of the dissolve substance. The
derivation of the approximation is based on the Frenkel theory of heterophase fluctuations. It was shown how the
parameters of this analytical approximation can be related to experimental data.
PACS: 05.10.Gg, 02.60.Ed, 64.60.A–
1. INTRODUCTION
It is known that generally the quantity of solute which
can be dissolved in a solvent under given tempera-
ture is limited. As a rule the solvent capability to
dissolve grows together with temperature. The sys-
tem, where solute fraction is less or equals the cor-
responding limit, is in equilibrium state under un-
changeable environment. In 1939 it was pointed out
by Frenkel [1] that in general case in the equilibrium
solution along with monomers there are clusters of
two and more solute atoms. Starting from the above
mentioned paper these clusters are considered as het-
erophase fluctuations. The cluster’s size distribution
function was obtained in paper [1] from the principle
of thermodynamic potential minimum. In paper [2]
much more simple way was proposed for obtaining the
distribution function. The method was based on the
stationary solution of the system of Becker-Doring
equations. In paper [2] it was demonstrated also a
causal relationship between heterophase fluctuations
and the characteristic times of successive stages of
nucleation in a supersaturated solution. The method
of study was numerical solution of the Fokker-Plank
equation.
In the present work the authors attempt to carry
the Frenkel’s idea of heterophase fluctuations through
to its logical completion. We choose the simplest kind
of such systems – a weak perfect single component
solution where solute and solvent do not react chem-
ically. This simplest system is still a good approach
to e.g. inorganic water dissolved salts.
2. BASIC EQUATIONS
It is known that the time evolution of solute clusters
size distribution function is described by a system of
Becker-Döring kinetic equations
∂f (x, t)
∂t
= I (x− 1, t) − I (x, t) , x ≥ 1 (1)
I (x, t) = W (x, x+ 1) f (x, t)
−W (x+ 1, x) f (x+ 1, t) .
(2)
Here the value f (x, t) is the number of clusters con-
sisting of x monomers normalized to the number of
atoms/molecules of solvent in unit volume in the
present moment of time t, I (x, t) is the cluster flux
in cluster size space, W (x, x+ 1) is the rate of ad-
sorption of monomers by solute cluster consisting of
x monomers, W (x+ 1, x) is the rate of desorption of
monomers off the solute cluster consisting of x + 1
monomers.
Equation (1) takes into account the fact of poly-
mer cluster immobility in physical space (at any case
the cluster mobility has to be much less than the
monomer mobility) and their interaction only by ad-
sorption or emission of mobile monomers. At the
same time the probability of an elementary process
with two and more monomers is negligibly small.
This is an analogy of the mean field approximation.
Analytical form of transient rates is given as a rule
by the following expressions
W (x, x+ 1) =
3D
a2
x1/3c (t) ,
W (x, x− 1) =
3D
a2
cex
1/3 exp
(
β1 · x−1/3
T
)
.
(3)
Here D is the diffusion coefficient of monomer in so-
lution, a is the lattice parameter of a cluster or, that
is the same, of a solute matter in parent phase, c (t)
is the monomers concentration, ce is it equilibrium
value, β1 =
8πa2γ
3 kB
is the cluster surface energy in
∗Corresponding author E-mail address: r v shapo@kipt.kharkov.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 273-276.
273
temperature units, γ is the specific surface energy,
kB is Boltzmann constant, and T is the temperature.
There is an evident and conventional approach to
the study and solution of equation (1). It consists in
the transformation of discrete variable into its analog
quantity and solving the corresponding Fokker-Plank
equation thus obtained from equation (1). The par-
abolic differential equation can be solved analytically
only approximately for some initial conditions and
certain supplementary considerations. The process of
its treatment becomes much more complicated if the
solute conservation law is taken into account. In that
case one can use numerical methods. It is also known
that studying relatively small time interval one needs
no conservation law and that the asymptotic solution
results from arbitrary physical reliable initial condi-
tions [3].
3. EQUILIBRIUM DISTRIBUTION
It is evident that the above equations can be ap-
plied to the stable state of the system. In this case the
initial conditions are at once the solution of equation
(1). If one considers a liquid in condition of limiting
saturation then there is no need in conservation law.
Actually the most evident way to obtain the satu-
rated solution is to bring into contact some quantity
of solvent with an a priori excess of solute substance.
It is evident that some time later the system comes
to equilibrium state. Due to excess in advance of sol-
vent the equilibrium system will consist of totally sat-
urated liquid solution being in thermodynamic bal-
ance with the residuum of solute substance. Without
loss of generality let us consider a flat interface of a
cluster. For weak perfect solutions it yields
ψs + T ln ce = ψb .
Therefore
ce = exp (−Δψ/T ) , (4)
where ψs is the specific thermodynamic potential of
solute in liquid and ψb is the specific thermodynamic
potential of solute in parent phase. The difference
Δψ = ψs − ψb is the variation of specific thermody-
namic potential under dilution.
Equations (1) with fluxes (2) have an evident im-
plication for equilibrium state. It is a coupling be-
tween distribution function values for different cluster
sizes,
f (x+ 1) =
W (x, x + 1)
W (x+ 1, x)
f (x) .
By means of the explicit form of transient rates (3)
some evident conclusions from this equation can be
received. First a coupling of distribution functions
for two values, x+1 and y respectively can be estab-
lished, and after simple transformation one obtains
the expression for distribution function of arbitrary
size expressed in terms of the stationary monomers
concentration c
f (x+ 1) =
W (x, x+ 1)
W (x+ 1, x)
f (x) =
k=x−y∏
k=0
(x− k)1/3
k=x−y∏
k=0
(x− k + 1)1/3 exp
(
β1
T
(x− k + 1)−1/3
)
(
c
ce
)x−y+1
f (y),
f (x+ 1) = exp
(
−β1
T
x+1∑
k=3
k−1/3
)
21/3
(x+ 1)1/3
(
c
ce
)x−1
W (1, 2)
W (2, 1)
f (1).
A probability of association of monomers into
dimers can have a different form in general case. Let
us suppose that
W (1, 2) =
3D
a2
· B · c (t) , (5)
where value B is a coefficient which defines an effi-
ciency of monomer collision. The coefficient modifies
the rate of monomer association. In paper [2] it was
accepted B = 2. In the present paper its numerical
value is not considered.
W (1, 2)
W (2, 1)
=
3D
a2
·B · c
3D
a2
ce · 21/3 · exp
(
β1
T
· 2−1/3
)
=
c
ce
2−1/3 · B · exp
(
−β1
T
· 2−1/3
)
.
Since f (1) ≡ c (t), then
f (x) = exp
(
−β1
T
x∑
k=2
k−1/3
)
Bce
x1/3
rx , (6)
where r = c/ce is a ratio of monomer concentration
to its equilibrium value. Since in the case of limiting
saturation c (t) = ce, therefore, cluster distribution
function can be written in the form
f (x) = exp
(
−β1
T
x∑
k=2
k−1/3
)
Bce
x1/3
. (7)
Total value of solute q (T ) is defined either as the
number of impurity atoms/molecules relative to the
number of solvent atoms/molecules or as volume spe-
274
cific value. It is defined by the next expression
q (T ) =
∞∑
x=1
x · f (x) . (8)
As it follows from equations (6) and (8) the limit sat-
uration is given by
qlim (T )= ce
[
1+B
∞∑
x=2
x2/3 exp
(
−β1
T
x∑
2
k−1/3
)]
.
(9)
Just the same expression was obtained in papers
[1] and [2]. Strictly speaking the main purpose of the
present paper is to transform relationship (9) into
more convenient form and to show the association of
its parameters with experimental data.
4. APPROXIMATION
Let us come over to summation of heterophase fluc-
tuation. It is necessary to find out a sum of the next
series
S (β) =
∞∑
x=2
x2/3rx exp
(
−β
x∑
2
k−1/3
)
. (10)
Here the ratio β = β1/T is a dimensionless coefficient.
The method of summation used for calculation
of the sum consists in step-by-step application of
Euler-MacLoren formula. As it is known [4] for ar-
bitrary function, thrice differentiable on a segment
M ≤ x ≤ N F (x), the next relationships is valid
i=N∑
i=M
F (i) = F (M) +
F (N) − F (M)
2
+
∫ N
M
F (x) dx+
F ′ (N) − F ′ (M)
12
+R3 , (11)
R3 ≤ 1
120
∫ N
M
|F ′′′ (x)|dx , (12)
x∑
2
k−1/3 = a (x) = 3
x2/3 − 22/3
2
+
x−1/3 + 2−1/3
2
− x−4/3 − 2−4/3
36
+ δ1 (x) , (13)
where R3 is the reminder.
The obtained relationship (13) shows that the
sought-for series (10) is convergent under condition
r ≤ 1. The limit saturation corresponds to the case
r = 1. Further argumentation is related to this case.
By means of formula (13) it is easy to show that the
absolute error is δ1 < 3 · 10−3. At the same time
one can expect that obtained formula is too compli-
cated if it will be repeatedly used in Euler-MacLoren
form. It can be shown the penultimate item in ex-
pression (13) is not negative and its absolute value
could not exceed 1/90. Taking this fact into account
one obtains an approximate reduced formula provid-
ing a good accuracy.
a (x) ≈ 3
x2/3 − 22/3
2
+
x−1/3 + 2−1/3
2
.
In that way expression (10) can be written with
a good accuracy in the next form
S (β) ≈ S̃ (β) exp
(
β · 5 · 2−4/3
)
, (14)
S̃ (β) =
∞∑
x=2
x2/3 exp
(
−β 3x2/3 + x−1/3
2
− λx
)
.
Here value λ = − ln r = ln ce/c is nonnegative one. In
the received expression slowly increasing power func-
tion is multiplied by fast decreasing exponent. Let
us transform formula (14) in the same way as (10)
but with some simplifications. In expression (11) one
keeps only two first terms and retains first term in
(14) with unchangeable form to compensate inaccu-
racy. In that way one gets
S̃ (β) ≈ 22/3 exp
(−β · 7 · 2−4/3 − λ · 2)+
+
32/3
2
exp
(
−β · 5 · 3−1/3 − λ · 3
)
+
+
∫ ∞
3
x2/3 exp
(
−β · 3x2/3 + x−1/3
2
− λ · x
)
dx.
The integral in the obtained relationship plays the
main role in the region of small values β. For region
β ≤ 10 one can use a good approximation for involved
integral in the next form
I (β) ≈ 3 exp
(
−3−1/3β
2
)
(15)
×
∫ ∞
31/3
z4 exp
(
−3z2β
2
− λ · z3
)
dz.
As the purpose of the work is an investigation of
the case of the limit saturated solution it is sufficient
to find out the value of expression (14) subject to con-
ditions that r = 1 → λ = 0. With that the right side
of (15) could be calculated in accordance with the
exact formula. This calculation allows us to obtain
the unknown sum S (β).∫ ∞
zmin
z4 exp
(−b · z2
)
dz
=
zmin
(
3 + 2bz2
min
)
4b2
e(−bz2
min)+
3
√
π
8b5/2
erfc
(
b1/2zmin
)
,
S (β) ≈ 22/3 exp
(
−2−1/3β
)
(16)
+
2 · 31/3+ 6β+ 32/3β2
2β2
exp
(
−5β
(
3−1/3 − 2−4/3
))
+
π1/2
61/2β5/2
exp
(
5β
24/3
− 3−1/35β
2
)
erfc
(
35/6β1/2
21/2
)
.
It is rather difficult to get an analytical estimate of
the error of this expression, but numerical calcula-
tion has shown that in a segment 10−2 ≤ β ≤ 102
the given formula provides an accuracy not less than
96%, minimal one has place at β ≈ 0.49. The calcu-
lations assumed that the upper limit of the sum 10
275
is 108. It can be shown that in this range of β this
value of upper limit provides sufficient accuracy.
The above calculations allow to write down the
maximum concentration of the equilibrium solution
at given temperature in the following simple form
qlim (T ) = ce [1 +B · S (β)] . (17)
For example, consider a water solution of nickel
nitrate Ni (NO3)2, it has a molecular mass 182.
The limit concentration of Ni (NO3)2 in aqueous
ammonia [5]
T, K 273 293 298 303 313
q·10−2 7.84 9.326 9.9 10.425 11.76
T, K 323 333 358.5 373
q·10−2 13.78 15.61 20.295 22.275
The fitting curve for data in the table
Using data from the table one can construct the
fitting curve by means of Levenberg-Marquardt algo-
rithm.
The figure illustrates the fact that the equation
(17) quite satisfactory describes the dependence of
limit solubility on temperature.
5. CONCLUSIONS
1. In the framework of a simple thermodynamic
model an analytical expression is obtained
which allows to determine the maximal quan-
tity of solute that can be dissolved in a solvent
as a function of temperature.
2. By an example of a water solution of nickel ni-
trate it is shown a good agreement of the found
approximation with experimental data.
References
1. J.I. Frenkel. A general theory of heterophase
fluctuations and pretransition phenomena // J.
Chem. Phys. 1939, v. 7, p. 538-547.
2. A.A. Turkin, A.S. Bakai. Modeling the precipita-
tion kinetics in systems with strong heterophase
fluctuations // Problems of Atomic Science and
Technology. 2007, iss. 3(2), p. 394-398.
3. I.M. Lifshitz, V.V. Slyozov. The kinetics of pre-
cipitation from supersaturated solid solutions //
J. Phys. Chem. Solids. 1961, v. 19, N 1-2, p. 35-
50.
4. Frank W.J. Olver. Asymptotic and Special Func-
tions. New York: “Academic Press”, 1974.
p. 279-289.
5. V.A. Rabinovich, Z.Y. Havin. An Abriged Chemi-
cal Reference Book. Leningrad: “Khimiya”, 1978,
p. 88 (in Russian).
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