ISSN 1028�3358, Doklady Physics, 2014, Vol. 59, No. 5, pp. 206–208. © Pleiades Publishing, Ltd., 2014.Original Russian Text © G.R. Allakhverdov, 2014, published in Doklady Akademii Nauk, 2014, Vol. 456, No. 3, pp. 287–289.

206

The concentration (equivalent) electric conductiv�ity Λ of electrolyte solutions can be generally deter�mined as

(1)

where σ is the specific conductance and с is the sto�ichiometric concentration. If, with regard to the inter�actions in a solution, the latter quantity in Eq. (1) isreplaced by effective concentration c0 that reflects thereal number of charged particles in the solution, then onecan determine quantity Λ0 obviously related to Λ as

(2)

where the concentration ratio can be expressed, gen�erally speaking, in an arbitrary yet identical scale. Thisconcentration ratio can also be expressed as the ratiobetween the effective, N0, and the stoichiometric, N,numbers of charged particles in the solution, whichcan be presented, using the canonic distribution, as

(3)

Here, N0 is the effective number of noninteracting ions(therefore, the interaction energy ε should be assumedas zero) and Z is the partition function that reflects theionic interactions in a system, which lead to a formalchange in a number of charged particles, i.e., associa�tion and electrostatic interaction of ions [1]. Assuming

Λ σc���,=

Λ Λ0c0

c���,=

N0

N����� e ε/kT–

Z���������� 1

Z��.= =

these interactions to be quasi�independent, we canwrite

(4)

and, combining Eqs. (2)–(4), determine the equiva�lent conductivity

(5)

where ZA and ZE are the partition function that corre�spond to the association and electrostatic interactionof ions.

In the solutions of strong electrolytes, the ionicassociation is insignificant and accompanied, as a rule,by the formation of molecular complexes at the differ�ent ion ratio 1:1 [2]. The equilibrium concentrationconstant of this process can be determined as

(6)

where , , and are the equilibrium numbers offree cations, anions, and associates, respectively, thatare contained in solution volume V. For each ion, e.g.,cation, we can write the material balance equation

(7)

or, with regard to Eq. (6),

(8)

where Np is the total (stoichiometric) content of cat�ions in the solution.

Under the conditions of weak association, quan�

tity in Eq. (8) can be assumed equal to total num�ber Nn of anions and partition function Zp for cationscan be determined, in accordance with Eq. (3), as

Z ZAZE=

Λ Λ0ZA1– ZE

1–,=

KVNa

0

Np0Nn

0����������,=

Np0 Nn

0 Na0

Np Np0 Na

0+=

Np Np0 1 K

Nn0

V�����+⎝ ⎠

⎛ ⎞ ,=

Nn0

The Conductivity Equation for Solutions of Strong Electrolytes

G. R. AllakhverdovPresented by Academician Yu.V. Gulyaev October 7, 2013

Received October 24, 2013

Abstract—Based on the statistical model of solutions, an equation for calculating the conductivity of solu�tions of strong electrolytes in a wide concentration range is derived. The calculated values of the key empiricalparameter of the equation are compared with the results of the calculation within the Onsager theory.

DOI: 10.1134/S1028335814050097

State Scientific and Research Institute of Chemical Reagents and High Purity Chemical Substances, Bogorodskii val 3, Moscow, 107076 Russiae�mail: grant.alver@yandex.ru

PHYSICS

DOKLADY PHYSICS Vol. 59 No. 5 2014

THE CONDUCTIVITY EQUATION FOR SOLUTIONS OF STRONG ELECTROLYTES 207

(9)

Writing the analogous expression for statistical sum Zn

for anions and using the approximation ln(1 + x) ≈ xacceptable under these conditions, we can presentpartition function ZA for the association process in theform

(10)

Partition function Ze that corresponds to the elec�trostatic interaction of ions can be determined as [1]

(11)

where Аe is the free Helmholtz energy [3] related tothis interaction,

(12)

where L = is the Landau length, D is the permit�

tivity, εi are the integers that characterize ion chargesin terms of elementary charge е0, r is the effective ionicradius, and

(13)

is the inverse Debye length.

To take into account hydrodynamic (electro�phoretic and relaxation) effects during charge transferin the solution, we may introduce empirical coeffi�cient β that characterizes the correction to the electro�static interaction energy and determine the total valueof ZE:

(14)

In the extremely diluted solution at C → 0, assum�ing ZA = 1 and limiting the consideration in expandingthe exponent in Eq. (12) to the linear term, we canexpress equivalent conductivity Λ, according to Eq. (5),in the form

(15)

where Λ0 is a constant parameter that represents thelimiting equivalent conductivity of the electrolyte.

On the other hand, in diluted solutions the value ofΛ can be written, according to the Onsager theory [4], as

ZpNp

Np0

����� 1 KNn

V�����.+≈=

ZAlnNp

N����� Zpln

Nn

N����� Znln+ 2K

NpNn

VN����������.= =

ZelnAe

NkT���������,–=

Ae13��NkTL εpεn

e κr– 1–r

��������������,=

e02

DkT���������

κ 4πL εpεnNV���⎝ ⎠

⎛ ⎞1/2

=

ZEln β Zeln βAe

NkT���������.–= =

Λ Λ0e

βAe

NkT���������–

Λ0 1 13��βL εpεn κ–⎝ ⎠

⎛ ⎞ ,≈=

(16)

where F is the Faraday number, η is the solvent viscos�ity, and NA is the Avogadro number.

The comparison of Eqs. (15) and (16) yields

(17)

where quantity q in the approximation of equality ofthe limiting equivalent conductivities of ions that formthe electrolyte is

(18)

(19)

Substituting numerical values of the constants forwater solutions at 298 K in Eq. (19), we have L = 7.156 ×10–8 cm and λ = 38.62 cm2 Ω–1 eq–1.

The ratio in Eq. (13) is the bulk electrolyte con�

centration, which can be presented as = νC, where

С is the molarity of the electrolyte (mole/L) and ν isthe stoichiometric coefficient. Analogously, we candetermine the ionic components that enter Eq. (10):

= νpC, = νnC, ν = νp + νn.

Thus, determining the values of ZA and ZE by Eqs.(10) and (14), respectively, and using Eq. (5), we deter�mine the equivalent conductivity as

(20)

Λ = Λ013��L εpεn Λ0

q

1 q+�������������κ– F2

6πηNA

�������������� εp εn+( )κ,–

β q

1 q+������������� λ

Λ0

�����εp εn+εpεn

�����������������,+=

q2 εpεn

εp εn+( )2�����������������������,=

λ F2

2πηLNA

�����������������.=

NV���

NV���

Np

V�����

Nn

V�����

Λ Λ013��βL εpεn

e κr– 1–

r������������– 2

νpνn

ν��������KC–⎝ ⎠

⎛ ⎞ ,exp=

Table 1. Equivalent conductivity of NaCl in water at 298 K

С, mole/LΛ, cm2 Ω–1 eq–1

exp. [5] calc. I calc. II

0.0005 124.51 124.59 124.36

0.005 120.64 120.61 120.55

0.05 111.06 111.01 111.19

0.5 93.62 93.85 93.89

1.0 85.76 85.99 85.92

2.0 74.71 74.51 74.41

3.0 65.57 65.12 65.08

4.0 57.23 57.03 57.06

5.0 49.46 49.98 50.08

208

DOKLADY PHYSICS Vol. 59 No. 5 2014

ALLAKHVERDOV

where the numerical value of κ for water solutions at298 K is

κ = 0.233 × 108 (ν|εpεn|C)1/2 cm–1.

Equation (20) can be used to describe the conduc�tivity of strong electrolytes within two models: with theuse of four parameters Λ0, K, r, β (I) and with the useof the theoretical calculation of value β by Eq. (17), thusreducing (20) to the three�parameter equation (II).The results of the calculations for typical electrolytes(Tables 1, 2) demonstrate that both models yield goodcoincidence of the calculated and experimental datain wide concentration and conductivity ranges. At thesame time, the empirical values of parameter β appearhigher than the corresponding values calculated withthe use of Eq. (17). The use of empirical model I yieldsbetter coincidence of the calculation and experiment;

however, in all cases, the difference between theobtained Λ0 and the experimental data determined bythe extrapolation in diluted solutions [5] is about 1%.

REFERENCES

1. G. R. Allakhverdov, Doklady Physics 53 (8), 420(2008).

2. V. Majer and K. Stulik, Talanta 29 (2), 145 (1982).

3. G. R. Allakhverdov, Doklady Physics 57 (6), 221(2012).

4. L. Onsager, Phys. Z. 28, 277 (1927).

5. R. Robinson and R. Stokes, Electrolyte Solutions (But�terworth, London, 1959).

Translated by E. Bondareva

Table 2. Parameters of Eq. (20) in water solutions of electrolytes at 298 K

Electrolyte Λ0, cm2 Ω–1 eq–1 r, nm K, L/mole β ΔC, mole/L δ, %

NaCl I 126.65 (126.45) 0.832 0.132 0.932 0.0005–5.0 0.4

NaCl II 126.30 0.780 0.130 [0.905] 0.0005–5.0 0.4

КС1 I 150.13 (149.85) 0.822 0.054 0.881 0.0005–4.0 0.2

КС1 II 149.52 0.732 0.051 [0.810] 0.0005–4.0 0.2

ВаС12 I 140.10 (139.98) 0.847 0.226 0.732 0.0005–1.0 0.2

ВаС12 II 139.39 0.760 0.210 [0.683] 0.0005–1.0 0.2

LaCl3 I 146.10 (146.05) 0.948 0.373 0.659 0.000167–1.0 0.4

LaCl3 II 144.33 0.820 0.349 [0.589] 0.000167–1.0 0.4

ΔC is the domain of definition, and δ is the mean approximation error; parentheses give the experimental data from [5], and brackets give thevalues of β calculated by Eq. (17).