Derivation of a formula for diffusion coefficients

case of NaCl (z₊=z₋= 1), the experimental coefficients are [JUS-97]: a_r equal to 0.4 nm, b equal to 0.055 l/mol, while the temperature dependent constants A and B are respectively 0.50925 l^1/2/mol^1/2 and 3.2864 l^1/2/mol^1/2.nm for water at 25°C.

Consequently γ_± is ranging from 0.68 to 1.00 with a minimum at a concentration of about 1 M.

2.3.1.2.1 Four different approaches for the derivation of D_j

a) Approach I

Considering only the first gradient in the absence of an electric field (∂φ_e/∂ =x 0in eq.

(2.26) ), this expression corresponds to Fick’s first law:

, , ,

, ,

ln _{s j s j s j}

s j j s j j j

C C C

q k C RT k RT D

x x x

∂ ∂ ∂

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= ⎜ ⎟= ⎜ ⎟= ⎜ ⎟

∂ ∂ ∂

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ^(2.27)

where the introduced diffusion coefficient is defined as:

j j

D = k RT (2.28)

b) Approach II

In a second approach we assume ∂C_s/∂ =x 0 in equation (2.26), and it means that the flux is expressed only as migration. The passage of charge i_cunder the influence of an electrical field becomes:

, , ,

2 2

, ,

e j e

s j j s j j s j j

j e

j s j c s j j

q k C z F C D z F

x RT x

z Fq i C D z F

RT x

φ φ

φ

⎛ ⎞

∂ ∂

⎛ ⎞ ⎛ ⎞

= ⎜⎝ ∂ ⎟⎠= ⎜⎝ ⎟⎠ ⎜⎝ ∂ ⎟⎠

⎛ ⎞ ⎛∂ ⎞

⇒ = = ⎜⎝ ⎟⎠ ⎜⎝ ∂ ⎟⎠

(2.29)

Next to this the conductivity can be expressed in the form κ_c = /l RΩ (S/m) with l the length (m), R the corresponding resistance (Ω) and Ω the area (m²) of the portion of solution studied. Substitution of Ohm’s law R = /E I , with Iis the current (A) and E the potential (V), gives:

c c

e

i I

E l x

κ φ

= Ω=

⎛∂ ⎞

⎜ ∂ ⎟

⎝ ⎠

(2.30)

So, the conductivity can be described as the ratio of current density to the potential gradient. By inserting the expression for i_c (equation (2.29)) into the equation of conductivity κ_j(equation (2.30)), we can write the molar conductivity λ_{c j,} as

2 2

, ,

,

c j j j

c j s j

D z F

C RT

λ =κ = (2.31)

Out of which the Nernst-Einstein expression for the diffusion coefficient can be derived:

,

2 2

c j j

j

D RT z F

= λ (2.32)

c) Approach III

Next the relation between the diffusion coefficient and mobility will be derived. The mobility uis the proportionality constant between the velocity and the field strengthE:

v=uE (2.33)

If migration of ions induced by an applied velocity field (2.33) exactly balances the diffusion the expression becomes:

,

, 0

s j

i s j j

D C C u E

x

∂

⎛ ⎞

− =

⎜ ∂ ⎟

⎝ ⎠ ^(2.34)

Using the Boltzmann distribution law

E

e⁻RT, and knowing that an ion in solution with a velocity v experiences a force zFE per mole, from an electric field of strength E and taking into account d

( )

^lnc 1dc

dx = c dx in equation (2.34), we get:

j

j j

D z F E u E RT

⎛ ⎞

⎜ ⎟ =

⎝ ⎠ ^(2.35)

Finally we get the Einstein equation for the diffusion coefficient in function of the mobility:

j j

j

D u RT

= z F (2.36)

d) Approach IV

The mobility and thus the diffusion constant may be related to the viscosity of the solvent medium. The viscous force on a single solvated ion of radius r_j under velocity v_jmay be set equal to the diffusion force experienced by that ion:

~

6 _{j j} 1 ^j

A

r v N x

π η μ

⎛∂ ⎞

⎜ ⎟

= ⎜⎜ ∂ ⎟⎟

⎝ ⎠

(2.37)

Substituting this expression in equation (2.24), taking into account k_j = D_j/ RT, yields:

~

j j 6

j j A

j

v RT r v N

x D

μ π η

⎛∂ ⎞

⎜ ⎟ = =

⎜ ∂ ⎟

⎜ ⎟

⎝ ⎠

(2.38)

The Stokes-Einstein equation for the diffusion coefficient related to the viscosity η becomes:

6

j

j j A

D RT

r v N π η

= (2.39)

2.3.1.2.2 Values for the ion versus molecular diffusion coefficient

The more fundamental approach, starting from the chemical potential of the chemical species as the driving gradient, resulted in four expressions for D_jin the previous paragraph:

- eq. (2.28) in absence of the electric field

- eq. (2.32) taking into account migration ( Nernst-Einstein expression) - eq. (2.36) using the mobility

- eq. (2.39) taking into account the viscosity (Stokes-Einstein expression)

These expressions help to provide insight into the factors which influence the free solution diffusion coefficient

Using for example the limiting equivalent conductivities (approach II) in water at 25°C [ROB-59], the diffusion coefficient values for several ions have been calculated based on equation (2.32) for Na⁺, K⁺, Cl^- and SO42- (Table 2.1).

Table 2.1 Self-diffusion coefficients for representative ions at infinite dilution in water at 25°C.

Ion type Ion specie Ion diffusion coefficient

Cation Na⁺ 1.33 x 10^-5 m²/s

Cation K⁺ 1.96 x 10^-5 m²/s

Anion Cl^- 2.03 x 10^-5 m²/s

Anion SO42- 1.07 x 10^-5 m²/s

These values are maximum values attained under ideal conditions (i.e. molecular scale and infinite dilution). Under non-ideal conditions (i.e. macroscopic scale, concentrated solution) a number of effects, as mentioned in the beginning of the discussion, should be added.

In a system consisting exclusively of a solution, four types of diffusion are commonly distinguished [SHA-91]: (a) self-diffusion, (b) tracer diffusion, (c) salt diffusion and (d) counterdiffusion.

Fig. 2.8 Different types of diffusion: (a) self-diffusion, (b) tracer diffusion, (c) salt diffusion and (d) counterdiffusion [SHA-91].

In the case of dilute binary mixtures, self-diffusion and tracer diffusion can be described by Fick’s first law. Self-diffusion (Fig. 2.8a) coefficients for anions and cations in infinitely dilute solutions are already shown in Table 2.1. For this test two half-cells are used. In such a system with equal concentrations the movements are truly random, but the motion of the molecules can never be traced. Therefore, in practice, in one half-cell a small amount of sodium Na⁺ has been replaced by its isotrope, ²²Na⁺. The tracer ions may be considered to be moving relative to a stationary background of non-diffusing ions. This movement is called “self-diffusion”.

(a) (b) (c) (d)

22NaCl + NaCl

42KCl + NaCl

NaCl NaCl NaCl water NaCl KCl

22Na^{+ 42}K⁺

Na⁺ Na⁺

Na⁺

Cl Na⁺

K⁺

(a) (b) (c) (d)

(a) (b) (c) (d)

22NaCl + NaCl

42KCl + NaCl

NaCl NaCl NaCl water NaCl KCl

22Na^{+ 42}K⁺

Na⁺ Na⁺

Na⁺

Cl Na⁺

K⁺

Tracer diffusion (Fig. 2.8b) is the same as self-diffusion except the isotropic species are of a different element. In the example the diffusion of ⁴²K⁺ is termed “tracer diffusion”. At infinite dilution, the tracer diffusion and self-diffusion coefficients are the same.

Salt diffusion (Fig. 2.8c) occurs when one half-cell contains a salt solution whereas the other half-cell contains only the solvent. Both oppositely charged ions will diffuse in the same direction. In this case, as explained before, an electrical potential is set up between the ions. Due to this electrochemical potential gradient, the slower moving ions speed up while the faster ions slow down. The combination of driving force by the chemical potential and draw-back force due to the counter-electrical field is referred to as

“electrochemical potential”. This leads to a free salt diffusion coefficient for an infinitely dilute solution of the form [LOW-81]:

( )

/ ,0

D D z z

D z D z D

+ − + −

+ −

+ + − −

= −

− ^(2.40)

For the electrolytes used in this study the calculated values of the resulting molecular diffusion coeffient are given in Table 2.2.

Table 2.2: Limiting free solution diffusion coefficients for representative simple electrolytes at 25°C (Robinson & Stokes, 1959)

Ion specie Molecular diffusion coefficient

NaCl 1.61 x 10^-9 m²/s

KCl 1.99 x 10^-9 m²/s

NaSO4 2.03 x 10^-5 m²/s

Counterdiffusion (Fig. 2.8d) is taking place when different ions are diffusing in opposite direction. Self- and tracer diffusion are limiting cases of counterdiffusion, and salt diffusion and counterdiffusion usually occur simultaneously.

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