Transport numbers

• The fraction of total current carried by the ions of a specified type.

• The limiting transport number, t0±, is defined for the limit of zero

concentration of the electrolyte solution.

• The relationship between transportation number and the mobility of an ion is:

• The relationship between transportation number and the conductivity is:

I

It

I

It

uvzuvz

uvzt0

vv

vt 0

The measurement of transport numbers

• Moving boundary method: the motion of a boundary between two ionic solutions with a common ion is observed as a current flows.

• Indicator solution: • Leading solution:• The mobility of the M ions

must be greater than that of N ions.

tI

clAFzt

Conductivities and ion-ion interactions

• To explain the c1/2 dependence in the Kohlrausch law.

Hückel-Onsager Theory

24.8 The thermodynamic view of diffusion

• The maximum amount of work can be done when moving a substance from local x to x+dx is:

• When expressed with an opposite force:

dw = - F dx

Then one gets:

Therefore: The slope of the chemical potential can be interpreted as an effect force, thermodynamic force. This force represents the spontaneous tendency of the molecules to disperse.

dxx

ddwTp,

TpxF

,

• Since μ = μө + RTlnα

• One get

• Using concentrations to replace the activity:

TpTp x

aRT

x

aRTuF

,,

ln}

)ln({

Connections between the thermodynamic force and the

concentration gradient

Tpx

c

c

RTF

,

Fick’s first law of diffusion revisit

• Fick’s law of diffusion discussed earlier was developed from the kinetic theory of gases.

• The flux of diffusing particles is due to a thermodynamic force arising from concentration gradient (i.e. the thermodynamic force is proportional to the concentration gradient).

• The drift speed is proportional to the thermodynamic force.

• The particle flux, J, is proportional to the drift speed.

• The chain of proportionalities (J ~ s, s ~ F, F ~ dc/dx) implies that J is proportional to concentration gradient.

The Einstein relation

• The flux is related to the drift speed by J = sc

• Comparing the above equation with the Fick’s law, one gets sc = -D (dc/dx)

• Express dc/dx in terms of F, one gets s = (DF)/(RT)

• The drift speed of an ions equals s = u E

• Therefore, u E = (DF)/(RT) = (zFED)/(RT)

• Reorganizing the above equation to D = (uRT)/(zF) (Einstein relation between the diffusion coefficient and the inonic mobility)

The Nernst – Einstein Equation

• Provides a link between the molar conductivity of an electrolyte and the diffusion coefficients.

• Can be applied to determine the ionic diffusion coefficients from conductivity measurement.

• For each type of ionλ = zuF = (z2DF2)/(RT)

• For electrolyte

Λm = (v+Z+2D+ + v-Z-

2D-)F2/(RT)

24.9 The diffusion equation

2

2

x

cD

t

c

Derivation of the diffusion equation

• The amount of particles enter the slab in the time interval dt equals: JAdt, where J is the matter flux

• The increase in molar concentration inside the slab is: JAdt / (Al t) = J/l

• Consider the outflow through the right-hand side:

-JAdt / (Al t) = J/l

• The net change is:

• Then

l

JJ

t

c '

2

2

x

cDlJJ

x

clc

xD

x

cD

x

cD

x

cDJJ

'

''

Solutions of the diffusion equation