Transport numbers
description
Transcript of Transport numbers
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