478
10 Transport Processes
where f + is the friction coefficient for the cations and f − is the friction coefficient for
the anions. The conductivity is
σ Fce
1
f +
+
1
f −
(if ν + ν − 1)
(10.5-17)
For a general binary electrolyte represented by M ν + X ν − , the analogue of Eq. (10.5-17)
is
σ Fe
c + z 2
+
f +
+
c − z 2
−
f −
Fce
ν + z 2
+
f +
+
ν − z 2
−
f −
(10.5-18)
Exercise 10.19
Show that Eq. (10.5-18) is correct.
Since the cations and anions in a given electrolyte will not generally have equal
friction coefficients, the two kinds of ions will not necessarily carry the same amount
of current. The fraction of the current that is carried by a given type of ion is called its
transference number, t:
t i
j i
j total
(10.5-19)
where j i is the magnitude of the current density due to ions of type i and j total is the
magnitude of the total current density. If there is only one type of cation and one type
of anion present, Eq. (10.5-18) implies that
t +
c + z 2
+ /f +
c + z 2
+ /f + + c − z 2
− /f −
ν + z 2
+ /f +
ν + z 2
+ /f + + ν − z 2
− /f −
(10.5-20)
and
t −
c − z 2
− /f −
c + z 2
+ /f + + c − z 2
− /f −
ν − z 2
− /f −
ν + z 2
+ /f + + ν − z 2
− /f −
(10.5-21)
The mobility u i of ions of type i is defined by
u i
v i
E E E
(10.5-22)
where v i is the magnitude of the mean drift velocity of this type of ion and where E E E
is the magnitude of the electric field. If we apply Stokes’ law,
u i
|z i |e
f i
|z i |e
6πηr ef f,i
(10.5-23)
where r eff ,i is the effective radius of the ions of type i. For a single electrolyte solute,
the conductivity can be written in terms of the mobilities:
σ F (c + z + u + + c − |z − | u − )
(10.5-24)
Exercise 10.20
Write the transference numbers in terms of the ion mobilities.