Physical Chemistry Third Edition

478 10 Transport Processes

wheref+is the friction coefficient for the cations andf−is the friction coefficient for

the anions. The conductivity is

σFce

(

1

f+

+

1

f−

)

(ifν+ν−1) (10.5-17)

For a general binary electrolyte represented by Mν+Xν−, the analogue of Eq. (10.5-17)

is

σFe

(

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:

ti

ji

jtotal

(10.5-19)

wherejiis the magnitude of the current density due to ions of typeiandjtotalis 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)

Themobilityuiof ions of typeiis defined by

ui

vi

EEE

(10.5-22)

whereviis the magnitude of the mean drift velocity of this type of ion and whereEEE

is the magnitude of the electric field. If we apply Stokes’ law,

ui

|zi|e

fi



|zi|e

6 πηref f,i

(10.5-23)

wherereff ,iis the effective radius of the ions of typei. For a single electrolyte solute,

the conductivity can be written in terms of the mobilities:

σF(c+z+u++c−|z−|u−) (10.5-24)

Exercise 10.20

Write the transference numbers in terms of the ion mobilities.