Physical Chemistry Third Edition

476 10 Transport Processes

Metal plate

of area 

Electrical

power

supply

Metal plate

of area 

d

Figure 10.9 An Electrically Conducting System (Schematic).

The conductivity has the units ohm−^1 m−^1. The ohm−^1 has been called themho(ohm

spelled backwards) but the SI name for ohm−^1 is thesiemens, denoted by S, so that

the conductivity has the units Siemens per meter (S m−^1 ).

We define thecurrent densityjas a vector with magnitude equal to the current per

unit area and with the same direction as the current:

j|j|

I

A

(10.5-4)

whereAis the cross-sectional area of the conductor andIis the current. Ohm’s law

can be written

j

I

A



V

RA



EEEd

rd



EEE

r

σEEE (10.5-5)

whereEEE is the magnitude of the electric field, equal toV/dfor an object like that of

Figure 10.9.

Consider a solution of an electrolyte solute with one cation and one anion. Let the

mean velocity of cations be denoted byv+and the mean velocity of anions be denoted

byv−. At equilibrium these mean velocities vanish because as many ions will be

moving in a given direction as in the opposite direction. In the presence of an electric

fieldv+andv−will be nonzero, in which case they are calleddrift velocities. The

current density is the sum of a cation contribution and an anion contribution. On the

average cations no farther from a fixed plane than a distance equal tov+times 1 second

will pass in 1 second. For unit area, the number of cations passing the fixed plane per

second is

number of cations per unit area per secondc+NAvv+ (10.5-6)

wherec+is the concentration of the cations in mol m−^3 andNAvis Avogadro’s constant.

The charge passing per second due to the cations is

charge per unit area per second due to cationsz+ec+NAvv+ (10.5-7)

whereeis the charge on a proton, 1. 6022 × 10 −^19 C, andz+is the valence of the

cation (the number of proton charges on one ion). A similar expression can be written