Ion transport number

In chemistry, ion transport number, also called the transference number, is the fraction of the total electric current carried in an electrolyte by a given ionic species i:^[1]

${\displaystyle t_{i}={\frac {I_{i}}{I_{\text{tot}}}}}$

Differences in transport number arise from differences in electrical mobility. For example, in an aqueous solution of sodium chloride, less than half of the current is carried by the positively charged sodium ions (cations) and more than half is carried by the negatively charged chloride ions (anions) because the chloride ions are able to move faster, i.e., chloride ions have higher mobility than sodium ions. The sum of the transport numbers for all of the ions in solution always equals unity:

${\displaystyle \sum _{i}t_{i}=1}$

The concept and measurement of transport number were introduced by Johann Wilhelm Hittorf in the year 1853.^[2] Liquid junction potential can arise from ions in a solution having different ion transport numbers.

At zero concentration, the limiting ion transport numbers may be expressed in terms of the limiting molar conductivities of the cation (⁠${\displaystyle \lambda _{0}^{+}}$⁠), anion (⁠${\displaystyle \lambda _{0}^{-}}$⁠), and electrolyte (⁠${\displaystyle \Lambda _{0}}$⁠):

${\displaystyle t_{+}=\nu ^{+}\cdot {\frac {\lambda _{0}^{+}}{\Lambda _{0}}}}$

and

${\displaystyle t_{-}=\nu ^{-}\cdot {\frac {\lambda _{0}^{-}}{\Lambda _{0}}},}$

where ⁠${\displaystyle \nu ^{+}}$⁠ and ⁠${\displaystyle \nu ^{-}}$⁠ are the numbers of cations and anions respectively per formula unit of electrolyte.^[1] In practice the molar ionic conductivities are calculated from the measured ion transport numbers and the total molar conductivity. For the cation ${\displaystyle \lambda _{0}^{+}=t_{+}\cdot {\tfrac {\Lambda _{0}}{\nu ^{+}}}}$, and similarly for the anion. In solutions, where ionic complexation or associaltion are important, two different transport/transference numbers can be defined.^[3]

The practical importance of high (i.e. close to 1) transference numbers of the charge-shuttling ion (i.e. Li+ in lithium-ion batteries) is related to the fact, that in single-ion devices (such as lithium-ion batteries) electrolytes with the transfer number of the ion near 1, concentration gradients do not develop. A constant electrolyte concentration is maintained during charge-discharge cycles. In case of porous electrodes a more complete utilization of solid electroactive materials at high current densities is possible, even if the ionic conductivity of the electrolyte is reduced.^[4][3]