Ion transport number

Ion transport number, also called the transference number, is the fraction of the total electrical current carried in an electrolyte by a given ionic species $i$ ,

t_{i}={\frac {I_{i}}{I_{tot}}}

Differences in transport number arise from differences in electrical mobility. For example, in a 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.

The concept and measurement of transport number were introduced by Johann Wilhelm Hittorf in the year 1853.^[1] 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 ( $\lambda _{0}^{+}$ ), anion ( $\lambda _{0}^{-}$ ), and electrolyte ( $\Lambda _{0}$ ):

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

and

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

,

where $\nu ^{+}$ and $\nu ^{-}$ are the numbers of cations and anions respectively per formula unit of electrolyte.^[2] In practice the molar ionic conductivities are calculated from the measured ion transport numbers and the total molar conductivity. For the cation $\lambda _{0}^{+}=t_{+}\cdot {\frac {\Lambda _{0}}{\nu ^{+}}}$ , and similarly for the anion.

Experimental measurement

There are two experimental techniques for the determination of transport numbers. The Hittorf method is based on measurements of ion concentration changes near the electrodes. The moving boundary method involves measuring the speed of displacement of the boundary between two solutions due to an electric current.^[3]

Hittorf method

In the Hittorf method, electrolysis is carried out in a cell with three compartments: anode, central, and cathode. Measurement of the concentration changes in the anode and cathode compartments determines the transport numbers.^[4] The exact relationship depends on the nature of the reactions at the two electrodes. For the electrolysis of aqueous copper(II) sulfate (CuSO₄) as an example, with Cu²⁺(aq) and SO²⁻
₄(aq) ions, the cathode reaction is the reduction Cu²⁺(aq) + 2 e⁻ → Cu(s) and the anode reaction is the corresponding oxidation of Cu to Cu²⁺. At the cathode, the passage of $Q$ coulombs of electricity leads to the reduction of $(Q/2F)$ moles of Cu²⁺, where $F$ is the Faraday constant. Since the Cu²⁺ ions carry a fraction t₊ of the current, the quantity of Cu²⁺ flowing into the cathode compartment is t₊ $(Q/2F)$ moles, so there is a net decrease of Cu²⁺ in the cathode compartment equal to (1–t₊) $(Q/2F)$ = t_– $(Q/2F)$ .^[5] This decrease may be measured by chemical analysis in order to evaluate the transport numbers. Analysis of the anode compartment gives a second pair of values as a check, while there should be no change of concentrations in the central compartment unless diffusion of solutes has led to significant mixing during the time of the experiment and invalidated the results.^[5]

Moving boundary method

This method was developed by British physicists Oliver Lodge in 1886 and William Cecil Dampier in 1893.^[3] It depends on the movement of the boundary between two adjacent electrolytes under the influence of an electric field. If a colored solution is used and the interface stays reasonably sharp, the speed of the moving boundary can be measured and used to determine the ion transference numbers.

For example, the transport numbers of hydrochloric acid (HCl(aq)) may be determined by electrolysis between a cadmium anode and a Ag-AgCl cathode. The anode reaction is Cd → Cd²⁺ + 2 e⁻ so that a cadmium chloride (CdCl₂) solution is formed near the anode and moves toward the cathode during the experiment. An acid-base indicator such as bromophenol blue is added to make visible the boundary between the acidic HCl solution and the near-neutral CdCl₂ solution.^[6] The boundary tends to remain sharp since the leading solution HCl has a higher conductivity that the indicator solution CdCl₂, and therefore a lower electric field to carry the same current. If a more mobile H⁺ ion diffuses into the CdCl₂ solution, it will rapidly be accelerated back to the boundary by the higher electric field; if a less mobile Cd²⁺ ion diffuses into the HCl solution it will decelerate in the lower electric field and return to the CdCl₂ solution. Also the apparatus is constructed with the anode below the cathode, so that the denser CdCl₂ solution forms at the bottom.^[2]

The cation transport number of the leading solution is then calculated as $t_{+}={\frac {z_{+}cLAF}{I\Delta t}}$ , where $z_{+}$ is the cation charge, $c$ the concentration, $L$ the distance moved by the boundary in time $\Delta t$ , $A$ the cross-sectional area, $F$ the Faraday constant, and $I$ the electric current.^[2]

Concentration cells

This quantity can be calculated from the slope of the function E_T = f(E) of two concentration cells, without or with ionic transport.

The EMF of transport concentration cell involves both the transport number of the cation and its activity coefficient:

E_{\mathrm {T} }=-z{\frac {RT}{F}}\int _{I}^{II}t_{+}d\ln a_{+/-}

where a₂ and a₁ are activities of HCl solutions of right and left hand electrodes respectively and t_M be transport number of Cl⁻

Notes

↑ Pathways to Modern Chemical Physics by Salvatore Califano (Springer 2012) p.61 ISBN 9783642281808
1 2 3 Peter Atkins and Julio de Paula, Physical Chemistry (8th ed. Oxford University Press, 2006) p.768-9 ISBN 0-7167-8759-8
1 2 Laidler K.J. and Meiser J.H., Physical Chemistry (Benjamin/Cummings 1982) p.276-280 ISBN 0-8053-5682-7
↑ Electrochemistry Dictionary - H Corrosion Doctors.
1 2 Principles and Applications of Electrochemistry D.R.Crow (4th ed., CRC Press 1994) p.165-169 ISBN 0748743782
↑ Transport numbers and ionic mobilities by the moving boundary method, G. A. Lonergan and D. C. Pepper, J. Chem. Educ., 1965, 42 (2), p. 82. doi:10.1021/ed042p82

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[1] Pathways to Modern Chemical Physics by Salvatore Califano (Springer 2012) p.61 ISBN 9783642281808

[Atk-2] 1 2 3 Peter Atkins and Julio de Paula, Physical Chemistry (8th ed. Oxford University Press, 2006) p.768-9 ISBN 0-7167-8759-8

[LM-3] 1 2 Laidler K.J. and Meiser J.H., Physical Chemistry (Benjamin/Cummings 1982) p.276-280 ISBN 0-8053-5682-7

[4] Electrochemistry Dictionary - H Corrosion Doctors.

[Crow-5] 1 2 Principles and Applications of Electrochemistry D.R.Crow (4th ed., CRC Press 1994) p.165-169 ISBN 0748743782

[6] Transport numbers and ionic mobilities by the moving boundary method, G. A. Lonergan and D. C. Pepper, J. Chem. Educ., 1965, 42 (2), p. 82. doi:10.1021/ed042p82