RELATIONSHIP BETWEEN GENERALISED RELATIVE IONIC MOBILITY AND LIMITING EQUIVALENT CONDUCTIVITY

First of all, it should be noted that the relative mobility of an ion, u, required for calculating liquid junction potentials (as listed in the above tables of mobilities and required in JPCalc calculations) represents the generalised (or absolute) mobility of an ion relative to K+. For example, if uX, is the relative mobility for ion X, with respect to K+, it will be given by:

uX = u*X / u*K

where u*X and u*K represent the absolute values of the generalised mobilities of ions  X and K+ respectively. The units of the relative mobility for ion X, uX, are (of course) dimensionless.

 

The following discussion indicates how the generalised mobilities of ions are in turn related to their limiting equivalent conductivities.

Since the velocity of an ion in solution, v, is related to the generalised (absolute) mobility, u*, and the generalised force, Fx, acting on it, then:

                                                            v = u* Fx

The force may be in Newtons/ mole or Newtons, depending on whether it is the force acting on a mole of ions or on a single ion (and whichever is chosen will affect the units of u*). The above generalised mobility is what is required for electrodiffusion flux equations, and would normally be that required for a force acting on a mole of ions.

In contrast, electrochemists, when measuring conductivity, use another definition of mobility, which may be defined as u', the electrical mobility, sometimes also called the conventional mobility (Bockris & Reddy, 1973; pp. 369-373), since they measure the mobility as the velocity/ electric field, E (e.g., in volts/m) as:

                                                            v = u' E

Since the actual force is zFE, we also have v = u*zFE, where z is the magnitude of the valency and F is the Faraday. Hence,

                                                            u* = u'/zF

We wish to know the relationship between generalised (absolute) mobility and the limiting equivalent conductivity, L0 (the conductivity of an electrolyte solution per equivalent, in the limit as the concentration goes to zero).  Now Lmakes allowance for the additional charge of polyvalent ions, so that

                                                            u' = L0 /F

[cp. Eqs. 4.156 - 4.160 in Bockris & Reddy, 1973 (p.373) for equivalent conductivity (L) and molar conductivity (Lm), where
L = Lm/z; N.B. error in sign of the anion subscript in Eq. 4.16)].  Hence, from the two equations above, the generalised mobility, u*, and L0 will be related by:

u* = L0 / (zF2)

[cf. the equation for the generalized mobility, u, for a single ion (e.g., Eq. A4 in Sugiharto et al., 2008), rather than for a mole of ions, u*, which is given by u = NL0 / (zF2), where N is Avagadro’s number]Hence, the relative mobility of ion X of valency z is given by:

uX = [L0X / z] / L0K

since, for K+, z = 1 and both limiting equivalent conductivities were measured at the same temperature.

For a monovalent ion Y, its relative mobility will simply be given by:

uY = L0Y / L0K

where L0Y is the limiting equivalent conductivity of Y at the same temperature as for L0K , normally 25 oC. 

For reference, L0K = 73.50 S.cm2.equiv-1 at 25 oC (Robinson & Stokes, 1965).

References

Bockris J. O'M and A.K.N. Reddy (1973).  Modern Electrochemistry, Vol 1, Plenum Press, New York

Sugiharto, S., T. M. Lewis, A. J. Moorhouse, P. R. Schofield, and P. H. Barry. (2008).  Anion-cation permeability correlates with hydrated counter-ion size in glycine receptor channels.  Biophys. J. 95:4698-4715.

 

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P. H. Barry  Feb 26, 2004

Additional discussion of derivation included, Aug 25, 2010

Feb 8, 2011