**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 u_{X},
is the relative mobility for ion X, with respect to K^{+}, it will be given by:

u

_{X}= 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, u_{X}, 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, F_{x}, acting on
it, then:

v = u* F_{x}

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,
L^{0} (the conductivity of an electrolyte solution per
equivalent, in the limit as the concentration goes to zero).
Now
L^{0 }makes allowance for the additional charge of
polyvalent ions, so that

u' =
L^{0} /F

[cp. Eqs.
4.156 - 4.160 in Bockris & Reddy, 1973 (p.373) for equivalent conductivity (L)
and molar conductivity (L_{m}),
where

L
= L_{m}/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
L^{0} will be related by:

u* = L

^{0}/ (zF^{2})

[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
= NL^{0}
/ (zF^{2}), where N is Avagadro’s number].
Hence, the
relative mobility of ion X of valency z is given by:

u

_{X}= [L^{0}_{X}/ z] / L^{0}_{K}

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:

u

_{Y}= L^{0}_{Y}/ L^{0}_{K}

where L^{0}_{Y} is the limiting equivalent conductivity of Y at the same
temperature as for L^{0}_{K} , normally 25 ^{o}C.

For reference,
L^{0}_{K} = 73.50 S.cm^{2}.equiv^{-1} at 25 ^{o}C
(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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Calculations, or for other information about such calculations and mobility
values, click here. **

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P. H. Barry Feb 26, 2004

Additional discussion of derivation included, Aug 25, 2010

Feb 8, 2011