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I am using the Nernst-Einsten relation as a way to calculate the dc ionic conductivity ($\sigma_{dc}$) of a membrane:

$$\sigma_{dc} = \frac{q^2CD }{k_BT}$$

where:
$q$ is the charge on ions [Coulombs]
$C$ is concentration or number density of ions [ions/$m^3$]
$D$ is the chemical Diffusion Coefficient ($m^2$/s)
$k_B$ is Boltzman's Constant [J/K]
$T$ is Temperature [K]

The units should work out to be $S/m$ or $A/Vm$, but I can only get it down to $A*ions/Vm$.

If I instead use $q$ as Coulombs/ion, the units work out to be $A/V*m*ions$.

I've seen the formula written this way in multiple sources so I'm convinced I am missing something obvious. Can someone please enlighten me? Perhaps I am misinterpreting what the 'ions' unit means in the context of ionic conductivity.

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    $\begingroup$ I would say that if "ions" means the number of ions, then it is a dimensionless quantity. For example, the "number density" of ions is measured in $m^{-3}$ $\endgroup$ – DelCrosB Sep 26 '16 at 20:29
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I don't see any problem with the dimensions. Let's have a look.

$$ \begin{align} \left[\sigma_{dc}\right]&=\frac{[q]^2[C][D]}{[k_B][T]}\\ &=\frac{C^2m^{-3}m^2s^{-1}}{JK^{-1}K}\\ &=\frac{C^2}{Jms}=\frac{C}{s}\frac{C}{Jm}\\ &=A\frac{1}{\left(J/C\right)m}\\ &=\frac{A}{Vm}=A/Vm \end{align} $$

The dimension is correct. The source of your error is that you used ions/$m^3$ as the dimension for concentration (number of ions per volume). All it means is that this much of ions are present in unit volume of your sample. As the number of ions is dimensionless, the dimension of concentration is $m^{-3}$.

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  • $\begingroup$ Thank you, this is exactly the clarification I was looking for $\endgroup$ – Ben Coscia Sep 27 '16 at 18:05

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