The ionic strength formula provides a way to estimate the Debye screening length. I’m having some trouble understanding the structure;

$$I = \begin{matrix}\frac{1}{2}\end{matrix}\sum_{{\rm i}=1}^{n} c_{\rm i}z_{\rm i}^{2}$$

It seem a bit tricky to find the derivation for this, but all I’m really looking for is an intuitive answer to a couple of questions;

  • Firstly; why is there a factor of a half out the front?

  • Secondly; why do we square the charge number of the ion?

I am a physicist, so please explain any chemistry concepts like I’m 5. Thank you for your patience.


Firstly; why is there a factor of a half out the front?

To avoid double-counting. For a salt such as $\ce{NaCl}$ comprising a monovalent cation and a monovalent anion, the ionic strength is defined such that it should be equal to the molar concentration of the salt, as opposed to the sum of both the anion and cation concentrations. Therefore, since such a salt has two constituents, each constituent contributes half of the total ionic strength.

Secondly; why do we square the charge number of the ion?

For this there's not much choice but to turn to the Poisson-Boltzman equation, which provides a differential expression for the electric field inside the electric double-layer. (Sorry, 5-year-old version of Jekowl!) Below is the version I have in my graduate transport book, Eq. (15.4-6):

$$ \nabla^2\Phi = -{F\over \epsilon} \sum_i{z_iC_{i\infty}\exp{\left(-{z_iF\over RT}\Phi\right)}} $$

  • $\Phi$ is the electric potential, $\mathrm{J} \over \mathrm{C}$ or $\mathrm{V}$
  • $F$ is Faraday's constant, $96,485\ {C \over \mathrm{mol}\ q}$
  • $\epsilon$ is the electrical permittivity, $\mathrm{C}\over \mathrm{V\ m}$
  • $z_i$ is the valence of ion $i$, $\mathrm{mol}\ q \over \mathrm{mol\ ion}$
  • $C_{i\infty}$ is the concentration of ion $i$ far from the surface of interest, $\mathrm{mol} \over \mathrm{m^3}$
  • $R$ is the ideal gas constant, $8.314\ {\mathrm{J} \over \mathrm{mol\ K}}$
  • $T$ is temperature, $\mathrm{K}$
  • $q$ is the elementary charge

If the magnitude of the electric field is small such that $\left|z_iF\Phi/RT\right|\ll 1$, the exponential can be expanded as a Taylor series:

$$ \nabla^2\Phi = -{F\over \epsilon} \sum_i{z_iC_{i\infty}\left(1-{z_iF\over RT}\Phi+\dots\right)} $$

If you then pull everything inside the parentheses, ignoring signs the second term becomes:

$$ C_{i\infty}z_i^2{F^2\over \epsilon RT}\Phi $$

The second-order dependence of $z_i$ is readily visible in the above.

  • 1
    $\begingroup$ Poission-Boltzmann is safe for physicists, because you can derive it from statistical mechanics :D $\endgroup$ – Jekowl Nov 18 '15 at 17:14

First the whole notion of solution activity is simplfied by the Debye–Hückel theory which describes dilute solutions in terms of the individual ions. The solution activity is needed to predict the colligative properties properties of solutions like:

  • Relative lowering of vapor pressure
  • Elevation of boiling point
  • Depression of freezing point
  • Osmotic pressure

The 1/2 factor is so that an electrolyte solution like NaCl (cation and anion only have one charge each) in dilute solutions reduces to it molar strength even though it creates two ions in solution. (The molal strength would be used in concentrated solutions.)

To understand the square term you'll need to read about the mathematical development of the Debye–Hückel theory.


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