# Homework Help: Electrical conductivity of seawater depends on salinity how?

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1. Aug 4, 2015

### H Smith 94

Hi! I am currently trying to determine how the salinity $S$ of a sample of seawater (or, objectively, a salt-water solution) changes its electrical conductivity $\sigma$.

It is clear that they are proportional since the mobility of the $\text{Na}^{+}$ and $\text{Cl}^{-}$ ions plays a huge role but I am unable to find any conclusive models which describe relationship between $\sigma$ and $S$. Does anyone have any pointers or know of any existing relations? It would also be useful to understand in what way these factors depend on the temperature $T$.

I have previously discovered the Kohlrausch ionic model (see also: Molar conductivity), which provides a semi-empirical determination of the specific conductivity. From this I determined that

$$\sigma_{25^\text{o}\text{C}}(C_n) = \left(\nu_+\lambda_+^{(0)}+\nu_-\lambda_-^{(0)}\right)\,C_n - KC_n^{3/2}$$
where:

$\nu_\pm$ is the number of moles of each ion;
$\lambda_\pm^{(0)}$ is the limiting molar conductivities of each ion,

for $\text{Na}^{+}$ and $\text{Cl}^{-}$:
$\lambda_+^{(0)} = 5.011\, \text{mS}\,\text{m}^2\,\text{mol}^{-1}$,
$\lambda_-^{(0)} = 7.634\, \text{mS}\,\text{m}^2\,\text{mol}^{-1}$;
$C_n(S) = S\rho_\text{water}/N_Am_i$ is the number concentration (i.e. number of ions per unit volume,) in which:

$S$ is salinity;
$\rho_\text{water}$ is the density of water;
$m_i$ is the mass of each salt particle ($m_i = m_\text{Na} + m_\text{Cl}$;)
$N_A$ is Avagadro's constant.

Making the assumption that $\nu_+ = \nu_- = \nu$ and using the substitution for $C_n(S)$ we find that

$$\sigma_{25^\text{o}\text{C}}(S) = \left(\lambda_{\text{Na}^{+}}^{(0)}+\lambda_{\text{Cl}^{-}}^{(0)}\right)\left(\frac{\rho_\text{water}}{N_Am_i}S\right)\nu - K\left(\frac{\rho_\text{water}}{N_Am_i}S\right)^{3/2}.$$
Although this model is great and seems to cover most bases it still has that pesky empirical $K$ value (the Kohlrausch constant,) so it's really not perfect.

Have I made any fatal assumptions/errors in deriving this? What are its limitations? Does anyone know of a way to expand this model to incorporate temperature, rather than simply being a specific conductivity?

Last edited: Aug 4, 2015
2. Aug 5, 2015

### Staff: Mentor

Beware - this is a quite difficult problem, which is easier to solve experimentally (that is - just measure the conductivity) than to get reasonable result from theoretical considerations. You should take ionic strength of the solution into account, and the fact that ions have tendency to travel in pairs (which is to some extent equivalent to assuming salts are not 100% dissociated). Thick books have been written and there is still no easy approach.

Unfortunately the only source I can suggest is a book that can be rather difficult to find - https://books.google.pl/books/about/Electrochemistry.html?id=FVF8kDHABH8C&hl=pl There are plenty of other books, it just happens I have this one on the shelf, so I am sure it contains an introduction to the theory needed.

3. Aug 5, 2015

### H Smith 94

Borek, thank you for moving this thread!
It certainly does seem that way! What's the best way to measure conductivity experimentally? I could try to tabulate some different values (like this one) and use those instead.
I was a little confused by the physical meaning of the ionic strength, so I just made a complete oversimplification so I didn't have to think about it haha. Maybe that was a bad idea. How would I factor this into the calculation?
I feared that would be the case! I'll have to root through the book shelves, too.

Thank you for the suggestion. Is this book written in English? My Polish is, let's say, nie dobry.

Last edited: Aug 5, 2015
4. Aug 5, 2015

### Staff: Mentor

Any serious electrochemistry book (or even some instrumental/analytical chemistry books) will discuss it.

Debye-Huckel theory. But I strongly suggest instead of asking about each things separately you start to read electrochemistry books, they always explain things in a reasonably complete and systematic manner, which is much better than the hit-and-run on the forum.

As far as I remember it was published in English, German and Polish, probably in Czech. But there are other similar books written in English (and probably much easier to find), I just don't know them.

5. Aug 5, 2015

### H Smith 94

Yes, you're right: I do seem a little leech-ey right now. Thank you for all your help, your pointers have been of invaluable assistance to me.

6. Aug 5, 2015