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In what way does the electrical conductivity $\sigma$ (or conversely, resistivity $\rho = 1/\sigma$) depend on the salinity $S$ of the saline- (or sea-) water?

I know they must be in some way proportional since the more free charge carriers (ions in this case) available, the more electric current is possible.

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Measuring the conductivity of water is actually one of the standard ways of measuring its salinity. The figure below (taken from this University of British Columbia page) shows that the relationship over many orders of magnitude is $$ \sigma\propto S, $$ where $\sigma$ is the conductivity and $S$ is the salinity. They further note that the conductivity also varies with temperature and the relationship is given approximately by $$ \frac{1}{\sigma(T)}=\frac{\frac{1}{\sigma_{18 ^\circ C}}}{1+0.025(T-18\ ^\circ C)}. $$

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Fondriest has a long tutorial on environmental conductivity measurements which is very informative. It does an excellent job of explaining the environmental factors which affect water salinity and therefore measurements of water conductivity.

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