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I have conducted an experiment measuring the conductivity of both hydrochloric acid and sulfuric acid in solution respectively, with varying concentration. Wikipedia as well as a question on the site suggests a $\sqrt{c}$ relationship is viable. However, my data resembles this:

enter image description here

Clearly not a simple $\sqrt{c}$. It reminds me of the inverse gamma distribution. I have tried looking at papers by Onsager on his conductivity theory, but they're unclear, and he doesn't propose a function. Are there any papers that look at accurately modelling conductivity as a function of concentration which exhibit this form? Obviously for low concentrations (e.g. 1-2 molar), a linear or $\sqrt{c}$ would suffice.

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At low concentration, conductivity is proportional to concentration (a linear relationship).

Each ion will have its own unique mobility, as discovered by Kohlrausch. $\ce{H+}$ has the highest mobility. As you can see in your graph the acids have higher conductivities than the salts. $\ce{OH-}$ is also highly mobile. As concentration increases, the linear relationship breaks down for two reasons.

Firstly, in infinitely dilute solution, for the strong electrolytes in the graph, there is complete dissociation into separately solvated ions. However, as concentration increases, a portion of the electrolyte exists as ion pairs. See for example Equations for Densities and Dissociation Constant of NaCl(aq) at 25°C from “Zero to Saturation” Based on Partial Dissociation J. Electrochem. Soc. 1997 vol. 144, pp. 2380-2384.

Secondly, the mobility of the ions that are solvated is decrease by the fact that they are no longer moving through water, but are instead moving past other ions as well.

To have a maximum in in the curves of the question, and to account for the above factors, it is necessary to subtract a term from the linear term.

You need a function of the form:

$$\text{Conductivity} = Ac - Bf(c),$$

where $A$ and $B$ are constants, $c$ is concentration, and $f(c)$ is some function of concentration. Historically, the function $Ac - Bc\sqrt{c}$ was the first and most simple to use.

I would start by trying to fit your data to that function.

More advanced treatments replace $\sqrt{c}$ with $\sqrt{I}$, where I is ionic strength. Further advances involve higher order $I$ in addition to $\sqrt{I}$, such as $I \ln (I)$, $I$, and $I^{3/2}$.

The $I \ln (I)$ and $I$ terms arise upon considering the ions as charged spheres of finite diameter, rather than a simple point charges, as explained in Electrolytic Conductance and Conductances of the Halogen Acids in Water and references cited therein. This is the reference to look at if you want the best function to fit the HCl curve in your graph as it includes numerical coefficients.

For more information see The Conductivity of Liquids by Olin Freeman Tower, which although not the most recent work, has the advantage of being understandable, or Calculating the conductivity of natural waters.

For a very advanced consideration, see "Electrical conductance of electrolyte mixtures of any type" Journal of Solution Chemistry, 1978, vol. 7, pp. 533-548.

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  • $\begingroup$ I completely agree with the discussion about ion pairing and the decreased mobility. We study charge transport, and similar curves exist - at high concentration, the mobile charges become "frustrated" because of electrostatic interactions. $\endgroup$ – Geoff Hutchison Mar 11 '15 at 16:57
  • $\begingroup$ I also wonder whether, particularly with sulfuric, if there's a density/viscosity effect at work at high concentrations. $\endgroup$ – Geoff Hutchison Mar 11 '15 at 16:59
  • $\begingroup$ @GeoffHutchison viscosity effect is mentioned in this document: nist.gov/data/nsrds/NSRDS-NBS331.pdf I'll try to add more to my answer from there. $\endgroup$ – DavePhD Mar 11 '15 at 17:30
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To expound on DavePhD's nice answer, molar conductivity follows Kolrausch's law, $\Lambda_m=\Lambda_m^{\circ} - \kappa \sqrt{c}$ at low concentrations, which means that extensive conductivity follows $\Lambda_m c = c\Lambda_m^{\circ} - \kappa c \sqrt{c}$ in this domain.

As you've discovered this relationship breaks down at high concentration. In a loose analogy to the virial equation of state for gases, I'd propose fitting your data to a modification of Kolrausch's law, specifically, to an expansion in $\sqrt{c}$:

$\Lambda_m=\Lambda_m^{\circ} - \kappa_1 \sqrt{c} - \kappa_2 (\sqrt{c})^2 - \kappa_3 (\sqrt{c})^3 + ...$

I'm not enough of Debye-Huckel theorist to know if this expansion has the same theoretical justification/derivation in terms of intermolecular forces that the virial equation for gases does. But, (a) it definitely introduces extra degrees of freedom that you will need to fit your more complex data, and (b) if your data did follow Kolrausch's law (and the data for e.g. sodium chloride doesn't look too far off, at least by eye), your fit would tell you so, meaning you wouldn't have to convert or translate your fit parameters to a new system of units to compare with tabulated values for Kolrausch's law -- your fit would tell you the Kolrausch parameters.

UPDATE: I dug around in the literature a bit. Wikipedia pointed me towards this 1960s JACS paper. There they give the Fuoss-Onsager equation, which can be written as:

$$\Lambda_m=\Lambda_m^{\circ} - \beta_1 \sqrt{c} - \beta_2 c - \beta_3 c \ln{c}$$

The $\beta_i$ parameters are expanded in the paper to show dependence on some underlying solvent and solute properties, but that isn't important for fitting compound-specific curves to your conductivity data. You can find papers on the Fuoss-Onsager equation and cite them if you like. And this equation still has the advantage that is reduces to Kolrausch's law for $\beta_2 = \beta_3 = 0$. So if that seems better to you than the expansion in $\sqrt{c}$, try that one.

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    $\begingroup$ I think the question would be why there are higher-order terms. $\endgroup$ – Geoff Hutchison Mar 11 '15 at 16:55
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    $\begingroup$ The easy answer is because the assumptions of Kolrausch's law break down at high concentrations...but I gather you (and the OP?) are asking why that is? That is a very different question than "what is a justifiable way to fit my experimental data". I worry that the OP wants a one-equation, globally applicable summary of the theory of the liquid state. That's an impossibly tall order. $\endgroup$ – Curt F. Mar 11 '15 at 18:23
  • $\begingroup$ I can't speak for the OP, but I suspect even a sense of the leading corrections (e.g., in Dave's answer) would be helpful. Clearly you can find an empirical fit, but the question would be what the higher order terms mean physically. $\endgroup$ – Geoff Hutchison Mar 11 '15 at 18:35
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    $\begingroup$ I updated my answer to explicitly refer to Fuoss-Onsager. $\endgroup$ – Curt F. Mar 11 '15 at 18:36
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Your carefully plotted results seem in line with this table from the Foxboro Company. Though there are references that give a simple square-root relationship, they are clearly wrong at high concentrations of many electrolytes.

There is a practical guide to conductivity measurement at Conductivity Theory and Practice, which mentions:

For some samples with high concentrations, the conductance = f (concentration) curve may show a maximum.

Wikipedia on Conductivity states:

Both Kohlrausch's law and the Debye-Hückel-Onsager equation break down as the concentration of the electrolyte increases above a certain value. The reason for this is that as concentration increases the average distance between cation and anion decreases, so that there is more inter-ionic interaction.

So, as you've done, actual conductivity curves are often determined experimentally, and formulae then are derived to fit the curves.

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    $\begingroup$ Yes, all of this is clear. The problem is finding a function to fit to the curve. Is there a more sophisticated theory that suggests a function, rather than I having to guess? Because there are many that can fit the curve 'decently.' $\endgroup$ – user1997744 Mar 1 '15 at 18:27
  • $\begingroup$ There are many tools to derive a formula to fit a curve. E.G. using Excel (cpp.edu/~seskandari/documents/Curve_Fitting_William_Lee.pdf), or MATLAB, online in Wolfram using it's Fit language (reference.wolfram.com/language/tutorial/CurveFitting.html), or this Regression Fitter (had2know.com/academics/…) $\endgroup$ – DrMoishe Pippik Mar 1 '15 at 18:34
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    $\begingroup$ That's not the problem; I have Mathematica that is capable of fitting any function to a curve, and determining the parameters. My point is that several functions may fit. I'd like a theory which derives or motivates a function, so that my choice of function is justified. $\endgroup$ – user1997744 Mar 1 '15 at 18:35
  • $\begingroup$ Sorry for the cop-out, but all the literature I've come across is pragmatic, without full numerical analysis for all the competing factors affecting conductivity. Determine the applicable range of concentrations, decide how close the fit must be, and see if you can derive a formula that fits within an acceptable limit. If someone out there has a better theoretical grasp, please answer. $\endgroup$ – DrMoishe Pippik Mar 1 '15 at 18:45

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