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Conductivity of electrolytes

T-105Home CU-307Threads energy · equilibrium
Statement

Kohlrausch's law of independent ionic migration.

Why it matters

debye-huckel treats how ion–ion interactions modify solution thermodynamics; Kohlrausch's law of independent ionic migration is the analogous, foundational result for solution transport properties, giving the conductivity of any strong electrolyte solution as a simple sum of independent contributions from its constituent ions. This matters practically because it is the standard route to a quantity, the limiting molar conductivity \(\Lambda_m^\circ\), that cannot be measured directly at all for weak electrolytes by simple extrapolation, and because the same ionic-conductivity data feeds directly into computing ionic mobilities and transport numbers, both essential for a full physical picture of the electrode–solution interface developed later in this unit.

Hypotheses
At infinite dilution, ions are far enough apart that inter-ionic electrostatic interactions (ion atmosphere effects, debye-huckel) become negligible.This is exactly the condition under which each ion's contribution to conductivity becomes a genuinely independent, additive quantity, unaffected by the presence of any other ion species; at finite, measurable concentrations, ion–ion interactions do reduce conductivity below this ideal additive limit, which is why the law is stated specifically for the extrapolated, infinite-dilution limiting conductivity \(\Lambda_m^\circ\). The electrolyte is fully dissociated into its constituent ions at the concentrations of interest.This holds essentially exactly for strong electrolytes across their whole practical concentration range, but not for weak electrolytes, whose degree of dissociation itself varies strongly and non-linearly with concentration (the Ostwald dilution law); this distinction is precisely why strong and weak electrolytes show such different \(\Lambda_m\) versus \(\sqrt{c}\) behaviour (Proof, Step 3–4).
Proof
1
\[ \Lambda_m = \frac{\kappa}{c} \]
Molar conductivity \(\Lambda_m\) is defined as the measured specific conductivity \(\kappa\) of a solution divided by its molar concentration \(c\), normalising away the trivial effect of simply having more or fewer ions per unit volume, so that \(\Lambda_m\) reflects each ion's intrinsic mobility and contribution more directly. A
2
\[ \Lambda_m^\circ = \nu_+\lambda_+^\circ + \nu_-\lambda_-^\circ \]
At infinite dilution (Hypotheses), each ion migrates independently under the applied field, contributing its own characteristic limiting molar ionic conductivity \(\lambda^\circ\), weighted by the number of that ion, \(\nu\), released per formula unit of electrolyte; the total limiting conductivity is simply the additive sum of these independent contributions — Kohlrausch's law itself. A
3
\[ \text{Strong electrolyte: } \Lambda_m = \Lambda_m^\circ - K\sqrt{c}\ (\text{Kohlrausch's empirical square-root law}) \]
For a fully dissociated strong electrolyte, molar conductivity falls off linearly in \(\sqrt{c}\) as concentration rises, a modest, empirically well-fit, and later theoretically justified (via Debye–Hückel-type ion-atmosphere reasoning) deviation from the ideal, infinite-dilution additive value; because this decline is linear in \(\sqrt{c}\), it is straightforward to extrapolate the data back to \(c\to0\) and read off \(\Lambda_m^\circ\) directly. A
4
\[ \text{Weak electrolyte: } \Lambda_m \text{ drops sharply, non-linearly, as } c \text{ increases; no reliable direct extrapolation to } \Lambda_m^\circ \text{ is possible.} \]
A weak electrolyte's degree of dissociation itself falls substantially as concentration rises (the Ostwald dilution law), so \(\Lambda_m\) reflects both changing ionic mobility and changing extent of dissociation simultaneously, giving a sharply curving, non-linear \(\Lambda_m\) vs. \(\sqrt c\) plot that cannot be reliably extrapolated by eye or simple fit back to \(c=0\). A
Result
\[ \Lambda_m^\circ(\text{weak electrolyte}) = \Lambda_m^\circ(\text{strong 1}) + \Lambda_m^\circ(\text{strong 2}) - \Lambda_m^\circ(\text{strong 3})\ \text{(Kohlrausch combination)} \]

Reading. Since limiting ionic conductivities are strictly additive (Step 2), a weak electrolyte's otherwise inaccessible \(\Lambda_m^\circ\) can always be obtained indirectly, by combining the easily, directly extrapolated \(\Lambda_m^\circ\) values of three suitably chosen strong electrolytes that together supply exactly the same ions in the same stoichiometric proportions.

Scope. The direct extrapolation route (Step 3) works cleanly only for strong electrolytes; weak electrolytes require the indirect combination route above, precisely because their own \(\Lambda_m\) vs. \(\sqrt c\) behaviour is not linear enough to extrapolate directly (Step 4).

Corollaries & converses
  • Once \(\Lambda_m^\circ\) for a weak electrolyte is obtained via the Result, comparing it against the electrolyte's actually measured \(\Lambda_m\) at a given concentration gives its degree of dissociation directly, \(\alpha=\Lambda_m/\Lambda_m^\circ\), a standard practical route to weak-acid or weak-base dissociation data from purely conductivity measurements.
  • debye-huckel's ion-atmosphere picture is the theoretical justification for the empirical square-root dependence in Step 3: a moving ion's surrounding ionic atmosphere lags slightly behind it and exerts a net retarding (relaxation) drag, an effect that grows with ionic strength in exactly the \(\sqrt c\) functional form found empirically by Kohlrausch decades earlier.
  • Converse: individual ionic conductivities \(\lambda^\circ_\pm\) determined from Kohlrausch's law feed directly into computing transport numbers (the fraction of total current carried by each ion) and ionic mobilities, both developed as this unit's subsequent results.
Fails without
  • Extrapolate a weak electrolyte's directly measured \(\Lambda_m\) vs. \(\sqrt c\) plot to \(c=0\), assuming the strong-electrolyte square-root law of Step 3 applies: because the weak electrolyte's dissociation fraction itself varies sharply and non-linearly with concentration (Hypotheses, Step 4), such a naive extrapolation gives a badly wrong value, far from the true \(\Lambda_m^\circ\); only the indirect Kohlrausch-combination route (Result) is reliable.
  • Apply the strict additivity of Step 2 at a finite, non-negligible concentration rather than at the infinite-dilution limit: ion–ion interactions (debye-huckel-type effects) genuinely reduce conductivity below the simple additive sum at any measurable concentration, so \(\Lambda_m^\circ\) as defined here is specifically an extrapolated, idealised limiting quantity, not the molar conductivity actually measured at any real, finite concentration.
Common errors
  • Attempting to extrapolate a weak electrolyte's \(\Lambda_m\) directly to \(c=0\) using the same linear-in-\(\sqrt c\) method valid for strong electrolytes (Fails without, first bullet).
  • Forgetting to weight each ion's limiting conductivity by its stoichiometric count \(\nu\) in Step 2 (e.g. treating \(\text{CaCl}_2\)'s \(\Lambda_m^\circ\) as \(\lambda^\circ(\text{Ca}^{2+})+\lambda^\circ(\text{Cl}^-)\) rather than \(\lambda^\circ(\text{Ca}^{2+})+2\lambda^\circ(\text{Cl}^-)\)).
  • Confusing specific conductivity \(\kappa\) (a bulk solution property, depending on total ion concentration) with molar conductivity \(\Lambda_m\) (normalised per mole, Step 1), which behaves very differently as concentration changes.
Discussion

Friedrich Kohlrausch established the empirical law of independent ionic migration in 1875–1876, from careful, systematic conductivity measurements across a wide range of electrolytes; the striking additivity he observed — that an ion's contribution to conductivity is essentially the same regardless of which counter-ion it is paired with, at infinite dilution — was itself an important early piece of evidence supporting the then-developing Arrhenius theory of electrolytic dissociation into independently mobile ions.

The theoretical justification for the empirical \(\sqrt c\) dependence in strong electrolytes (Step 3) came decades later, from Debye and Hückel's ion-atmosphere theory (debye-huckel) together with Onsager's subsequent refinement explicitly incorporating ionic mobility and the resulting relaxation and electrophoretic retardation effects — another instance of the historically recurring empirical-law-before-theoretical-derivation pattern seen elsewhere in this network.

Common misconception: that molar conductivity is a fixed, concentration-independent property of an electrolyte, in the way a rate constant or equilibrium constant might loosely be thought of as fixed for a given system. \(\Lambda_m\) is explicitly concentration-dependent for every real electrolyte (Steps 3–4); only the extrapolated, infinite-dilution limit \(\Lambda_m^\circ\) is a fixed, characteristic reference value.

Worked examples
1
\[ \Lambda_m^\circ(\text{HCl})=426.2,\ \Lambda_m^\circ(\text{CH}_3\text{COONa})=91.0,\ \Lambda_m^\circ(\text{NaCl})=126.5\ (\text{all }\text{S cm}^2\text{mol}^{-1}) \]
To find \(\Lambda_m^\circ(\text{CH}_3\text{COOH})\), a weak acid, three strong electrolytes are chosen whose ion combination sums to exactly the acetic acid ion pair: \(\text{HCl}+\text{CH}_3\text{COONa}-\text{NaCl}\) supplies \(\text{H}^++\text{CH}_3\text{COO}^-\) once the common \(\text{Na}^+\) and \(\text{Cl}^-\) exactly cancel. A
2
\[ \Lambda_m^\circ(\text{CH}_3\text{COOH}) = 426.2+91.0-126.5 = 390.7\ \text{S cm}^2\text{mol}^{-1} \]
Applying the Result's combination rule directly with the three tabulated strong-electrolyte values gives acetic acid's limiting molar conductivity, a quantity that could never be obtained by direct extrapolation of acetic acid's own conductivity data (Step 4). A
\[ \Lambda_m^\circ(\text{CH}_3\text{COOH}) \approx 390.7\ \text{S cm}^2\text{mol}^{-1} \]

Reading. Three easily, directly measured strong-electrolyte limiting conductivities combine, via simple addition and subtraction, to give a weak electrolyte's limiting conductivity that is inaccessible to direct measurement.

Scope. The identical combination technique applies to any weak electrolyte, provided a suitable set of strong electrolytes supplying exactly its constituent ions (with any common "extra" ions cancelling) can be identified.

Further example. Determine the limiting molar conductivity of calcium chloride, \(\Lambda_m^\circ(\text{CaCl}_2)\), given the limiting ionic conductivities \(\lambda^\circ(\text{Ca}^{2+}) = 119.0\ \text{S cm}^2\text{mol}^{-1}\) and \(\lambda^\circ(\text{Cl}^-) = 76.4\ \text{S cm}^2\text{mol}^{-1}\). Additionally, calculate the degree of dissociation \(\alpha\) of a \(0.010\ \text{mol L}^{-1}\) solution of \(\text{CaCl}_2\) at \(25^\circ\text{C}\), given that its measured molar conductivity is \(\Lambda_m = 218.0\ \text{S cm}^2\text{mol}^{-1}\).

1
\[ \text{Dissociation: } \text{CaCl}_2 \rightarrow \text{Ca}^{2+} + 2\text{Cl}^- \]
Identify the stoichiometry of the ions produced per formula unit. One mole of calcium chloride yields one mole of calcium ions and two moles of chloride ions. A
2
\[ \Lambda_m^\circ(\text{CaCl}_2) = \lambda^\circ(\text{Ca}^{2+}) + 2\lambda^\circ(\text{Cl}^-) \]
Apply Kohlrausch's law of independent ionic migration. The limiting molar conductivity is the sum of the limiting ionic conductivities, weighted by their stoichiometric coefficients in the dissociation equation. A
3
\[ \Lambda_m^\circ(\text{CaCl}_2) = 119.0 + 2(76.4) = 119.0 + 152.8 = 271.8\ \text{S cm}^2\text{mol}^{-1} \]
Substitute the given ionic conductivities into the expression. This value represents the conductivity at infinite dilution where ion-ion interactions are negligible. A
4
\[ \alpha = \frac{\Lambda_m}{\Lambda_m^\circ} \]
Use the definition of the degree of dissociation for an electrolyte. It is the ratio of the molar conductivity at a specific concentration to the limiting molar conductivity. A
5
\[ \alpha = \frac{218.0}{271.8} \approx 0.802 \]
Calculate the degree of dissociation. The result indicates that at \(0.010\ \text{mol L}^{-1}\), approximately \(80.2\%\) of the calcium chloride is effectively dissociated into free ions contributing to conductivity. A
\[ \Lambda_m^\circ(\text{CaCl}_2) = 271.8\ \text{S cm}^2\text{mol}^{-1}, \quad \alpha \approx 0.802 \]

Reading. The limiting molar conductivity is determined by the sum of the individual ionic contributions, accounting for the two chloride ions per formula unit. The degree of dissociation quantifies the extent to which the electrolyte is ionized at the given concentration, showing that even strong electrolytes do not reach 100% effective dissociation at finite concentrations due to ion pairing and interionic attractions.

Scope. This calculation holds for dilute aqueous solutions where Kohlrausch's law applies. The degree of dissociation calculated here is an "apparent" degree based on conductivity, which may differ slightly from the true chemical degree of dissociation due to the formation of ion pairs that do not contribute fully to conductivity.

Problems
  1. Given \(\lambda^\circ(\text{Na}^+)=50.1\) and \(\lambda^\circ(\text{Cl}^-)=76.4\) (S cm\(^2\)mol\(^{-1}\)), compute \(\Lambda_m^\circ(\text{NaCl})\) using Step 2, and compare with the value used in Worked Example 1.
    SolutionBy Step 2, \(\Lambda_m^\circ(\text{NaCl}) = \lambda^\circ(\text{Na}^+)+\lambda^\circ(\text{Cl}^-) = 50.1+76.4 = 126.5\ \text{S cm}^2\text{mol}^{-1}\), matching the value used directly in Worked Example 1 exactly, confirming the additive relationship.
  2. Using \(\Lambda_m^\circ(\text{KOH})=271.5\), \(\Lambda_m^\circ(\text{NH}_4\text{Cl})=149.7\), and \(\Lambda_m^\circ(\text{KCl})=149.9\) (all S cm\(^2\)mol\(^{-1}\)), find \(\Lambda_m^\circ(\text{NH}_4\text{OH})\), a weak base.
    SolutionChoosing the combination that supplies \(\text{NH}_4^++\text{OH}^-\) once common ions cancel: \(\text{KOH}+\text{NH}_4\text{Cl}-\text{KCl}\) supplies \(\text{K}^++\text{OH}^-+\text{NH}_4^++\text{Cl}^--\text{K}^+-\text{Cl}^- = \text{NH}_4^++\text{OH}^-\). \(\Lambda_m^\circ(\text{NH}_4\text{OH}) = 271.5+149.7-149.9 = 271.3\ \text{S cm}^2\text{mol}^{-1}\).
  3. Explain, using Steps 3 and 4, why a plot of \(\Lambda_m\) vs. \(\sqrt c\) for acetic acid curves sharply upward as \(c\to0\), while the same plot for HCl remains essentially linear across a comparable concentration range.
    SolutionHCl, a strong electrolyte, is essentially fully dissociated at every concentration in the range, so its \(\Lambda_m\) reflects only the modest ion–ion interaction (Debye–Hückel-type) retardation captured by Step 3's linear square-root law. Acetic acid, a weak electrolyte, has a degree of dissociation that itself rises sharply as concentration falls toward zero (per the Ostwald dilution law, referenced in the Hypotheses); this additional, strongly concentration-dependent rise in the fraction of dissociated ions, superimposed on the mobility effects alone, produces the characteristic sharp, non-linear upward curvature of \(\Lambda_m\) as \(c\to0\) described in Step 4, in clear contrast to HCl's near-linear behaviour.
  4. Calculate the limiting ionic conductivity of the nitrate ion, \(\lambda^\circ(\text{NO}_3^-)\), given \(\Lambda_m^\circ(\text{KNO}_3)=145.0\) and \(\Lambda_m^\circ(\text{KCl})=149.9\) \(\text{S cm}^2\text{mol}^{-1}\), and the known value \(\lambda^\circ(\text{Cl}^-)=76.4\ \text{S cm}^2\text{mol}^{-1}\).
    SolutionFirst, determine \(\lambda^\circ(\text{K}^+)\) from the KCl data: \(\lambda^\circ(\text{K}^+) = \Lambda_m^\circ(\text{KCl}) - \lambda^\circ(\text{Cl}^-) = 149.9 - 76.4 = 73.5\ \text{S cm}^2\text{mol}^{-1}\). Then, use the KNO\(_3\) data: \(\lambda^\circ(\text{NO}_3^-) = \Lambda_m^\circ(\text{KNO}_3) - \lambda^\circ(\text{K}^+) = 145.0 - 73.5 = 71.5\ \text{S cm}^2\text{mol}^{-1}\).
  5. A student claims that \(\Lambda_m^\circ(\text{BaCl}_2)\) can be calculated as \(2\lambda^\circ(\text{Ba}^{2+}) + 2\lambda^\circ(\text{Cl}^-)\). Explain why this is incorrect and provide the correct expression for \(\Lambda_m^\circ(\text{BaCl}_2)\) in terms of ionic conductivities.
    SolutionThe claim is incorrect because the stoichiometric coefficients in the chemical formula do not multiply the ionic conductivities in the summation for molar conductivity. \(\Lambda_m^\circ\) is defined per mole of electrolyte. One mole of \(\text{BaCl}_2\) yields one mole of \(\text{Ba}^{2+}\) and two moles of \(\text{Cl}^-\). However, the ionic conductivity \(\lambda^\circ\) is already defined per mole of that specific ion. The correct expression is the sum of the conductivities of the ions produced by one mole of the electrolyte: \(\Lambda_m^\circ(\text{BaCl}_2) = \lambda^\circ(\text{Ba}^{2+}) + 2\lambda^\circ(\text{Cl}^-)\). The factor of 2 applies to the chloride ion because there are two chloride ions per formula unit, but the barium ion coefficient remains 1.