Concentration Dependence of Molar Conductivity

Dissolve two salts in water and dilute both a thousandfold. One shows conductivity that climbs gently and predictably. The other shoots up almost without bound. That contrast — strong versus weak electrolyte behavior — is one of the cleanest diagnostic tools in solution electrochemistry. Molar conductivity, $\Lambda_m$ , measures how well one mole of dissolved ions carries current. It is not constant. It depends on concentration, and the way it depends on concentration reveals the underlying chemistry. Strong electrolytes obey a simple square-root law. Weak electrolytes obey a completely different relationship tied to their degree of dissociation. This article builds the theory from first principles. You will see why strong electrolytes follow Kohlrausch's empirical square-root law. You will see why weak electrolytes do not. You will learn how to extract the limiting molar conductivity, $\Lambda_m^{\circ}$ , for each class by extrapolation. Two worked numerical examples fix the arithmetic, and the discussion sets up Kohlrausch's law of independent migration and the Debye–Hückel–Onsager theory.

Key takeaways

Molar conductivity $\Lambda_m$ decreases as concentration increases, for both strong and weak electrolytes, but for different physical reasons.
Strong electrolytes follow Kohlrausch's law, $\Lambda_m = \Lambda_m^{\circ} - K\sqrt{c}$ , a straight line against $\sqrt{c}$ .
Weak electrolytes show a steep, curved drop because $\Lambda_m$ tracks the degree of dissociation, $\alpha$ , which itself falls as $c$ rises.
$\Lambda_m^{\circ}$ for strong electrolytes comes from linear extrapolation to $c \to 0$ ; for weak electrolytes it must come from Kohlrausch's law of independent migration instead.
Ostwald's dilution law links $\Lambda_m/\Lambda_m^{\circ}$ to the acid dissociation constant $K_a$ for weak electrolytes.
Key idea: the shape of the $\Lambda_m$ versus $\sqrt{c}$ curve — gently sloped line or steep unbounded rise — is a direct fingerprint of whether an electrolyte is fully or only partly dissociated.

The core insight

Molar conductivity falls with rising concentration in every electrolyte, but strong electrolytes fall gently and linearly in $\sqrt{c}$ because ionic mobility is throttled by interionic drag, while weak electrolytes fall steeply and non-linearly because rising concentration suppresses the fraction of molecules that dissociate at all.

The physical picture

Molar conductivity is defined as

\Lambda_m = \frac{\kappa}{c}

where $\kappa$ is the measured conductivity of the solution and $c$ is the molar concentration of the electrolyte. Naively one might expect $\Lambda_m$ to be a fixed property of a salt, since it is already normalized per mole of dissolved substance. Experiment says otherwise: $\Lambda_m$ decreases as $c$ increases, for every electrolyte, but the rate and shape of that decrease split cleanly into two families. For a strong electrolyte — NaCl, KCl, HCl, most soluble salts — dissociation is essentially complete at all measurable concentrations. Every formula unit contributes fully mobile ions. Yet $\Lambda_m$ still drops as $c$ rises, because each ion's motion is retarded by the electrostatic pull of its surrounding cloud of oppositely charged neighbors, the "ionic atmosphere." As concentration increases, this atmosphere becomes denser and more distorted by the moving ion, dragging on its mobility through two effects: the relaxation effect (the atmosphere lags and pulls the ion backward) and the electrophoretic effect (the ion must drag solvent molecules moving oppositely with the counter-ion atmosphere). Friedrich Kohlrausch found empirically that this drag makes $\Lambda_m$ fall linearly in $\sqrt{c}$ , not in $c$ itself — a signature later explained theoretically by Debye, Hückel, and Onsager. For a weak electrolyte — acetic acid, ammonia, weak organic bases — the story is entirely different. These substances dissociate only partially, and the fraction that dissociates, the degree of dissociation $\alpha$ , itself depends on concentration through the equilibrium constant $K_a$ . Diluting a weak acid shifts its equilibrium toward more complete dissociation (Le Chatelier), so $\alpha$ rises sharply as $c \to 0$ . Because $\Lambda_m$ is essentially proportional to $\alpha$ , the molar conductivity of a weak electrolyte shoots up steeply at low concentration and never approaches a finite slope in $\sqrt{c}$ — it diverges toward $\Lambda_m^{\circ}$ in a curved, asymptotic fashion rather than a straight line.

Figure

Molar conductivity versus √c for a strong electrolyte KCl and a weak electrolyte CH₃COOH

The second picture worth holding in mind is mechanistic: what actually happens to a weak electrolyte's population of molecules as it is diluted.

Figure

Dissociation equilibrium shifting with dilution for a weak acid HA

The core relations

\Lambda_m = \frac{\kappa}{c}

where $\kappa$ is conductivity (S m⁻¹ or S cm⁻¹) and $c$ is molar concentration.

\Lambda_m = \Lambda_m^{\circ} - K\sqrt{c} \qquad \text{(Kohlrausch's law, strong electrolytes)}

where $\Lambda_m^{\circ}$ is the limiting molar conductivity at infinite dilution and $K$ is an empirical constant depending on the electrolyte and solvent.

\alpha \approx \frac{\Lambda_m}{\Lambda_m^{\circ}} \qquad \text{(weak electrolytes, low concentration)}

where $\alpha$ is the degree of dissociation.

K_a = \frac{c\,\alpha^2}{1-\alpha} \qquad \text{(Ostwald's dilution law)}

where $K_a$ is the acid (or base) dissociation constant and $c$ is the nominal concentration.

\frac{1}{\Lambda_m} = \frac{1}{\Lambda_m^{\circ}} + \frac{\Lambda_m\, c}{K_a\,(\Lambda_m^{\circ})^2} \qquad \text{(Ostwald linearized form)}

where the left-hand side, plotted against $\Lambda_m c$ , gives a straight line whose intercept is $1/\Lambda_m^{\circ}$ .

Derivation AFrom Ostwald's dilution law to the shape of the weak-electrolyte curve

Goal. Show why $\Lambda_m$ for a weak electrolyte rises steeply and non-linearly as $c \to 0$ , in contrast to the linear-in- $\sqrt c$ behavior of a strong electrolyte, and obtain a working formula relating $\Lambda_m$ , $c$ , and $K_a$ .

Step 1 — set up the equilibrium. Consider a weak acid HA dissociating as

\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-

with nominal concentration $c$ and degree of dissociation $\alpha$ . At equilibrium the concentrations are $c(1-\alpha)$ for HA and $c\alpha$ for each ion, so the equilibrium constant is

K_a = \frac{(c\alpha)(c\alpha)}{c(1-\alpha)} = \frac{c\,\alpha^2}{1-\alpha}

This is Ostwald's dilution law.

Step 2 — connect $\alpha$ to $\Lambda_m$ . Only the dissociated fraction carries current, so the conductivity contributed per mole of nominal solute scales with $\alpha$ :

\Lambda_m \approx \alpha\, \Lambda_m^{\circ}

This approximation neglects the small interionic-drag correction present even in the free ions, which is legitimate here because that correction is far smaller than the huge swing in $\alpha$ itself. Rearranging,

\alpha = \frac{\Lambda_m}{\Lambda_m^{\circ}}

Step 3 — substitute into Ostwald's law. Insert $\alpha = \Lambda_m/\Lambda_m^{\circ}$ into the dissociation constant expression:

K_a = \frac{c\left(\dfrac{\Lambda_m}{\Lambda_m^{\circ}}\right)^2}{1 - \dfrac{\Lambda_m}{\Lambda_m^{\circ}}}

Multiply numerator and denominator by $(\Lambda_m^{\circ})^2$ :

K_a = \frac{c\,\Lambda_m^2}{\Lambda_m^{\circ}\left(\Lambda_m^{\circ}-\Lambda_m\right)}

Step 4 — solve for $\Lambda_m$ as a function of $c$ . Cross-multiplying and collecting terms in $\Lambda_m$ gives a quadratic:

c\,\Lambda_m^2 + K_a\Lambda_m^{\circ}\,\Lambda_m - K_a(\Lambda_m^{\circ})^2 = 0

Solving by the quadratic formula (taking the physically meaningful positive root) shows $\Lambda_m$ is not a linear function of $c$ , nor of $\sqrt{c}$ . As $c \to 0$ , the dominant balance forces $\alpha \to 1$ and $\Lambda_m \to \Lambda_m^{\circ}$ , but the approach is governed by a $1/\sqrt{c}$ -type divergence in the slope, not a finite one. Explicitly, for small $\alpha$ (moderate $c$ ), $K_a \approx c\alpha^2$ , so

\alpha \approx \sqrt{\frac{K_a}{c}}

and therefore

\Lambda_m \approx \Lambda_m^{\circ}\sqrt{\frac{K_a}{c}}

Step 5 — contrast with the strong electrolyte. For a strong electrolyte, $\alpha = 1$ always; the only concentration dependence comes from interionic drag, captured empirically as a term linear in $\sqrt{c}$ subtracted from a constant $\Lambda_m^{\circ}$ . For a weak electrolyte, $\Lambda_m$ itself scales as $1/\sqrt{c}$ near moderate concentrations — a completely different, steep, unbounded-looking curve that only flattens out at extremely low $c$ where $\alpha \to 1$ .

\boxed{\Lambda_m \approx \Lambda_m^{\circ}\sqrt{\dfrac{K_a}{c}} \ \ (\text{weak electrolyte, moderate } c) \qquad \text{vs.} \qquad \Lambda_m = \Lambda_m^{\circ} - K\sqrt{c}\ \ (\text{strong electrolyte})}

Worked examples

Example 1 — Extrapolating $\Lambda_m^{\circ}$ for a strong electrolyte (KCl). Conductivity data for aqueous KCl at 298 K (illustrative, consistent with standard tables):

$c$ / mol L⁻¹	$\sqrt{c}$	$\Lambda_m$ / S cm² mol⁻¹
0.0005	0.02236	148.9
0.0020	0.04472	148.0
0.0100	0.10000	141.3

Kohlrausch's law says $\Lambda_m = \Lambda_m^{\circ} - K\sqrt{c}$ . Use the first two points to find the slope:

K = -\frac{\Lambda_m(0.0020) - \Lambda_m(0.0005)}{\sqrt{0.0020} - \sqrt{0.0005}} = -\frac{148.0 - 148.9}{0.04472 - 0.02236}

K = -\frac{-0.9}{0.02236} = 40.25\ \text{S cm}^2\,\text{mol}^{-1}\,(\text{mol L}^{-1})^{-1/2}

Extrapolate to $\sqrt{c} = 0$ using the first data point:

\Lambda_m^{\circ} = \Lambda_m(0.0005) + K\sqrt{0.0005} = 148.9 + 40.25(0.02236) = 148.9 + 0.90 = 149.8\ \text{S cm}^2\,\text{mol}^{-1}

Check against the third point: predicted $\Lambda_m = 149.8 - 40.25(0.1000) = 149.8 - 4.03 = 145.8$ , reasonably close to the tabulated 141.3 (real data show mild curvature at higher $c$ , which is why extrapolation should lean on the most dilute points). The accepted literature value is $\Lambda_m^{\circ}(\text{KCl}) = 149.9\ \text{S cm}^2\,\text{mol}^{-1}$ at 298 K — our extrapolation from dilute data lands right on it.

Example 2 — Degree of dissociation and $K_a$ for acetic acid from conductivity. A 0.0100 mol L⁻¹ aqueous solution of acetic acid at 298 K has a measured molar conductivity $\Lambda_m = 16.18$ S cm² mol⁻¹. Using Kohlrausch's law of independent migration (foreshadowed, derived properly in the next article), the limiting molar conductivity of acetic acid is obtained by combining tabulated ionic values:

\Lambda_m^{\circ}(\text{CH}_3\text{COOH}) = \lambda^{\circ}(\text{H}^+) + \lambda^{\circ}(\text{CH}_3\text{COO}^-) = 349.6 + 40.9 = 390.5\ \text{S cm}^2\,\text{mol}^{-1}

Step A — degree of dissociation.

\alpha = \frac{\Lambda_m}{\Lambda_m^{\circ}} = \frac{16.18}{390.5} = 0.04144

So only about 4.14% of the acetic acid molecules are dissociated at this concentration — consistent with it being a weak electrolyte.

Step B — dissociation constant via Ostwald's law.

K_a = \frac{c\,\alpha^2}{1-\alpha} = \frac{(0.0100)(0.04144)^2}{1 - 0.04144} = \frac{(0.0100)(1.717\times10^{-3})}{0.9586}

K_a = \frac{1.717\times10^{-5}}{0.9586} = 1.79\times10^{-5}

This matches the accepted literature value $K_a(\text{CH}_3\text{COOH}) = 1.8\times10^{-5}$ at 298 K almost exactly — a strong internal consistency check, confirming both the conductivity-based $\alpha$ and the tabulated ionic conductivities are mutually compatible.

Step C — sanity check by comparing to the derived approximate formula. From Step 5 of the derivation, $\Lambda_m \approx \Lambda_m^{\circ}\sqrt{K_a/c}$ :

\Lambda_m \approx 390.5\sqrt{\frac{1.8\times10^{-5}}{0.0100}} = 390.5\sqrt{1.8\times10^{-3}} = 390.5(0.04243) = 16.57\ \text{S cm}^2\,\text{mol}^{-1}

This is within 2.4% of the measured 16.18 S cm² mol⁻�1 — good agreement, given the approximation $1-\alpha\approx1$ used in deriving that shortcut formula.

Limiting cases & common traps

Situation	What happens	Common error
Strong electrolyte, $c \to 0$	$\Lambda_m \to \Lambda_m^{\circ}$ linearly in $\sqrt{c}$	Plotting against $c$ instead of $\sqrt{c}$ , producing a curve and a false "no clean limit" conclusion
Weak electrolyte, $c \to 0$	$\alpha \to 1$ , $\Lambda_m \to \Lambda_m^{\circ}$ , but curve is steep and non-linear	Trying to extrapolate $\Lambda_m^{\circ}$ directly from weak-electrolyte data — the curve diverges too fast near $c=0$ to extrapolate reliably
Strong electrolyte, moderate $c$	Interionic drag (relaxation + electrophoretic effects) dominates	Attributing the decrease in $\Lambda_m$ to "incomplete dissociation" — strong electrolytes are still essentially 100% dissociated
Weak electrolyte, moderate $c$	Equilibrium suppresses $\alpha$ ; $\Lambda_m$ falls fast with rising $c$	Confusing $\Lambda_m^{\circ}$ obtained via Kohlrausch's independent-migration law (needed for weak electrolytes) with one obtained via direct extrapolation (valid only for strong electrolytes)

The chemist behind it

Friedrich Kohlrausch spent decades in the late nineteenth century making exquisitely precise conductivity measurements, using a self-designed alternating-current bridge to avoid the electrode polarization that had corrupted earlier direct-current attempts. His painstaking dilution series on dozens of salts revealed the square-root law that bears his name, long before any microscopic theory of ionic atmospheres existed. He also discovered that limiting molar conductivities could be built additively from independent ionic contributions — the law of independent migration of ions, the natural next topic after this one. The theoretical explanation of why the square-root law holds came only decades later, from Peter Debye, Erich Hückel, and Lars Onsager, who modeled the ionic atmosphere explicitly and derived the Debye–Hückel–Onsager equation for the slope $K$ .

Origins · 1 minFriedrich Kohlrausch and how this was discovered

Friedrich Kohlrausch

1840–1910 · German physicist whose precision AC conductivity bridge revealed the √c law and the additivity of ionic conductivities

Ask the tutor: How does Kohlrausch's law of independent migration let us predict Λm° for a weak electrolyte we've never measured directly?

Can a weak electrolyte ever show a straight-line Λm vs √c plot?

Only over a very narrow, very dilute concentration window, and even then it is an approximation, not an exact law. The true behavior of a weak electrolyte is controlled by the dissociation equilibrium through Ostwald's dilution law, which predicts a steep, curved rise in $\Lambda_m$ as $c \to 0$ . Treating that curve as linear in $\sqrt c$ over any appreciable range will give a poor, unreliable extrapolated intercept.

How is Λm° actually measured for a weak electrolyte if extrapolation fails?

Chemists invoke Kohlrausch's law of independent migration of ions: at infinite dilution, each ion migrates independently and contributes a fixed limiting ionic conductivity, $\lambda^{\circ}$ , regardless of its partner ion. Summing tabulated $\lambda^{\circ}$ values for H⁺ and CH₃COO⁻, for instance, gives $\Lambda_m^{\circ}$ for acetic acid directly, sidestepping the need to extrapolate the poorly behaved weak-electrolyte curve.

Does temperature change these relationships qualitatively?

No, the qualitative shapes persist at any fixed temperature, but the numbers shift. Higher temperature lowers water's viscosity, raising ionic mobilities and therefore $\Lambda_m^{\circ}$ for both classes. For weak electrolytes, $K_a$ itself is temperature-dependent through its own reaction thermodynamics, so the curved Ostwald relationship shifts as well, but the strong-versus-weak contrast in shape remains intact at every temperature.

Is there a clean numerical cutoff between "strong" and "weak" based on α?

Not a rigid one, but a useful rule of thumb: strong electrolytes have $\alpha$ effectively equal to 1 (typically above about 0.95–0.99) at all experimentally accessible concentrations, while weak electrolytes show $\alpha$ varying substantially with concentration, often falling well below 0.1 at ordinary concentrations like 0.01–0.1 mol/L, as seen for acetic acid above.