Start — why salt water conducts and pure water barely does. Dip two wires from a battery into a glass of pure water and almost no current flows. Stir in a spoon of table salt and a bulb in the circuit lights up. The difference is that salt breaks apart into charged particles — ions — that are free to drift: positive ions creep toward the negative wire, negative ions toward the positive one. That drifting of ions is the electric current inside a liquid. No free ions, no current.

Build — measuring "how well it conducts." We rate a solution with $\kappa$ (kappa), its specific conductivity: picture a 1-cm cube of the liquid and ask how easily current crosses it. More dissolved ions means a bigger $\kappa$. But $\kappa$ secretly mixes two things — the kind of ions and how crowded they are. To compare different electrolytes fairly, chemists divide out the crowding by the concentration $c$ to get the molar conductivity: $$\Lambda_m = \frac{\kappa}{c}$$ Worked number: 0.10 M KCl measures $\kappa \approx 0.0129$ S/cm. Converting $c$ to mol·cm⁻³ ($0.10/1000$), $\Lambda_m = 0.0129 / 10^{-4} \approx 129$ S·cm²/mol — a little below its infinite-dilution ceiling $\Lambda_m^\circ \approx 149.9$.

Deepen — strong vs weak, and one rule-breaking ion. Two patterns show up. Strong electrolytes (KCl, HCl — fully split into ions) sag only gently as you concentrate them, following Kohlrausch's square-root law $\Lambda_m = \Lambda_m^\circ - K\sqrt{c}$, because crowded ions drag on each other through their surrounding "ionic atmosphere." Weak electrolytes (acetic acid) only partly break up: a fraction $\alpha$ of molecules form ions, so $\Lambda_m = \alpha\,\Lambda_m^\circ$ and $\alpha = \Lambda_m/\Lambda_m^\circ$ falls fast as $c$ rises. One ion cheats: H⁺ is about $7\times$ more mobile than Na⁺ because it doesn't physically swim — it "hops" along a chain of water molecules (the Grotthuss mechanism). The sliders map straight onto this: Concentration slides your point along the $\sqrt{c}$ curve, Temperature lifts $\Lambda_m$ (~2% per K as the water thins), and cell distance L and area A set the geometric cell constant $L/A$ that turns a measured resistance $R$ into $\kappa$.

Close — try this in the sim above. (1) Pick the KCl preset and drag Concentration from 0.0001 M up to 1 M while watching "Λm vs √c" — the Kohlrausch line slopes gently downward. (2) Load CH₃COOH and compare its α readout to KCl's α ≈ 1: the weak acid sits far below 1 and drops steeply with c. (3) Open the Mobility Comparison view and see how far the H⁺ and OH⁻ bars overshoot Na⁺ and Li⁺ — proton-hopping made visible.

1From Ohm's law to specific conductivity. For a uniform solution sample of length $L$ and cross-section $A$: $$R = \rho\frac{L}{A}\;\Longrightarrow\;\kappa = \frac{1}{\rho} = \frac{L}{AR}$$ The geometric ratio $L/A$ is called the cell constant; it depends only on the conductivity cell, not the solution.

2Molar conductivity definition. Since $\kappa$ depends on concentration (more ions → higher $\kappa$), divide by molar concentration $c$ to get an intensive property: $$\Lambda_m = \frac{\kappa}{c}$$ Units must agree: $\kappa$ in S·m⁻¹, $c$ in mol·m⁻³, gives $\Lambda_m$ in S·m²·mol⁻¹. (Often tabulated in S·cm²·mol⁻¹ — beware unit conversions.)

3Kohlrausch's Law of Independent Ion Migration. At infinite dilution, ions move independently: $$\Lambda_m^\circ = \nu_+\lambda_+^\circ + \nu_-\lambda_-^\circ$$ For example, for NaCl: $\Lambda_m^\circ$(NaCl) = $\lambda^\circ$(Na⁺) + $\lambda^\circ$(Cl⁻) = 50.1 + 76.4 = 126.5 S·cm²/mol.

4Square-root dependence (Kohlrausch). For strong electrolytes at low concentration: $$\Lambda_m = \Lambda_m^\circ - K\sqrt{c}$$ The $-K\sqrt{c}$ correction comes from Debye-Hückel-Onsager theory: ions slow each other down through ionic atmosphere relaxation and electrophoretic effects.

5Weak electrolytes — Ostwald's Dilution Law. For a weak acid HA dissociating into H⁺ and A⁻ with degree $\alpha$: $$K_a = \frac{c\alpha^2}{1-\alpha},\qquad \alpha = \frac{\Lambda_m}{\Lambda_m^\circ}$$ Substituting: $$\boxed{\;K_a = \frac{c\Lambda_m^2}{\Lambda_m^\circ(\Lambda_m^\circ - \Lambda_m)}\;}$$ Plotting $1/\Lambda_m$ vs $c\Lambda_m$ gives a straight line whose intercept is $1/\Lambda_m^\circ$.

6Ionic mobility connection. $\lambda_i^\circ = z_iFu_i$ where $u_i$ is the ionic mobility (m²/V·s). H⁺ has anomalously high mobility (~3.62×10⁻⁷ m²/V·s) due to Grotthuss mechanism — proton hopping along H-bond chains rather than physical migration of an H⁺ ion.

Problem: $\Lambda_m^\circ$(CH₃COOH) = 390.7 S·cm²/mol (from Kohlrausch's law). At 0.0100 M, the measured $\Lambda_m$ = 16.2 S·cm²/mol. Find $K_a$.
$$\alpha = \frac{16.2}{390.7} = 0.0415$$ $$K_a = \frac{c\alpha^2}{1-\alpha} = \frac{(0.0100)(0.0415)^2}{1-0.0415} = \frac{1.72\times10^{-5}}{0.9585} = 1.80\times10^{-5}$$ Excellent agreement with the literature value $K_a \approx 1.75\times10^{-5}$ for acetic acid.