Ostwald Dilution Law

Dissolve acetic acid in water and the solution barely conducts. Dissolve the same molar amount of hydrochloric acid, and the current surges. Both are monoprotic acids at the same concentration, yet they behave completely differently. The difference is dissociation: HCl splits apart almost completely, while acetic acid barely ionizes at all. The Ostwald Dilution Law explains this quantitatively. It connects three measurable quantities: the degree of dissociation of a weak electrolyte, its equilibrium dissociation constant, and the molar conductivity we measure in the lab. This link matters because conductivity is easy to measure but dissociation constants are the thermodynamically meaningful quantity chemists actually want. This article derives the law from the dissociation equilibrium and from the conductivity definition of degree of dissociation. You will see the derivation step by step. Then you will apply it to real numbers for acetic acid. Finally, we connect it to Kohlrausch's Law, which fixes the one input the whole law needs: the limiting molar conductivity.

Key takeaways

Ostwald's Dilution Law relates the dissociation constant $K_a$ of a weak electrolyte to its degree of dissociation $\alpha$ and concentration $c$ .
The degree of dissociation is measured operationally as $\alpha = \Lambda_m/\Lambda_m^\circ$ , the ratio of molar conductivity to its value at infinite dilution.
The law predicts $\alpha$ increases as concentration decreases, approaching complete dissociation only in the limit of infinite dilution.
It applies strictly to weak electrolytes; strong electrolytes show no such simple relation because they are already fully dissociated.
Kohlrausch's Law of independent migration of ions supplies the $\Lambda_m^\circ$ value that the Ostwald law needs, since weak electrolytes cannot be extrapolated directly.
Key idea: Treating dissociation of a weak electrolyte as a genuine chemical equilibrium, with conductivity measuring the fraction dissociated, turns Ohm's-law-type data into a thermodynamic equilibrium constant.

The core insight

A weak electrolyte dissociates only partially, and because the fraction dissociated, $\alpha$ , is exactly what fixes how many ions are free to carry current, the mass-action expression for the dissociation equilibrium translates directly into a relation between the measured conductivity ratio $\Lambda_m/\Lambda_m^\circ$ and the equilibrium constant $K_a$ , divided out over the total concentration $c$ .

The physical picture

Consider a weak monobasic acid HA dissolved in water at total (formal) concentration $c$ . Unlike a strong electrolyte such as HCl, HA does not dissociate completely. Instead it establishes a genuine chemical equilibrium:

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

At equilibrium, only a fraction $\alpha$ of the original HA molecules has split into ions; the rest remains as intact, electrically neutral molecules. This $\alpha$ is called the degree of dissociation, and it lies between 0 (no dissociation) and 1 (complete dissociation). The crucial experimental link is this: molar conductivity $\Lambda_m$ of the solution depends on how many ions are actually present and free to migrate under an applied field. If dissociation were complete, we would measure the limiting molar conductivity $\Lambda_m^\circ$ (the value at infinite dilution, where every ion moves independently and unhindered by neighbors — see Molar Conductivity). Because only a fraction $\alpha$ of the electrolyte is actually ionized at concentration $c$ , the observed molar conductivity is reduced in proportion:

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

This operational definition of $\alpha$ is what turns a conductivity measurement into a chemical equilibrium calculation. Once $\alpha$ is known at a given $c$ , the equilibrium constant $K_a$ follows from the mass-action law applied to the dissociation reaction above. The picture below shows the microscopic contrast: dilute weak acid solutions contain mostly intact molecules with only a few free ions, while strong electrolytes are essentially fully ionized at any ordinary concentration.

Figure

Dissociation equilibrium of a weak electrolyte HA at concentration c, degree of dissociation α

The core relations

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

where: $K_a$ is the dissociation constant, $c$ the formal (total) concentration, $\alpha$ the degree of dissociation.

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

where: $\Lambda_m$ is the molar conductivity measured at concentration $c$ , $\Lambda_m^\circ$ is the limiting (infinite-dilution) molar conductivity.

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

where: this is the same law written entirely in terms of measurable conductivities, eliminating $\alpha$ explicitly.

\alpha \approx \sqrt{\frac{K_a}{c}} \quad (\alpha \ll 1)

where: valid in the dilute limit, showing $\alpha$ scales as $c^{-1/2}$ — the qualitative signature of a weak electrolyte.

Derivation AFrom dissociation equilibrium to the dilution law

Goal: derive the relation between $K_a$ , $\alpha$ , and $c$ , then express it in terms of conductivities alone.

Step 1. Set up the equilibrium concentrations. Let the weak acid HA have total stoichiometric concentration $c$ (mol/L). At equilibrium, a fraction $\alpha$ has dissociated, so:

[\text{HA}] = c(1-\alpha), \qquad [\text{H}^+] = c\alpha, \qquad [\text{A}^-] = c\alpha

Step 2. Write the mass-action expression. The thermodynamic equilibrium constant for HA ⇌ H⁺ + A⁻, treating activity coefficients as unity (a good approximation only in dilute solution), is:

K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}

Step 3. Substitute the equilibrium concentrations.

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

This is Ostwald's Dilution Law in its standard form: a single algebraic relation between $K_a$ , $c$ , and $\alpha$ .

Step 4. Bring in conductivity to make $\alpha$ measurable. We cannot see molecules dissociate directly, but we can measure $\Lambda_m$ , the molar conductivity at concentration $c$ . By definition:

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

where $\kappa$ is the measured specific conductivity. At infinite dilution the electrolyte is (in the ideal limit) fully dissociated and ions move without interionic drag, giving $\Lambda_m^\circ$ . Because conductivity is (to good approximation) proportional to the concentration of free ions actually present, the ratio of the two conductivities equals the fraction of molecules that are ionized:

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

Step 5. Substitute $\alpha$ into the mass-action result. Replacing every $\alpha$ in Step 3 with $\Lambda_m/\Lambda_m^\circ$ :

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

Step 6. Clear the compound fraction. 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 7. Extract the dilute-solution limit. When $\alpha \ll 1$ (very weak electrolyte, or not too dilute), $1-\alpha \approx 1$ , so $K_a \approx c\alpha^2$ , giving:

\alpha \approx \sqrt{K_a/c}

This shows that as $c \to 0$ , $\alpha \to 1$ : dilution always drives dissociation toward completion, which is the qualitative content of the "dilution" in the law's name — diluting the solution shifts the equilibrium toward more ions, exactly as Le Chatelier's principle demands for a reaction that increases particle number.

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

Worked examples

Example 1 — Foundational: finding $\alpha$ and $K_a$ for acetic acid from conductivity data. A 0.0100 mol/L aqueous solution of acetic acid (CH₃COOH) has measured molar conductivity $\Lambda_m = 16.2\ \text{S cm}^2\text{mol}^{-1}$ at 25 °C. The limiting molar conductivity, obtained from Kohlrausch's Law using tabulated ionic conductivities, is:

\Lambda_m^\circ = \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 1 — degree of dissociation:

\alpha = \frac{\Lambda_m}{\Lambda_m^\circ} = \frac{16.2}{390.5} = 0.0415

So only about 4.15% of the acetic acid molecules are ionized at this concentration — consistent with acetic acid being a weak acid.

Step 2 — dissociation constant:

K_a = \frac{c\alpha^2}{1-\alpha} = \frac{(0.0100)(0.0415)^2}{1-0.0415}

K_a = \frac{(0.0100)(1.72\times10^{-3})}{0.9585} = \frac{1.72\times10^{-5}}{0.9585} = 1.80\times10^{-5}

Sanity check: the accepted literature value for acetic acid's $K_a$ at 25 °C is $1.75\times10^{-5}$ , essentially matching our result (small discrepancy from rounding and from neglecting activity coefficients at this concentration, which are not exactly 1). The order of magnitude and the small $\alpha$ both confirm acetic acid is weak, as expected.

Example 2 — Applied: predicting conductivity at a new concentration and testing the $\alpha \propto c^{-1/2}$ scaling. Using the $K_a = 1.75\times10^{-5}$ (accepted value) and the same $\Lambda_m^\circ = 390.5\ \text{S cm}^2\text{mol}^{-1}$ , predict $\Lambda_m$ for a tenfold more dilute solution, $c = 1.00\times10^{-3}$ mol/L, and compare with the dilute-limit approximation.

Step 1 — solve the quadratic exactly. From $K_a = c\alpha^2/(1-\alpha)$ , rearrange:

c\alpha^2 + K_a\alpha - K_a = 0

With $c = 1.00\times10^{-3}$ and $K_a = 1.75\times10^{-5}$ :

(1.00\times10^{-3})\alpha^2 + (1.75\times10^{-5})\alpha - 1.75\times10^{-5} = 0

Using the quadratic formula with $a = 1.00\times10^{-3}$ , $b = 1.75\times10^{-5}$ , $c' = -1.75\times10^{-5}$ :

\alpha = \frac{-b+\sqrt{b^2-4ac'}}{2a} = \frac{-1.75\times10^{-5}+\sqrt{(1.75\times10^{-5})^2+4(1.00\times10^{-3})(1.75\times10^{-5})}}{2(1.00\times10^{-3})}

The discriminant term dominates: $4ac'$ magnitude $= 7.00\times10^{-8}$ , so $\sqrt{7.00\times10^{-8}+3.06\times10^{-10}} \approx \sqrt{7.03\times10^{-8}} = 2.652\times10^{-4}$ .

\alpha = \frac{-1.75\times10^{-5}+2.652\times10^{-4}}{2.00\times10^{-3}} = \frac{2.477\times10^{-4}}{2.00\times10^{-3}} = 0.1239

Step 2 — compare with the dilute approximation.

\alpha_{\text{approx}} = \sqrt{K_a/c} = \sqrt{1.75\times10^{-5}/1.00\times10^{-3}} = \sqrt{0.0175} = 0.1323

The approximation overestimates $\alpha$ by about 6.8%, since $\alpha = 0.124$ is no longer negligibly small compared to 1, so dropping $(1-\alpha)$ introduces visible error.

Step 3 — predicted conductivity.

\Lambda_m = \alpha \Lambda_m^\circ = (0.1239)(390.5) = 48.4\ \text{S cm}^2\text{mol}^{-1}

Sanity check on scaling: going from $c = 0.0100$ to $c=0.00100$ (tenfold dilution) increased $\alpha$ from 0.0415 to 0.124, roughly a factor of 3 — close to the $\sqrt{10} \approx 3.16$ predicted by the $\alpha \propto c^{-1/2}$ law, confirming the qualitative trend even though the exact quadratic is needed for precision.

Limiting cases & common traps

Situation	What happens	Why
$c \to 0$	$\alpha \to 1$	Dilution favors the side with more particles (Le Chatelier)
$c$ large, weak acid	$\alpha$ small, $K_a \approx c\alpha^2$	$(1-\alpha)\approx 1$ valid dilute-limit approximation
Strong electrolyte (HCl, NaOH)	Law does not apply	Already essentially fully dissociated at all normal concentrations; no equilibrium to model
Weak electrolyte, activities ignored	Small systematic error in $K_a$	True thermodynamic $K_a$ uses activities, not concentrations; error grows with $c$
Need $\Lambda_m^\circ$ for a weak acid	Cannot extrapolate $\Lambda_m$ vs $\sqrt c$ directly	Weak electrolyte curves are too steep near $c=0$ ; use Kohlrausch's Law of independent ionic migration instead

The chemist behind it

Friedrich Wilhelm Ostwald derived this law in 1888 as part of his broader program applying the law of mass action, newly formalized by Guldberg and Waage, to ionic equilibria in solution. Ostwald was a fierce advocate of the Arrhenius theory of electrolytic dissociation at a time when many chemists still doubted that salts and acids split into charged ions in solution. By showing that conductivity data for weak acids obeyed a simple mass-action relation with concentration-independent $K_a$ , Ostwald provided some of the most convincing quantitative evidence that dissociation was a real equilibrium process, not merely a mathematical convenience. This work helped cement physical chemistry as a rigorous quantitative discipline distinct from purely descriptive chemistry.

Origins · 1 minFriedrich Wilhelm Ostwald and how this was discovered

Friedrich Wilhelm Ostwald

1853–1932 · Co-founder of physical chemistry; 1909 Nobel Laureate in Chemistry for work on catalysis, chemical equilibrium, and reaction rates

Ask the tutor: How does Kohlrausch's Law of independent ionic migration let us construct Λm° for a weak acid we can never fully dissociate?

Why does diluting a weak electrolyte increase its degree of dissociation?

Dissociation of HA into H⁺ and A⁻ increases the total number of solute particles in solution. By Le Chatelier's principle, diluting the solution — effectively reducing all concentrations — shifts the equilibrium toward the side with more particles, since that side's concentration terms fall less steeply under dilution. Mathematically, $K_a$ is fixed at a given temperature, but $c$ in $K_a = c\alpha^2/(1-\alpha)$ shrinks, forcing $\alpha$ to grow so the product stays constant. In the limit $c \to 0$ , $\alpha \to 1$ : the electrolyte becomes fully dissociated only at infinite dilution.

Why do we need Kohlrausch's Law to find Λm° for a weak electrolyte?

For strong electrolytes, $\Lambda_m^\circ$ can be found by plotting $\Lambda_m$ against $\sqrt{c}$ and extrapolating linearly to $c=0$ , since the Debye–Hückel–Onsager theory predicts this linear form. Weak electrolytes do not follow this linear relation; their $\Lambda_m$ rises steeply and nonlinearly as $c \to 0$ , because $\alpha$ itself is changing rapidly. Direct extrapolation is unreliable. Instead, Kohlrausch's Law of independent migration of ions lets us build $\Lambda_m^\circ$ for the weak electrolyte from tabulated limiting ionic conductivities of its ions, measured independently from well-behaved strong electrolytes.

Does Ostwald's Dilution Law hold exactly, or are there corrections?

It holds only approximately, because the derivation uses concentrations in the mass-action expression rather than thermodynamic activities. At higher concentrations, ion-ion interactions make activity coefficients deviate from 1, so the "constant" $K_a$ extracted from the simple formula drifts slightly with concentration. For dilute weak electrolytes (roughly below 0.01–0.1 mol/L, depending on charge type), the deviation is small enough that the law is an excellent working approximation, which is why it remains standard in introductory treatments.

How is this law used experimentally to find a dissociation constant?

A chemist measures the specific conductivity κ of the weak electrolyte solution at a known concentration $c$ , computes $\Lambda_m = \kappa/c$ , and obtains $\Lambda_m^\circ$ from tabulated limiting ionic conductivities via Kohlrausch's Law. Then $\alpha = \Lambda_m/\Lambda_m^\circ$ gives the degree of dissociation, and substitution into $K_a = c\alpha^2/(1-\alpha)$ yields the dissociation constant. Repeating at several concentrations and checking that $K_a$ stays roughly constant is itself a classic test for confirming that a compound behaves as a weak electrolyte.