= Ideal weak electrolyte dilution law
{title2=$K_a=c\alpha^2/(1-\alpha)$}
Consider a neutral <monoprotic acid> $\mathrm{HA}$ dissolved in <water>, with no added salts or other acids. Assume dilute ideal <thermodynamic activities>, fixed <temperature>, <water> <thermodynamic activity> one, and negligible <self-ionization of water>. Write $C_T$ for its <analytical concentration>, $c=C_T/c^\circ$ with $c^\circ=1\,\mathrm{mol\,L^{-1}}$, and $\alpha$ for its <weak acid dissociation fraction>. Conservation of acid units and <electric charge> gives
$$
[\mathrm{HA}]=C_T(1-\alpha),\qquad
[\mathrm A^-]=[\mathrm H_3\mathrm O^+]=C_T\alpha.
$$
Here $\mathrm A^-$ is the <conjugate base>, and $\mathrm H^+$ is conventional shorthand for hydrated <hydrogen ions>, represented by <hydronium> in this model. Substitution into the <dimensionless> <acid dissociation constant> gives
$$
K_a=\frac{[\mathrm H_3\mathrm O^+][\mathrm A^-]}{c^\circ[\mathrm{HA}]}
=\frac{c\alpha^2}{1-\alpha},\qquad
c\alpha^2+K_a\alpha-K_a=0.
$$
Solving this <quadratic equation> and rejecting its negative <root of a polynomial> gives an exact result within the stated model:
$$
\alpha=\frac{\sqrt{K_a^2+4K_ac}-K_a}{2c}
=\frac{2K_a}{K_a+\sqrt{K_a^2+4K_ac}}.
$$
The second form avoids subtracting nearly equal numbers. Unlike the approximation $\alpha\approx\sqrt{K_a/c}$, it retains depletion of neutral <weak acid> and always satisfies $0<\alpha<1$ for positive $K_a,c$. The broader <weak acid mass and charge balances> restore <self-ionization of water> when that omitted contribution matters.
\Image[/weak-electrolyte-fraction-and-ion-concentration.png]
{title=<Dilution> increases the ionized fraction while decreasing the absolute <hydronium> <concentration> in an ideal <weak electrolyte> model}
{description=Original calculated curves for stipulated <dimensionless> <acid dissociation constant> $K_a=10^{-5}$ at fixed <temperature>, not measurements of a named <chemical acid>. The shared horizontal <analytical concentration> axis decreases to the right. Separate panels show <weak acid dissociation fraction> and $[\mathrm H_3\mathrm O^+]=C_T\alpha$; a dashed comparison shows a completely ionized <monoprotic acid> at the same $C_T$. Both omit <self-ionization of water>, other <ions> and nonideal <thermodynamic activities>.}
For an original worked example, stipulate $K_a=10^{-5}$ without identifying a real substance. At $C_T=10^{-2}\,\mathrm{mol\,L^{-1}}$, the <weak acid dissociation fraction> is approximately $0.0311267$, or $3.11267\%$, and $[\mathrm H_3\mathrm O^+]=3.11267\times10^{-4}\,\mathrm{mol\,L^{-1}}$. A hundredfold <dilution> to $C_T=10^{-4}\,\mathrm{mol\,L^{-1}}$ raises the fraction to $0.270156$, or $27.0156\%$, but lowers $[\mathrm H_3\mathrm O^+]$ to $2.70156\times10^{-5}\,\mathrm{mol\,L^{-1}}$. A completely ionized <strong acid> at $C_T=10^{-4}\,\mathrm{mol\,L^{-1}}$ would have $[\mathrm H_3\mathrm O^+]\approx10^{-4}\,\mathrm{mol\,L^{-1}}$: less than the more concentrated <weak acid> in the first calculation, despite its greater converted fraction.
The directions of both changes follow without relying on the plot. Put $x=[\mathrm H_3\mathrm O^+]/c^\circ=c\alpha$. Rearranging the <chemical equilibrium> relation gives
$$
c=x+\frac{x^2}{K_a},\qquad
\frac{\mathrm dc}{\mathrm dx}=1+\frac{2x}{K_a}>0,\qquad
\alpha=\frac{K_a}{K_a+x}.
$$
Thus smaller $c$ forces smaller $x$, which in turn makes $\alpha$ larger. The <derivative> proves that total <hydronium> <concentration> and converted fraction move in opposite directions during <dilution>. In this model, adding <solvent> also makes additional acid units ionize at fixed total acid <amount of substance>, even though their <ions> are spread over more <volume (thermodynamics)>. That distinction explains why “more ionized” alone does not predict greater <electrical conductivity>.
Do not extend the no-background model to unlimited <dilution>: its prediction $[\mathrm H_3\mathrm O^+]\to0$ omits <water>'s own <ions>. The <weak acid mass and charge balances> include that background. Added salts, other acids or bases, and nonideal <activity coefficients> also change the relation between <analytical concentration>, converted fraction and <hydronium> <concentration>.
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