Adding <sodium hydroxide> to acidified <glycine> successively removes its two titratable <protons>. Two broad <buffer solution> regions appear around the two <pKa> values, separated by a steep <pH> rise near the <isoelectric point>. This is useful evidence for two separated <acid-base chemistry> steps even though the changing <chemical species> are invisible. In the approximation that added <hydroxide> only changes <glycine protonation>, $b=1-\bar z$ is the added base <amount of substance> per initial <amount of substance> of <glycine>. Then $b\approx0.5$, $1$ and $1.5$ correspond to $\mathrm pK_{a1}$, $\mathrm{pI}$ and $\mathrm pK_{a2}$; endpoints are not exactly pure <chemical species>. <Organic Chemistry by John McMurry>[§26.2, Figure 26.2: calculated two-leg titration and half-equivalent interpretation]{ref}

\Image[/glycine-titration-mean-charge-model.png]
{title=Calculated <glycine titration> and mean <electric charge> models}
{description=Upper panel: solid curve includes <self-ionization of water> and <dilution> for $20\,\mathrm{mL}$ of $0.100\,\mathrm{mol\,L^{-1}}$ <glycine> with one equivalent of <hydrochloric acid>, titrated with $0.100\,\mathrm{mol\,L^{-1}}$ <sodium hydroxide>. Dashed curve retains only the <glycine> proton inventory. Lower panel: $\bar z$ against <pH> from the <glycine speciation model>. Both are stipulated ideal models, not laboratory data.}

For the solid-curve model, start with $n_G=0.00200\,\mathrm{mol}$ of <glycine> in $V_0=0.0200\,\mathrm L$ and exactly $n_G$ of fully dissociated <hydrochloric acid>; this fixes the <chloride> inventory, but does not force every <glycine> <molecule> to remain $G^+$. Let the <sodium hydroxide> <concentration> be $C_B=0.100\,\mathrm{mol\,L^{-1}}$ and $b=n_{\mathrm{NaOH}}/n_G$. Assume <volume (thermodynamics)> is additive and no other <ions> are present; stipulate $K_w=10^{-14}$ for <self-ionization of water> at a <temperature> of $25\,\mathrm{{}^\circ C}$. Then
$$
V=V_0+\frac{bn_G}{C_B},\qquad C_T=\frac{n_G}{V},\qquad
[\mathrm{Na^+}]=bC_T,\qquad[\mathrm{Cl^-}]=C_T.
$$
Balancing all positive and negative <electric charges> requires
$$
c^\circ\left(h-\frac{K_w}{h}\right)+C_T\left(b+\bar z-1\right)=0.
$$
Solving this equation gives the solid curve. It remains finite at $b=0$ and beyond $b=2$, where ignoring free <hydronium> and <hydroxide> would fail. At the first <pKa>, the full model gives $b\approx0.436$, rather than $0.5$: appreciable free <hydronium> makes that half-equivalent shortcut inaccurate. At the second <pKa>, $b\approx1.502$. Known acid inventory, <dilution>, and <thermodynamic activity> assumptions are therefore part of any quantitative interpretation, not optional corrections inferred from the curve's shape.


 Glycine titration (source code)