IChO 2003 · Theory problem 17

Working curves for a weak acid

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The absorbance of solutions of the weak acid \(\ce{HX}\) were obtained. Associate the expected form of the resulting working curve with those shown in figure, under the following conditions:

Figure 1

a) Pure aqueous solutions of \(\ce{HX}\) were used. Only the undissociated species \(\ce{HX}\) absorb.

Curve A ( ), Curve B ( ), Curve C ( ), Curve D ( )

b) Pure aqueous solutions of \(\ce{HX}\) were used. Only the anionic species \(\ce{X^-}\) absorb.

Curve A ( ), Curve B ( ), Curve C ( ), Curve D ( )

c) All solutions of \(\ce{HX}\) contain an excess of a strong base. Only the undissociated \(\ce{HX}\) species absorb.

Curve A ( ), Curve B ( ), Curve C ( ), Curve D ( )

d) All solutions of \(\ce{HX}\) contain an excess of a strong acid. Only the undissociated \(\ce{HX}\) species absorb.

Curve A ( ), Curve B ( ), Curve C ( ), Curve D ( )

e) Pure aqueous solutions of \(\ce{HX}\) were used. Both \(\ce{HX}\) and \(\ce{X^-}\) absorb. Measurements were obtained at a wavelength where the molar absorptivities of \(\ce{X^-}\) and \(\ce{HX}\) are equal and different than zero.

Curve A ( ), Curve B ( ), Curve C ( ), Curve D ( )

Official English text of the problem.

Answers

  • a) Curve A — increasing, concave upward.
  • b) Curve B — increasing, concave downward.
  • c) Curve D — essentially zero absorbance.
  • d) Curve C — straight line through the origin.
  • e) Curve C — straight line through the origin.

Solution

Let the total analytical concentration be \[ c=[\ce{HX}]+[\ce{X^-}]. \] The acid dissociation equilibrium is \[ \ce{HX + H2O <=> H3O+ + X^-}, \qquad K_a=\frac{[\ce{H3O+}][\ce{X^-}]}{[\ce{HX}]}. \]

Here, \(\ce{HX}\) is an unspecified monoprotic weak acid and \(\ce{X^-}\) its conjugate base; their structures are not given, so unique molecular SMILES cannot be assigned. The other species used are water \(\ce{H2O}\), O; hydronium \(\ce{H3O+}\), [OH3+]; and hydroxide \(\ce{OH^-}\), [OH-].

At a fixed wavelength and optical path length \(\ell\), the Beer–Lambert law gives the absorbance \(\mathcal A\): \[ \mathcal A =\ell\left( \varepsilon_{\ce{HX}}[\ce{HX}] +\varepsilon_{\ce{X^-}}[\ce{X^-}] \right). \] With concentrations in \(\mathrm{mol\,L^{-1}}\), \(\ell\) in \(\mathrm{cm}\), and molar absorptivities in \(\mathrm{L\,mol^{-1}\,cm^{-1}}\), absorbance is dimensionless.

a) In pure aqueous acid solutions, neglecting water’s contribution to the hydronium concentration, put \[ x=[\ce{X^-}]=[\ce{H3O+}], \qquad [\ce{HX}]=c-x. \] Then \[ K_a=\frac{x^2}{c-x}, \qquad x=\frac{\sqrt{K_a^2+4K_ac}-K_a}{2}. \] This function increases with \(c\) but is concave downward: \[ \frac{d^2x}{dc^2} =-\frac{2K_a^2}{(K_a^2+4K_ac)^{3/2}}<0. \] Consequently, \(c-x=[\ce{HX}]\) is concave upward. Since only undissociated acid absorbs, \[ \mathcal A=\varepsilon_{\ce{HX}}\ell(c-x). \] The absorbance therefore increases with an increasing slope: curve A. Chemically, increasing the acid concentration decreases its fraction dissociated, so the fraction present as absorbing \(\ce{HX}\) increases.

b) Only the conjugate base absorbs: \[ \mathcal A=\varepsilon_{\ce{X^-}}\ell x. \] The expression for \(x\) above is increasing and concave downward. Although the absolute concentration of \(\ce{X^-}\) increases, its fraction of the total acid decreases as \(c\) increases. Thus the absorbance rises with a decreasing slope: curve B.

c) Excess strong base converts the acid essentially completely into its conjugate base: \[ \ce{HX + OH^- -> X^- + H2O}. \] Under the intended complete-neutralization approximation, \[ [\ce{HX}]\approx0. \] Because only \(\ce{HX}\) absorbs, \[ \mathcal A=\varepsilon_{\ce{HX}}\ell[\ce{HX}]\approx0. \] The working curve lies along the horizontal concentration axis: curve D.

d) Excess strong acid suppresses dissociation of the weak acid: \[ \frac{[\ce{X^-}]}{[\ce{HX}]} =\frac{K_a}{[\ce{H3O+}]}\ll1. \] Hence essentially all the analytical acid remains undissociated: \[ [\ce{HX}]\approx c, \qquad \mathcal A\approx\varepsilon_{\ce{HX}}\ell c. \] The absorbance is proportional to total concentration, giving a straight line through the origin: curve C.

e) Let the equal, nonzero molar absorptivities be \[ \varepsilon_{\ce{HX}}=\varepsilon_{\ce{X^-}}=\varepsilon>0. \] The absorbances of the two species add: \[ \begin{aligned} \mathcal A &=\varepsilon\ell[\ce{HX}] +\varepsilon\ell[\ce{X^-}]\\ &=\varepsilon\ell\bigl([\ce{HX}]+[\ce{X^-}]\bigr)\\ &=\varepsilon\ell c. \end{aligned} \] Thus dissociation does not affect the absorbance at this wavelength. The working curve is a straight line through the origin: curve C.

The answers were checked against the official answers published by the organisers.