IChO 1995 · Theory problem 2

Nitrate in milk serum: ion-selective electrode

Download PDFIChO 1995

To control the quality of milk serum, a dairy by-product, the concentration of \(\ce{NO3-}\) ion in serum is monitored by means of an ion selective electrode. Generally there is about 15 mg \(\ce{NO3-}\) ion per litre in serum, measured on the basis of nitrogen mass.

2.1 For a nitrate ion selective electrode a calibration curve as shown below was obtained using a series of standard nitrate solutions containing \(0.5\ \mathrm{mol\,dm^{-3}}\) \(\ce{K2SO4}\), \(1.0\times10^{-3}\ \mathrm{mol\,dm^{-3}}\) \(\ce{H2SO4}\) and \(2.6\times10^{-3}\ \mathrm{mol\,dm^{-3}}\) \(\ce{Cl-}\) ion as the background. Decide whether it is feasible to measure concentration \(\ce{NO3-}\) in serum under the above conditions.

Figure 1

2.2 Given are selective coefficients of \(\ce{Cl-}\), \(\ce{SO4^2-}\) and \(\ce{ClO4-}\) versus \(\ce{NO3-}\) as follow

\[ K_{\ce{NO3-},\ce{Cl-}} =\frac{c_{\ce{NO3-}}}{c_{\ce{Cl-}}} =4.9\times10^{-2} \qquad K_{\ce{NO3-},\ce{SO4^2-}} =\frac{c_{\ce{NO3-}}}{c_{\ce{SO4^2-}}^{1/2}} =4.1\times10^{-3} \]

\[ K_{\ce{NO3-},\ce{ClO4-}} =\frac{c_{\ce{NO3-}}}{c_{\ce{ClO4-}}} =1.0\times10^{-3} \]

where the units of the concentrations are in \(\mathrm{mol\,dm^{-3}}\) which is the best to reduce the interference of \(\ce{Cl-}\) to \(\ce{NO3-}\) determination, so as to control the error in the \(\ce{NO3-}\) concentration within 1 %, when there are \(1.40\times10^{-3}\ \mathrm{mol\,dm^{-3}}\) \(\ce{NO3-}\) and \(1.60\times10^{-2}\ \mathrm{mol\,dm^{-3}}\) \(\ce{Cl-}\) in serum:

\[ \text{(a)}\quad\ce{AgNO3} \qquad \text{(b)}\quad\ce{Ag2SO4} \qquad \text{(c)}\quad\ce{AgClO4} \]

Calculate the amount of the salt that should be added to \(1\ \mathrm{dm^3}\) of the sample solution to be measured.

2.3 The \(\ce{NO3-}\) ion concentration was determined by this method at 298 K. For \(25.00\ \mathrm{cm^3}\) sample solution the electronic potential, \(E\), is measured to be \(-160\ \mathrm{mV}\). After adding \(1.00\ \mathrm{cm^3}\) \(1.00\times10^{-3}\ \mathrm{mol\,dm^{-3}}\) \(\ce{NO3-}\) standard solution to the above solution, \(E\) changes to \(-130\ \mathrm{mV}\). Find the \(\mathrm{pNO_3}\) of the serum.

2.4 The selective coefficient of \(\ce{CH3COO-}\) versus \(\ce{NO3-}\), \(K(\ce{NO3-}/\ce{CH3COO-})=2.7\times10^{-3}\). If \(\ce{AgCH3COO}\) instead of \(\ce{Ag2SO4}\) is added to the sample solution of question 2.2, find the upper limit of the pH value below which the same requirement in question 2.2 can be met.

\[ K_{\mathrm{sp}}(\ce{AgCl})=3.2\times10^{-10} \qquad K_{\mathrm{sp}}(\ce{Ag2SO4})=8.0\times10^{-5} \]

\[ K_{\mathrm{sp}}(\ce{AgCH3COO})=8.0\times10^{-3} \qquad K_a(\ce{CH3COOH})=2.2\times10^{-5} \]

\[ A_{\mathrm r}(\ce{N})=14.00 \]

Official English text of the problem.

Answers

  • 2.1: Yes. The expected nitrate concentration is \(1.07\times10^{-3}\ \mathrm{mol\,dm^{-3}}\), near the beginning of the approximately linear part of the calibration curve.
  • 2.2: Choose silver sulfate, \(\ce{Ag2SO4}\), SMILES [Ag+].[Ag+].[O-]S(=O)(=O)[O-]. The minimum addition to \(1\ \mathrm{dm^3}\) is \(7.86\times10^{-3}\ \mathrm{mol}\), approximately \(2.45\ \mathrm g\).
  • 2.3: \(\boxed{\mathrm{pNO_3}=4.77}\), corresponding to \(c_{\ce{NO3-}}=1.71\times10^{-5}\ \mathrm{mol\,dm^{-3}}\).
  • 2.4: The usual quantitative-precipitation approximation gives \(\boxed{\mathrm{pH}_{\max}\approx4.34}\). Retaining both residual chloride and acetate interference gives the slightly stricter limit \(\boxed{\mathrm{pH}_{\max}\approx4.33}\).

Solution

2.1 — Feasibility of the measurement

Nitrate, \(\ce{NO3-}\), has SMILES [O-][N+](=O)[O-]. Each nitrate ion contains one nitrogen atom. Because the stated concentration is expressed as nitrogen mass,

\[ c_{\ce{NO3-}} =\frac{15\times10^{-3}\ \mathrm{g\,dm^{-3}}} {14.00\ \mathrm{g\,mol^{-1}}} =1.07\times10^{-3}\ \mathrm{mol\,dm^{-3}}. \]

The graph becomes approximately linear in \(E\) versus \(\log c_{\ce{NO3-}}\) around \(10^{-3}\ \mathrm{mol\,dm^{-3}}\), and remains so at higher concentrations. The expected serum concentration therefore lies in the useful, concentration-sensitive region.

Thus, measurement is feasible using the supplied calibration.

2.2 — Choice and minimum amount of silver salt

Silver(I) ion, [Ag+], precipitates chloride, [Cl-], as silver chloride, [Ag+].[Cl-]:

\[ \ce{Ag+(aq) + Cl-(aq) -> AgCl(s)}. \]

The candidate salts are:

Salt Name and SMILES Assessment
\(\ce{AgNO3}\) Silver nitrate, [Ag+].[O-][N+](=O)[O-] Adds the analyte itself in a large amount.
\(\ce{Ag2SO4}\) Silver sulfate, [Ag+].[Ag+].[O-]S(=O)(=O)[O-] Suitable because sulfate is already the large, calibrated background.
\(\ce{AgClO4}\) Silver perchlorate, [Ag+].[O-]Cl(=O)(=O)=O Introduces too much additional perchlorate interference.

Here sulfate is [O-]S(=O)(=O)[O-], and perchlorate is [O-]Cl(=O)(=O)=O. These oxyanion SMILES represent individual resonance forms. The disconnected SMILES for salts specify ionic composition; they do not describe crystal lattices.

For silver perchlorate, nearly complete removal of the initial chloride would introduce approximately \(0.0160\ \mathrm{mol\,dm^{-3}}\) perchlorate. Its nitrate-equivalent contribution alone would be

\[ (1.0\times10^{-3})(1.60\times10^{-2}) =1.60\times10^{-5}\ \mathrm{mol\,dm^{-3}}, \]

which exceeds the permitted error,

\[ B=0.01(1.40\times10^{-3}) =1.40\times10^{-5}\ \mathrm{mol\,dm^{-3}}. \]

Leaving more chloride cannot solve this problem, because chloride has the larger selectivity coefficient.

For silver sulfate, the sulfate response is treated as part of the existing \(0.5\ \mathrm{mol\,dm^{-3}}\) background, since the solution already contains 0.5 M potassium sulfate. The precipitation reaction is

\[ \ce{Ag2SO4(s) + 2Cl-(aq) -> 2AgCl(s) + SO4^2-(aq)}. \]

The maximum permissible residual chloride concentration satisfies

\[ K_{\ce{NO3-},\ce{Cl-}}c_{\ce{Cl-}}\le B. \]

Consequently,

\[ c_{\ce{Cl-},\max} =\frac{1.40\times10^{-5}}{4.9\times10^{-2}} =2.857\times10^{-4}\ \mathrm{mol\,dm^{-3}}. \]

At this boundary, the dissolved silver concentration is

\[ c_{\ce{Ag+}} =\frac{K_{\mathrm{sp}}(\ce{AgCl})}{c_{\ce{Cl-}}} =\frac{3.2\times10^{-10}}{2.857\times10^{-4}} =1.12\times10^{-6}\ \mathrm{mol\,dm^{-3}}. \]

The silver added must supply both the precipitated silver chloride and this small dissolved remainder. Neglecting the volume change on adding solid salt,

\[ 2n_{\ce{Ag2SO4}} =V\left(c_{\ce{Cl-},0}-c_{\ce{Cl-},\max}+c_{\ce{Ag+}}\right). \]

For \(V=1\ \mathrm{dm^3}\),

\[ n_{\ce{Ag2SO4},\min} =\frac{1}{2} \left(1.60\times10^{-2}-2.857\times10^{-4} +1.12\times10^{-6}\right) =7.8577\times10^{-3}\ \mathrm{mol}. \]

Therefore,

\[ \boxed{n_{\ce{Ag2SO4},\min}=7.86\times10^{-3}\ \mathrm{mol}}. \]

Using \(M(\ce{Ag2SO4})\approx311.8\ \mathrm{g\,mol^{-1}}\), this is approximately \(2.45\ \mathrm g\).

The dose is compatible with complete dissolution of the added silver sulfate: at the calculated endpoint,

\[ c_{\ce{Ag+}}^2c_{\ce{SO4^2-}} \approx(1.12\times10^{-6})^2(0.51) \approx6.4\times10^{-13} \ll8.0\times10^{-5}. \]

Thus, saturation with solid silver sulfate does not determine the minimum dose.

2.3 — Standard addition

For a monovalent ion at \(298\ \mathrm K\), the magnitude of the Nernst slope is

\[ S=\frac{2.303RT}{F} =0.05913\ \mathrm V. \]

Using the measured-potential polarity shown in the problem, increasing nitrate concentration increases \(E\). Hence

\[ E_2-E_1=S\log_{10}\frac{c_2}{c_1}. \]

Since \(\Delta E=30\ \mathrm{mV}\),

\[ r=\frac{c_2}{c_1} =10^{30/59.13} =3.216. \]

Let \(c\) be the original nitrate concentration. Including dilution by the added standard,

\[ c_2=\frac{cV_0+c_sV_s}{V_0+V_s}. \]

Therefore,

\[ c=\frac{c_sV_s}{r(V_0+V_s)-V_0}. \]

Substitution gives

\[ c= \frac{(1.00\times10^{-3}\ \mathrm{mol\,dm^{-3}}) (0.00100\ \mathrm{dm^3})} {3.216(0.02600\ \mathrm{dm^3})-0.02500\ \mathrm{dm^3}} =1.706\times10^{-5}\ \mathrm{mol\,dm^{-3}}. \]

Thus,

\[ \boxed{ \mathrm{pNO_3} =-\log_{10}\left( \frac{c_{\ce{NO3-}}}{1\ \mathrm{mol\,dm^{-3}}} \right) =4.77 }. \]

2.4 — Suppressing acetate interference by protonation

The relevant species are silver acetate, \(\ce{AgCH3COO}\), [Ag+].CC(=O)[O-]; acetate, \(\ce{CH3COO-}\), CC(=O)[O-]; acetic acid, \(\ce{CH3COOH}\), CC(=O)O; and hydrogen ion, \(\ce{H+}\), [H+].

Silver acetate removes chloride through

\[ \ce{AgCH3COO(s) + Cl-(aq) -> AgCl(s) + CH3COO-(aq)}. \]

Acidification suppresses the concentration of interfering acetate:

\[ \ce{CH3COOH(aq) <=> H+(aq) + CH3COO-(aq)}. \]

The combined reaction can be written as

\[ \ce{AgCH3COO(s) + Cl-(aq) + H+(aq) -> AgCl(s) + CH3COOH(aq)}. \]

Usual quantitative-precipitation approximation. Removing approximately \(0.0160\ \mathrm{mol\,dm^{-3}}\) chloride introduces the same total concentration of acetate species:

\[ C_{\mathrm{Ac}} =c_{\ce{CH3COO-}}+c_{\ce{CH3COOH}} \approx1.60\times10^{-2}\ \mathrm{mol\,dm^{-3}}. \]

If residual chloride interference is neglected, the maximum free acetate concentration is

\[ c_{\ce{CH3COO-},\max} =\frac{B}{K_{\ce{NO3-},\ce{CH3COO-}}} =\frac{1.40\times10^{-5}}{2.7\times10^{-3}} =5.185\times10^{-3}\ \mathrm{mol\,dm^{-3}}. \]

Using

\[ K_a=\frac{c_{\ce{H+}}c_{\ce{CH3COO-}}} {c_{\ce{CH3COOH}}}, \]

the minimum hydrogen-ion concentration is

\[ c_{\ce{H+},\min} =K_a\frac{C_{\mathrm{Ac}}-c_{\ce{CH3COO-},\max}} {c_{\ce{CH3COO-},\max}} \]

\[ =(2.2\times10^{-5}) \frac{0.0160-0.005185}{0.005185} =4.589\times10^{-5}\ \mathrm{mol\,dm^{-3}}. \]

Therefore,

\[ \boxed{\mathrm{pH}_{\max}\approx4.34}. \]

Lower pH increases protonation and reduces acetate interference.

Retaining both interferences. The chloride concentration allowed at the minimum sulfate dose in 2.2 already consumes the entire 1% error allowance. With acetate present, chloride must be precipitated more completely.

To find the strict maximum pH, let \(x=c_{\ce{Cl-}}\), and let

\[ \alpha=\frac{K_a}{K_a+c_{\ce{H+}}} \]

be the fraction of total acetate present as acetate ion. Silver chloride equilibrium and the silver balance give

\[ c_{\ce{Ag+}}=\frac{K_{\mathrm{sp}}(\ce{AgCl})}{x}, \]

\[ C_{\mathrm{Ac}} =0.0160-x+\frac{K_{\mathrm{sp}}(\ce{AgCl})}{x}. \]

The combined nitrate-equivalent interference is consequently

\[ \Delta c =k_{\mathrm{Cl}}x+ k_{\mathrm{Ac}}\alpha \left(0.0160-x+\frac{K_{\mathrm{sp}}}{x}\right), \]

where \(k_{\mathrm{Cl}}=0.049\) and \(k_{\mathrm{Ac}}=0.0027\).

Writing \(b=k_{\mathrm{Ac}}\alpha\),

\[ \Delta c =0.0160b+(k_{\mathrm{Cl}}-b)x+\frac{bK_{\mathrm{sp}}}{x}. \]

For a fixed pH, its minimum over the silver acetate dose is

\[ \Delta c_{\min} =0.0160b+ 2\sqrt{(k_{\mathrm{Cl}}-b)bK_{\mathrm{sp}}}. \]

Setting this equal to \(B=1.40\times10^{-5}\) gives

\[ \alpha=0.31874, \qquad c_{\ce{H+}}=K_a\frac{1-\alpha}{\alpha} =4.702\times10^{-5}\ \mathrm{mol\,dm^{-3}}, \]

and therefore

\[ \boxed{\mathrm{pH}_{\max}=4.3277\approx4.33}. \]

This boundary is attainable with approximately

\[ c_{\ce{Cl-}}=2.39\times10^{-6}\ \mathrm{mol\,dm^{-3}}, \qquad C_{\mathrm{Ac}}=0.01613\ \mathrm{mol\,dm^{-3}}, \]

corresponding to \(0.01613\ \mathrm{mol}\) silver acetate per litre. Its dissolved-ion product is only

\[ c_{\ce{Ag+}}c_{\ce{CH3COO-}} \approx6.9\times10^{-7} < K_{\mathrm{sp}}(\ce{AgCH3COO}), \]

so silver acetate solubility does not prevent this result.

The solution was checked against the official solution. Where this solution differs from the official solution, the official solution contains an error or a misprint.