IChO 2012 · Theory problem 3

Thiomolybdate ions

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Thiomolybdate ions are derived from molybdate ions, \(\ce{MoO4^2-}\), by replacing oxygen atoms with sulfur atoms. In nature, thiomolybdate ions are found in such places as the deep waters of the Black Sea, where biological sulfate reduction generates \(\ce{H2S}\). The molybdate to thiomolybdate transformation leads to rapid loss of dissolved \(\ce{Mo}\) from seawater to underlying sediments, depleting the ocean in \(\ce{Mo}\), a trace element essential for life.

The following equilibria control the relative concentrations of molybdate and thiomolybdate ions in dilute aqueous solution:

Figure 1

3.1 If at equilibrium the concentrations of \(\ce{MoO4^2-}\) and \(\ce{H2S(aq)}\) are equal to \(1\cdot10^{-7}\) and \(1\cdot10^{-6}\), respectively, what would be the equilibrium concentration of \(\ce{MoS4^2-}\)?

Solutions containing \(\ce{MoO2S2^2-}\), \(\ce{MoOS3^2-}\) and \(\ce{MoS4^2-}\) display absorption peaks in the visible wavelength range at 395 and 468 nm. The other ions, as well as \(\ce{H2S}\), absorb negligibly in the visible wavelength range. The molar absorptivities (\(\varepsilon\)) at these two wavelengths are given in the following table:

\(\varepsilon\) at 468 nm
\(\mathrm{dm^3\,mol^{-1}\,cm^{-1}}\)		 \(\varepsilon\) at 395 nm
\(\mathrm{dm^3\,mol^{-1}\,cm^{-1}}\)
\(\ce{MoS4^2-}\) 11870 120
\(\ce{MoOS3^2-}\) 0 9030
\(\ce{MoO2S2^2-}\) 0 3230

3.2 A solution not at equilibrium contains a mixture of \(\ce{MoS4^2-}\), \(\ce{MoOS3^2-}\) and \(\ce{MoO2S2^2-}\) and no other \(\ce{Mo}\)-containing species. The total concentration of all species containing \(\ce{Mo}\) is \(6.0\cdot10^{-6}\ \mathrm{mol\,dm^{-3}}\). In a 10.0 cm absorption cell, the absorbance of the solution at 468 nm is 0.365 and that at 395 nm is 0.213. Calculate the concentrations of all three \(\ce{Mo}\)-containing anions in this mixture.

A solution with initial concentration of \(\ce{MoS4^2-}\) equal to \(2.0\cdot10^{-7}\ \mathrm{mol\,dm^{-3}}\) hydrolyzes in a closed system. The \(\ce{H2S}\) product accumulates until equilibrium is reached. When calculating the final equilibrium concentrations of \(\ce{H2S(aq)}\), and all five \(\ce{Mo}\)-containing anions (that is, \(\ce{MoO4^2-}\), \(\ce{MoO3S^2-}\), \(\ce{MoO2S2^2-}\), \(\ce{MoOS3^2-}\) and \(\ce{MoS4^2-}\)) ignore the possibility that \(\ce{H2S}\) might ionize to \(\ce{HS-}\) under certain pH conditions.

3.3 Write the six independent equations that determine the system.

3.4 Calculate the above mentioned six concentrations making reasonable approximations, giving your answers to two significant figures.

Official English text of the problem.

Answers

  • 3.1 \[ \boxed{[\ce{MoS4^2-}]=7.4\times10^{-12}\ \mathrm{mol\,dm^{-3}}} \]

  • 3.2 To two significant figures: \[ \boxed{ \begin{aligned} [\ce{MoS4^2-}]&=3.1\times10^{-6}\ \mathrm{mol\,dm^{-3}},\\ [\ce{MoOS3^2-}]&=2.0\times10^{-6}\ \mathrm{mol\,dm^{-3}},\\ [\ce{MoO2S2^2-}]&=9.5\times10^{-7}\ \mathrm{mol\,dm^{-3}}. \end{aligned}} \]

  • 3.3 Let \(h=[\ce{H2S(aq)}]\), and let \(x_0,x_1,x_2,x_3,x_4\) denote the concentrations of \(\ce{MoO4^2-}\), \(\ce{MoO3S^2-}\), \(\ce{MoO2S2^2-}\), \(\ce{MoOS3^2-}\), and \(\ce{MoS4^2-}\), respectively. With \(C=2.0\times10^{-7}\ \mathrm{mol\,dm^{-3}}\), the six equations are \[ \boxed{ \begin{aligned} \frac{x_3h}{x_4}&=K_1,& \frac{x_2h}{x_3}&=K_2,\\ \frac{x_1h}{x_2}&=K_3,& \frac{x_0h}{x_1}&=K_4,\\ x_0+x_1+x_2+x_3+x_4&=C,\\ h+x_1+2x_2+3x_3+4x_4&=4C. \end{aligned}} \]

  • 3.4 \[ \boxed{ \begin{aligned} [\ce{H2S(aq)}]&=7.8\times10^{-7}\ \mathrm{mol\,dm^{-3}},\\ [\ce{MoO4^2-}]&=1.8\times10^{-7}\ \mathrm{mol\,dm^{-3}},\\ [\ce{MoO3S^2-}]&=2.1\times10^{-8}\ \mathrm{mol\,dm^{-3}},\\ [\ce{MoO2S2^2-}]&=1.0\times10^{-9}\ \mathrm{mol\,dm^{-3}},\\ [\ce{MoOS3^2-}]&=8.0\times10^{-11}\ \mathrm{mol\,dm^{-3}},\\ [\ce{MoS4^2-}]&=4.8\times10^{-12}\ \mathrm{mol\,dm^{-3}}. \end{aligned}} \]

Solution

The species and notation used throughout are:

Concentration Ion SMILES
\(x_0\) Molybdate, \(\ce{MoO4^2-}\) [O-][Mo](=O)(=O)[O-]
\(x_1\) Monothiomolybdate, \(\ce{MoO3S^2-}\) [O-][Mo](=O)(=S)[O-]
\(x_2\) Dithiomolybdate, \(\ce{MoO2S2^2-}\) [S-][Mo](=O)(=O)[S-]
\(x_3\) Trithiomolybdate, \(\ce{MoOS3^2-}\) [S-][Mo](=O)(=S)[S-]
\(x_4\) Tetrathiomolybdate, \(\ce{MoS4^2-}\) [S-][Mo](=S)(=S)[S-]

These SMILES give formal Lewis representations; the placement of localized bonds and charges is not unique because of resonance. The four oxygen/sulfur ligands have approximately tetrahedral coordination around molybdenum(VI), which these SMILES do not explicitly specify.

Water, \(\ce{H2O}\), has SMILES O; hydrogen sulfide, \(\ce{H2S}\), has SMILES S; and hydrosulfide, \(\ce{HS-}\), has SMILES [SH-]. Hydrosulfide formation is excluded as instructed.

The given equilibrium constants are used in their dilute-solution concentration form, with liquid water omitted from the expressions.

3.1

Adding the four hydrolysis reactions cancels the intermediate ions:

\[ \ce{MoS4^2- + 4H2O(l) <=> MoO4^2- + 4H2S(aq)}. \]

The equilibrium constant for this overall reaction is the product of the four constants:

\[ K_1K_2K_3K_4 =(1.3\times10^{-5})(1.0\times10^{-5}) (1.6\times10^{-5})(6.5\times10^{-6}) =1.352\times10^{-20}. \]

Consequently,

\[ K_1K_2K_3K_4 =\frac{[\ce{MoO4^2-}][\ce{H2S}]^4} {[\ce{MoS4^2-}]}. \]

Rearranging and substituting the specified concentrations gives

\[ [\ce{MoS4^2-}] =\frac{(1\times10^{-7})(1\times10^{-6})^4} {1.352\times10^{-20}} =7.396\times10^{-12}\ \mathrm{mol\,dm^{-3}}. \]

Thus,

\[ \boxed{[\ce{MoS4^2-}]\approx7.4\times10^{-12}\ \mathrm{mol\,dm^{-3}}}. \]

3.2

For a mixture, the Beer–Lambert law is

\[ A_\lambda=l\sum_i\varepsilon_{i,\lambda}c_i. \]

At 468 nm, only tetrathiomolybdate absorbs appreciably. Hence,

\[ 0.365=(10.0)(11870)x_4, \]

so

\[ x_4=3.0749789\times10^{-6}\ \mathrm{mol\,dm^{-3}}. \]

The molybdenum balance gives

\[ x_3+x_2=6.0\times10^{-6}-x_4. \]

At 395 nm, all three species contribute:

\[ 0.213=10.0(120x_4+9030x_3+3230x_2). \]

Substituting \(x_2=6.0\times10^{-6}-x_4-x_3\),

\[ x_3= \frac{0.213/10.0-120x_4-3230(6.0\times10^{-6}-x_4)} {9030-3230} =1.9798594\times10^{-6}\ \mathrm{mol\,dm^{-3}}. \]

Therefore,

\[ x_2=6.0\times10^{-6}-x_4-x_3 =9.4516167\times10^{-7}\ \mathrm{mol\,dm^{-3}}. \]

Rounded to two significant figures, these are \(3.1\times10^{-6}\), \(2.0\times10^{-6}\), and \(9.5\times10^{-7}\ \mathrm{mol\,dm^{-3}}\), respectively. No equilibrium relations are used because this mixture is explicitly not at equilibrium.

3.3

There are six unknown concentrations: \(x_0,x_1,x_2,x_3,x_4,h\).

The four equilibrium relations are

\[ \begin{aligned} K_1&=\frac{x_3h}{x_4}=1.3\times10^{-5},\\ K_2&=\frac{x_2h}{x_3}=1.0\times10^{-5},\\ K_3&=\frac{x_1h}{x_2}=1.6\times10^{-5},\\ K_4&=\frac{x_0h}{x_1}=6.5\times10^{-6}. \end{aligned} \]

Each anion contains one molybdenum atom, so conservation of molybdenum gives

\[ \boxed{x_0+x_1+x_2+x_3+x_4=C =2.0\times10^{-7}\ \mathrm{mol\,dm^{-3}}}. \]

Initially, every molybdenum atom is associated with four sulfur atoms. At equilibrium, sulfur is distributed between the anions and dissolved hydrogen sulfide. Thus,

\[ \boxed{h+x_1+2x_2+3x_3+4x_4 =4C=8.0\times10^{-7}\ \mathrm{mol\,dm^{-3}}}. \]

These are the required six independent equations. Combining the two elemental balances also gives the useful equivalent relation

\[ \boxed{h=4x_0+3x_1+2x_2+x_3}. \]

The coefficients count the number of hydrogen sulfide molecules released when one initial tetrathiomolybdate ion reaches each hydrolysis state.

3.4

The sulfur balance bounds the hydrogen sulfide concentration:

\[ 0<h\leq4C=8.0\times10^{-7}\ \mathrm{mol\,dm^{-3}}. \]

The equilibrium relations give the successive concentration ratios

\[ \frac{x_1}{x_0}=\frac{h}{K_4},\qquad \frac{x_2}{x_1}=\frac{h}{K_3},\qquad \frac{x_3}{x_2}=\frac{h}{K_2},\qquad \frac{x_4}{x_3}=\frac{h}{K_1}. \]

Even at the upper bound for \(h\), these ratios are only approximately \(0.12\), \(0.050\), \(0.080\), and \(0.062\). Molybdate therefore predominates, and nearly complete hydrolysis provides a reasonable first approximation:

\[ h^{(0)}=4C=8.0\times10^{-7}\ \mathrm{mol\,dm^{-3}}. \]

To obtain reliable two-significant-figure results for the minor species, refine this approximation using both balances. Express every molybdenum concentration in terms of \(x_0\) and \(h\):

\[ \begin{aligned} x_1&=x_0\frac{h}{K_4},\\ x_2&=x_0\frac{h^2}{K_3K_4},\\ x_3&=x_0\frac{h^3}{K_2K_3K_4},\\ x_4&=x_0\frac{h^4}{K_1K_2K_3K_4}. \end{aligned} \]

The molybdenum balance then gives

\[ x_0= \frac{C}{ 1+\dfrac{h}{K_4} +\dfrac{h^2}{K_3K_4} +\dfrac{h^3}{K_2K_3K_4} +\dfrac{h^4}{K_1K_2K_3K_4} }. \]

For each estimate of \(h\), calculate these concentrations and update \(h\) from

\[ h_{\mathrm{new}}=4C-x_1-2x_2-3x_3-4x_4. \]

Successive approximations are

\[ \begin{aligned} h^{(0)}&=8.0000000\times10^{-7},\\ h^{(1)}&=7.7574998\times10^{-7},\\ h^{(2)}&=7.7647978\times10^{-7},\\ h^{(3)}&=7.7645781\times10^{-7}, \end{aligned} \qquad \mathrm{mol\,dm^{-3}}. \]

The concentrations have stabilized sufficiently for the requested rounding:

Species Value before final rounding \((\mathrm{mol\,dm^{-3}})\) Two significant figures \((\mathrm{mol\,dm^{-3}})\)
\(\ce{H2S(aq)}\) \(7.7646\times10^{-7}\) \(\boxed{7.8\times10^{-7}}\)
\(\ce{MoO4^2-}\) \(1.7766\times10^{-7}\) \(\boxed{1.8\times10^{-7}}\)
\(\ce{MoO3S^2-}\) \(2.1223\times10^{-8}\) \(\boxed{2.1\times10^{-8}}\)
\(\ce{MoO2S2^2-}\) \(1.0299\times10^{-9}\) \(\boxed{1.0\times10^{-9}}\)
\(\ce{MoOS3^2-}\) \(7.9968\times10^{-11}\) \(\boxed{8.0\times10^{-11}}\)
\(\ce{MoS4^2-}\) \(4.7763\times10^{-12}\) \(\boxed{4.8\times10^{-12}}\)

Using unrounded values, the molybdenum concentrations sum to \(2.0\times10^{-7}\ \mathrm{mol\,dm^{-3}}\), and the total sulfur concentration is \(8.0\times10^{-7}\ \mathrm{mol\,dm^{-3}}\), as required.

The solution was checked against the official solution.