IChO 2024 · Theory problem 4

Potassium

Download PDFIChO 2024

Elementary potassium was the first metal synthesized by electrolysis (1807). Humphry Davy used voltaic cells and electrodes immersed into molten \(\ce{KOH}\).

4.1 Give balanced equations for the processes taking place on the cathode (1) and the anode (2). 2 pt

It is little known that only a year after Davy, French chemists have produced potassium in a chemical reduction. One of the methods involved a mixture of potassium carbonate and finely powdered carbon. This mixture was heated to very high temperatures in a gun barrel. Carbon monoxide and potassium vapors were leaving the end of the barrel, where the metal could be condensed to a solid.

4.2 Write down the balanced equation for the reaction giving only these two products. 1 pt

The chemical production of potassium had a low yield. Depending on the circumstances, a considerable amount of reddish paste was left behind in the gun barrel. Gmelin isolated an interesting compound (X) from this residue in 1825 by recrystallization in water. The exact structure of the byproduct X was determined more than one and a half century later.

X is a hydrated salt (2 mol water : 1 mol of salt) forming golden yellow crystals. Its anion has an exceptionally symmetric (shown by vibrational spectra) and stable planar cyclic structure.

The diprotic acid (Y) that can be formed from the salt X has surprisingly (at least to the superficial observer) low \(\mathrm{p}K_a\) values: 0.8 and 2.2.

4.3 Give the pH where 95% of all dissolved acid Y is in the neutral acid form. 3 pt

4.4 Calculate the pH of a \(0.01\ \mathrm{mol\,dm^{-3}}\) solution of the salt X in pure water. 4 pt

The key step in the structure identification of X was the treatment of the anhydrous salt with methyl iodide. The salt reacted with two equivalents of methyl iodide. The somewhat volatile compound that was obtained had a molar mass of \(170.12\ \mathrm{g/mol}\) (measured in a mass spectrometer).

A number of analogous salts were also identified in the mixture in smaller quantities. The appropriate potassium salts when treated with two equivalent of \(\ce{MeI}\) gave derivatives with molar masses of: \(142.11\ \mathrm{g/mol}\), \(114.10\ \mathrm{g/mol}\), \(198.13\ \mathrm{g/mol}\).

4.5 Give the empirical formula of Gmelin’s compound X. 4 pt

4.6 Draw the structure of the free acid Y. 3 pt

The extraordinary stability of the series of anions in the salts analogous to X can be attributed to electron delocalization.

4.7 Draw one resonance structure with a significant contribution for the anion in each of the four salts. 4 pt

Avogadro constant: \(N_A=6.022\cdot10^{23}\ \mathrm{mol^{-1}}\) Gas constant: \(R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}\)
Ionic product of water at 298.15 K: \(K_w=10^{-14}\) Zero of the Celsius scale: \(0^\circ\mathrm{C}=273.15\ \mathrm{K}\)
Faraday constant: \(F=96485\ \mathrm{C\,mol^{-1}}\) Planck constant: \(h=6.626\cdot10^{-34}\ \mathrm{J\,s}\)
Speed of light: \(c=2.998\cdot10^8\ \mathrm{m\,s^{-1}}\) Standard pressure: \(p^\circ=1\cdot10^5\ \mathrm{Pa}=1\ \mathrm{bar}\)
Ideal gas law: \(pV=nRT\) Beer-Lambert law: \(A=\log\left(\frac{I_0}{I}\right)=\varepsilon\cdot l\cdot c\)
Enthalpy: \(H=U+pV\) Gibbs energy: \(G=H-TS\)
\(\Delta_rG^\circ=-RT\ln K=-nFE^\circ_{\mathrm{cell}}\) Henderson-Hasselbalch equation: \(\mathrm{pH}=\mathrm{p}K_a+\log\left(\frac{[\ce{A-}]}{[\ce{HA}]}\right)\)
Arrhenius equation: \(k=A\exp\left(-\frac{E_A}{RT}\right)\) Zeroth order integrated rate law: \([\ce{A}]=[\ce{A}]_0-kt\)
Surface of a sphere: \(A=4\pi R^2\) First order integrated rate law: \(\ln[\ce{A}]=\ln[\ce{A}]_0-kt\)
Volume of a sphere: \(V=\frac{4\pi}{3}R^3\) \(\displaystyle\sum_{k=0}^{\infty}ar^k=\frac{a}{1-r}\quad\text{for }\lvert r\rvert<1\)

Periodic table

Figure 1

Official English text of the problem.

Answers

  • 4.1 Cathode (1): \(\boxed{\ce{K+ + e- -> K}}\). Anode (2): \(\boxed{\ce{4OH- -> O2 + 2H2O + 4e-}}\).
  • 4.2 \(\boxed{\ce{K2CO3 + 2C -> 2K(g) + 3CO(g)}}\).
  • 4.3 \(\boxed{\mathrm{pH}=-0.48}\).
  • 4.4 \(\boxed{\mathrm{pH}=7.21}\), including water autoionization at \(298.15\ \mathrm{K}\).
  • 4.5 X is potassium croconate dihydrate, \(\boxed{\ce{K2C5O5.2H2O}}\), or \(\ce{C5H4K2O7}\) as an elemental empirical formula. SMILES: [K+].[K+].[O-]C1=C([O-])C(=O)C(=O)C1=O.O.O.
  • 4.6 Y is croconic acid, \(\boxed{\ce{C5H2O5}}\), also named 4,5-dihydroxycyclopent-4-ene-1,2,3-trione. SMILES: OC1=C(O)C(=O)C(=O)C1=O.
  • 4.7 One significant resonance contributor for each anion is specified below. Each has two adjacent, singly bonded negatively charged oxygen atoms, a double bond between their ring carbons, and carbonyl groups on all remaining ring carbons.
Anion Formula SMILES of one resonance contributor
Croconate \(\ce{C5O5^2-}\) [O-]C1=C([O-])C(=O)C(=O)C1=O
Squarate \(\ce{C4O4^2-}\) [O-]C1=C([O-])C(=O)C1=O
Deltate \(\ce{C3O3^2-}\) [O-]C1=C([O-])C1=O
Rhodizonate \(\ce{C6O6^2-}\) [O-]C1=C([O-])C(=O)C(=O)C(=O)C1=O

Solution

4.1

In molten potassium hydroxide, potassium cations gain electrons at the cathode:

\[ \boxed{\ce{K+ + e- -> K}} \]

Hydroxide ions lose electrons at the anode, forming oxygen and water:

\[ \boxed{\ce{4OH- -> O2 + 2H2O + 4e-}} \]

Both atoms and charge are balanced. Multiplying the cathode equation by four and adding gives the overall reaction:

\[ \ce{4KOH -> 4K + O2 + 2H2O} \]

The species are potassium hydroxide, [K+].[OH-]; potassium cation, [K+]; potassium, [K]; hydroxide ion, [OH-]; dioxygen, O=O; and water, O. The potassium atom SMILES does not describe bulk metallic bonding.

4.2

One formula unit of potassium carbonate supplies two potassium atoms and three oxygen atoms. Since carbon monoxide is the only oxygen-containing product, three molecules of carbon monoxide must form. They require two carbon atoms in addition to the carbonate carbon:

\[ \boxed{\ce{K2CO3 + 2C -> 2K(g) + 3CO(g)}} \]

Potassium carbonate has SMILES [K+].[K+].[O-]C(=O)[O-]; carbon monoxide has SMILES [C-]#[O+]. Elemental carbon is an extended solid and has no single molecular SMILES.

4.3

Write the diprotic acid as \(\ce{H2A}\), with equilibria

\[ \ce{H2A <=> H+ + HA-}, \qquad \ce{HA- <=> H+ + A^2-}. \]

Here \(\ce{H+}\), proton SMILES [H+], is the conventional aqueous acid–base shorthand. The specific acid and anions are identified below.

Using the usual ideal-solution concentration treatment,

\[ K_{a1}=10^{-0.8}=0.158489, \qquad K_{a2}=10^{-2.2}=0.00630957. \]

Let \(h=[\ce{H+}]\), with concentrations expressed in \(\mathrm{mol\,dm^{-3}}\). The equilibrium ratios are

\[ \frac{[\ce{HA-}]}{[\ce{H2A}]}=\frac{K_{a1}}{h}, \qquad \frac{[\ce{A^2-}]}{[\ce{H2A}]}=\frac{K_{a1}K_{a2}}{h^2}. \]

Consequently, the fraction present as neutral acid is

\[ \alpha_0 = \frac{[\ce{H2A}]} {[\ce{H2A}]+[\ce{HA-}]+[\ce{A^2-}]} = \frac{h^2}{h^2+K_{a1}h+K_{a1}K_{a2}}. \]

Setting \(\alpha_0=0.95\) gives

\[ h^2-19K_{a1}h-19K_{a1}K_{a2}=0. \]

The positive root is

\[ h= \frac{19K_{a1}+ \sqrt{(19K_{a1})^2+76K_{a1}K_{a2}}}{2} = 3.01759\ \mathrm{mol\,dm^{-3}}. \]

Thus

\[ \boxed{\mathrm{pH}=-\log_{10}(3.01759)=-0.47966\approx-0.48}. \]

The corresponding fractions are \(0.950000\), \(0.049896\), and \(0.000104\) for neutral acid, monoanion, and dianion. A negative pH is permissible.

4.4

The dipotassium salt supplies the dianion at analytical concentration

\[ C=0.010\ \mathrm{mol\,dm^{-3}}, \qquad [\ce{K+}]=2C. \]

Its main hydrolysis equilibrium is

\[ \ce{A^2- + H2O <=> HA- + OH-}, \]

with

\[ K_b=\frac{K_w}{K_{a2}} =\frac{10^{-14}}{10^{-2.2}} =1.58489\times10^{-12}. \]

The estimate \(\sqrt{K_bC}=1.26\times10^{-7}\ \mathrm{mol\,dm^{-3}}\) is comparable to the hydroxide concentration in pure water. Water autoionization must therefore be retained.

The material and charge balances are

\[ C=[\ce{H2A}]+[\ce{HA-}]+[\ce{A^2-}] \]

and

\[ 2C+h=[\ce{HA-}]+2[\ce{A^2-}]+[\ce{OH-}]. \]

Eliminating \(C\),

\[ [\ce{OH-}]=h+[\ce{HA-}]+2[\ce{H2A}]. \]

At approximately neutral pH, both acid dissociation constants are much greater than \(h\). Thus nearly all the acid-derived material remains as dianion:

\[ [\ce{A^2-}]\simeq C, \qquad [\ce{HA-}]\simeq\frac{Ch}{K_{a2}}, \qquad [\ce{H2A}]\ll[\ce{HA-}]. \]

Using \([\ce{OH-}]=K_w/h\),

\[ \frac{K_w}{h} \simeq h+\frac{Ch}{K_{a2}}, \]

so

\[ h\simeq \sqrt{\frac{K_w}{1+C/K_{a2}}} = \sqrt{\frac{10^{-14}}{1+0.010/0.00630957}} = 6.22\times10^{-8}\ \mathrm{mol\,dm^{-3}}. \]

Therefore,

\[ \boxed{\mathrm{pH}=7.2062\approx7.21}. \]

As a check,

\[ [\ce{OH-}]\approx1.61\times10^{-7}\ \mathrm{mol\,dm^{-3}}, \qquad [\ce{HA-}]\approx9.86\times10^{-8}\ \mathrm{mol\,dm^{-3}}. \]

Only about \(9.86\times10^{-6}\) of the analytical acid concentration is monoanion, validating the approximation \([\ce{A^2-}]\simeq C\). Solving the full equilibria and charge balance gives the same pH to the precision reported.

4.5

Reaction with two equivalents of methyl iodide introduces two methyl groups into the dianion. Using the supplied atomic masses,

\[ M(\ce{CH3})=12.01+3(1.008)=15.034\ \mathrm{g\,mol^{-1}}. \]

The mass of the anionic framework in X is therefore

\[ 170.12-2(15.034)=140.052\ \mathrm{g\,mol^{-1}}. \]

The derivative masses differ by multiples of

\[ 28.01\ \mathrm{g\,mol^{-1}} = 12.01+16.00, \]

the mass of one carbon–oxygen unit. Removing the two methyl groups from each derivative gives:

Derivative molar mass / \(\mathrm{g\,mol^{-1}}\) Framework molar mass / \(\mathrm{g\,mol^{-1}}\) Number of carbon–oxygen units Anion
170.12 140.05 5 Croconate, \(\ce{C5O5^2-}\)
142.11 112.04 4 Squarate, \(\ce{C4O4^2-}\)
114.10 84.03 3 Deltate, \(\ce{C3O3^2-}\)
198.13 168.06 6 Rhodizonate, \(\ce{C6O6^2-}\)

Together with the stated planar cyclic symmetry, these masses identify the cyclic oxocarbon dianions: rings of carbon atoms bearing one exocyclic oxygen per carbon.

For X, the framework is \(\ce{C5O5^2-}\). Two potassium cations balance its charge, and the stated hydration adds two water molecules:

\[ \boxed{\ce{X = K2C5O5.2H2O}}. \]

Its elemental empirical formula is equivalently

\[ \boxed{\ce{C5H4K2O7}}. \]

The compound is potassium croconate dihydrate, SMILES [K+].[K+].[O-]C1=C([O-])C(=O)C(=O)C1=O.O.O. This notation specifies the components and one anion resonance contributor; it does not specify the crystal lattice or water coordination.

The methylation is

\[ \ce{K2C5O5 + 2CH3I -> C5O3(OCH3)2 + 2KI}. \]

The organic product is dimethyl croconate, SMILES COC1=C(OC)C(=O)C(=O)C1=O. Methyl iodide is CI, and potassium iodide is [K+].[I-]. The calculated product mass is

\[ M(\ce{C7H6O5}) =7(12.01)+6(1.008)+5(16.00) =170.118\ \mathrm{g\,mol^{-1}}, \]

which rounds to the given \(170.12\ \mathrm{g\,mol^{-1}}\).

4.6

Protonating the two negatively charged oxygen atoms of a croconate resonance contributor gives croconic acid, \(\ce{C5H2O5}\):

\[ \ce{C5O5^2- + 2H+ -> C5H2O5}. \]

Its name is 4,5-dihydroxycyclopent-4-ene-1,2,3-trione, and its SMILES is:

OC1=C(O)C(=O)C(=O)C1=O

To draw it precisely:

  1. Draw a five-carbon ring numbered consecutively 1–5.
  2. Put an exocyclic carbonyl oxygen on each of carbons 1, 2, and 3.
  3. Put a double bond between carbons 4 and 5.
  4. Attach one hydroxyl group to carbon 4 and one to carbon 5.
  5. All remaining ring bonds are single; no hydrogen is bonded directly to a ring carbon.

The acid–base species used in 4.3–4.4 are therefore croconic acid, hydrogen croconate, and croconate. Hydrogen croconate, \(\ce{C5HO5-}\), has the representative SMILES [O-]C1=C(O)C(=O)C(=O)C1=O.

The low \(\mathrm{p}K_a\) values are consistent with stabilization of the conjugate bases by electron delocalization through the conjugated carbon–oxygen framework.

4.7

For every anion, a significant contributor can be drawn by the same construction:

  • Draw a ring of \(n\) carbon atoms.
  • Choose two adjacent ring carbons and join them by a double bond.
  • Attach a singly bonded \(\ce{O-}\) to each of those two carbons.
  • Attach an exocyclic carbonyl oxygen to every remaining ring carbon.
  • Make every other ring bond single.

This gives the following precise drawings:

Anion Carbon ring Groups on the two double-bonded ring carbons Groups on the remaining ring carbons
Croconate Five-membered Two singly bonded \(\ce{O-}\) atoms Three carbonyls
Squarate Four-membered Two singly bonded \(\ce{O-}\) atoms Two carbonyls
Deltate Three-membered Two singly bonded \(\ce{O-}\) atoms One carbonyl
Rhodizonate Six-membered Two singly bonded \(\ce{O-}\) atoms Four carbonyls

The corresponding names, formulas, and explicit SMILES are given in Answers, 4.7. In each contributor, every carbon and oxygen has an octet, the net charge is \(-2\), and both formal negative charges lie on oxygen.

Electron-pair shifts generate other resonance contributors without moving atoms. For example, an anionic oxygen lone pair can form a carbonyl bond while the neighboring ring double bond shifts and another carbonyl \(\pi\)-electron pair moves onto oxygen. This relocates the formal negative charges around the conjugated framework.

The actual ions are resonance hybrids. The localized SMILES do not show their planar geometry, delocalized charge, or bond equalization; in particular, a single localized contributor does not display the full symmetry of croconate.

The solution was checked against the official solution.