IChO 1992 · Theory problem 6

Carbon dioxide in the atmosphere and in water

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The concentration of carbon dioxide in the atmosphere has increased substantially during this century and is predicted to continue to increase. The \([\ce{CO2}]\) is expected to be about 440 ppm (\(440\times10^{-6}\ \mathrm{atm}\)) in the year 2020.

6.1 Calculate the concentration (in \(\mathrm{mol\,dm^{-3}}\)) of \(\ce{CO2}\) dissolved in distilled water equilibrated with the atmosphere in the year 2020.

6.2 Calculate the \(pH\)-value of the solution in 6.1.

6.3 Calculate the enthalpy of reaction between \(\ce{CO2(aq)}\) and \(\ce{H2O}\).

6.4 If the temperature of an equilibrated solution of \(\ce{CO2}\) is increased and the concentration of dissolved carbon dioxide is maintained constant, the \(pH\) of the solution may change. Predict whether the \(pH\) will increase or decrease.

Data:

Henry's Law constant for \(\ce{CO2}\) at 298 K: \(0.0343\ \mathrm{dm^3\,mol^{-1}\,atm^{-1}}\)

Thermodynamic values, in kJ/mol at 298 K are:

\(\Delta_{\mathrm f}G^0\) \(\Delta_{\mathrm f}H^0\)
\(\ce{CO2(aq)}\) −386.2 −412.9
\(\ce{H2O(l)}\) −237.2 −285.8
\(\ce{HCO3^-(aq)}\) −587.1 −691.2
\(\ce{H+(aq)}\) 0.00 0.00

Official English text of the problem.

Answers

  • 6.1: \(\boxed{[\ce{CO2(aq)}]=1.51\times10^{-5}\ \mathrm{mol\,dm^{-3}}}\) at 298 K.
  • 6.2: \(\boxed{\mathrm{pH}=5.59}\).
  • 6.3: \(\boxed{\Delta_{\mathrm r}H^\circ=+7.5\ \mathrm{kJ\,mol^{-1}}}\) for \(\ce{CO2(aq) + H2O(l) <=> H+(aq) + HCO3^-(aq)}\); the reaction is endothermic.
  • 6.4: The \(\boxed{\mathrm{pH}\text{ decreases}}\).

Solution

The species involved are carbon dioxide, \(\ce{CO2}\), O=C=O; water, \(\ce{H2O}\), O; hydrogen carbonate (bicarbonate), \(\ce{HCO3^-}\), O=C(O)[O-]; and the hydrogen ion, \(\ce{H+}\), [H+]. In water, the hydrogen ion is solvated; its hydration is not represented by [H+].

6.1

The dissolution equilibrium is

\[ \ce{CO2(g) <=> CO2(aq)}. \]

Henry’s law gives

\[ [\ce{CO2(aq)}]=k_Hp_{\ce{CO2}}. \]

For this equation, the given numerical constant is interpreted as \(k_H=0.0343\ \mathrm{mol\,dm^{-3}\,atm^{-1}}\). Thus,

\[ \begin{aligned} [\ce{CO2(aq)}] &=(0.0343\ \mathrm{mol\,dm^{-3}\,atm^{-1}}) (440\times10^{-6}\ \mathrm{atm})\\ &=1.5092\times10^{-5}\ \mathrm{mol\,dm^{-3}}\\ &\approx\boxed{1.51\times10^{-5}\ \mathrm{mol\,dm^{-3}}}. \end{aligned} \]

This is the concentration of molecular dissolved carbon dioxide. Equilibration with the atmosphere maintains this concentration while some dissolved carbon dioxide reacts with water.

6.2

The supplied thermodynamic data describe the net acid dissociation:

\[ \ce{CO2(aq) + H2O(l) <=> H+(aq) + HCO3^-(aq)}. \]

Its standard Gibbs energy change is

\[ \begin{aligned} \Delta_{\mathrm r}G^\circ &=\sum \Delta_{\mathrm f}G^\circ(\text{products}) -\sum \Delta_{\mathrm f}G^\circ(\text{reactants})\\ &=[0.00+(-587.1)]-[(-386.2)+(-237.2)]\\ &=+36.3\ \mathrm{kJ\,mol^{-1}}. \end{aligned} \]

Using \(\Delta_{\mathrm r}G^\circ=-RT\ln K\), with \(R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}\),

\[ K=\exp\!\left(-\frac{36.3\times10^3}{8.314\times298}\right) =4.33\times10^{-7}. \]

Taking the activity of water as unity and treating the solution as ideal,

\[ K\simeq \frac{[\ce{H+}][\ce{HCO3^-}]} {[\ce{CO2}]c^\circ}, \qquad c^\circ=1\ \mathrm{mol\,dm^{-3}}. \]

Neglecting water’s autoionization and further dissociation of hydrogen carbonate, electroneutrality gives

\[ [\ce{H+}]\simeq[\ce{HCO3^-}]=x. \]

Therefore,

\[ \begin{aligned} x&=\sqrt{K[\ce{CO2}]c^\circ}\\ &=\sqrt{(4.33\times10^{-7}) (1.5092\times10^{-5}\ \mathrm{mol\,dm^{-3}}) (1\ \mathrm{mol\,dm^{-3}})}\\ &=2.56\times10^{-6}\ \mathrm{mol\,dm^{-3}}. \end{aligned} \]

Hence,

\[ \boxed{\mathrm{pH} =-\log_{10}\!\left(\frac{[\ce{H+}]}{c^\circ}\right) =5.59}. \]

6.3

For the same balanced reaction,

\[ \ce{CO2(aq) + H2O(l) <=> H+(aq) + HCO3^-(aq)}, \]

Hess’s law gives

\[ \begin{aligned} \Delta_{\mathrm r}H^\circ &=\sum\Delta_{\mathrm f}H^\circ(\text{products}) -\sum\Delta_{\mathrm f}H^\circ(\text{reactants})\\ &=[0.00+(-691.2)]-[(-412.9)+(-285.8)]\\ &=\boxed{+7.5\ \mathrm{kJ\,mol^{-1}}}. \end{aligned} \]

The formation of hydrogen ions and hydrogen carbonate from dissolved carbon dioxide and water is therefore endothermic.

6.4

The van ’t Hoff equation gives

\[ \frac{d\ln K}{dT} =\frac{\Delta_{\mathrm r}H^\circ}{RT^2}>0. \]

Thus, increasing temperature increases the equilibrium constant for the endothermic reaction. Since the dissolved carbon dioxide concentration is maintained constant,

\[ [\ce{H+}]\simeq\sqrt{K[\ce{CO2}]c^\circ} \]

increases as \(K\) increases. Consequently, the pH decreases.

The solution was checked against the official solution.