IChO 1972 · Theory problem 4

Neutralisation heat and final temperature of mixed solutions

Amount of 50 g of a 4 % sodium hydroxide solution and 50 g of a 1.825 % solution of hydrochloric acid were mixed in a heat insulated vessel at a temperature of 20 °C. The temperature of the solution obtained in this way increased to 23.4 °C. Then 70 g of a 3.5 % solution of sulphuric acid at a temperature of 20 °C were added to the above solution.

Problems:

4.1 Calculate the final temperature of the resulting solution.

4.2 Determine the amount of a dry residue that remains after evaporation of the solution.

In calculating the first problem use the heat capacity value \(c = 4.19\ \mathrm{J\,g^{-1}\,K^{-1}}\).

Relative atomic masses:

\(A_{\mathrm r}(\ce{H}) = 1;\quad A_{\mathrm r}(\ce{O}) = 16;\quad A_{\mathrm r}(\ce{Na}) = 23;\quad A_{\mathrm r}(\ce{S}) = 32;\quad A_{\mathrm r}(\ce{Cl}) = 35.5.\)

Official English text of the problem.

Answers

  • 4.1: \(\boxed{T_{\mathrm f}=24.0\,^\circ\mathrm C}\).
  • 4.2: \(\boxed{m_{\mathrm{dry}}=3.55\ \mathrm g}\) of anhydrous sodium sulfate, \(\ce{Na2SO4}\), SMILES [Na+].[Na+].[O-]S(=O)(=O)[O-], assuming complete expulsion of hydrogen chloride during evaporation and drying.

Solution

4.1

The initial reactants are sodium hydroxide, \(\ce{NaOH}\), SMILES [Na+].[OH-], and hydrochloric acid, an aqueous solution of hydrogen chloride, \(\ce{HCl}\), SMILES Cl.

Their amounts are

\[ n(\ce{NaOH}) =\frac{(50\ \mathrm g)(0.04)}{40\ \mathrm{g\,mol^{-1}}} =0.0500\ \mathrm{mol}, \]

\[ n(\ce{HCl}) =\frac{(50\ \mathrm g)(0.01825)}{36.5\ \mathrm{g\,mol^{-1}}} =0.0250\ \mathrm{mol}. \]

Neutralisation produces sodium chloride, \(\ce{NaCl}\), SMILES [Na+].[Cl-], and water, \(\ce{H2O}\), SMILES O:

\[ \ce{NaOH + HCl -> NaCl + H2O}. \]

Hydrochloric acid is the limiting reactant. Thus \(0.0250\ \mathrm{mol}\) of water forms, and the same amount of sodium hydroxide remains:

\[ n_{\mathrm{remaining}}(\ce{NaOH}) =0.0500-0.0250 =0.0250\ \mathrm{mol}. \]

The heat released in this first reaction is

\[ Q_1=mc\Delta T =(100\ \mathrm g) (4.19\ \mathrm{J\,g^{-1}\,K^{-1}}) (3.4\ \mathrm K) =1424.6\ \mathrm J. \]

Consequently, the molar enthalpy of neutralisation inferred from the experiment is

\[ \Delta_{\mathrm{neut}}H =-\frac{1424.6\ \mathrm J}{0.0250\ \mathrm{mol}} \approx-57.0\ \mathrm{kJ\,mol^{-1}}. \]

The added sulfuric acid, \(\ce{H2SO4}\), SMILES OS(=O)(=O)O, has amount

\[ n(\ce{H2SO4}) =\frac{(70\ \mathrm g)(0.035)}{98\ \mathrm{g\,mol^{-1}}} =0.0250\ \mathrm{mol}. \]

It reacts with the remaining sodium hydroxide in a \(1:1\) molar ratio. The formal salt product is sodium hydrogen sulfate, \(\ce{NaHSO4}\), SMILES [Na+].[O-]S(=O)(=O)O:

\[ \ce{NaOH + H2SO4 -> NaHSO4 + H2O}. \]

A further \(0.0250\ \mathrm{mol}\) of water forms. Using the same molar neutralisation heat and neglecting dilution effects,

\[ Q_2=Q_1=1424.6\ \mathrm J. \]

The final heat balance must include warming the added acid solution from \(20\,^\circ\mathrm C\):

\[ (100\ \mathrm g)c(T_{\mathrm f}-23.4\,^\circ\mathrm C) +(70\ \mathrm g)c(T_{\mathrm f}-20.0\,^\circ\mathrm C) =Q_2. \]

Equivalently, all three original solutions started at \(20\,^\circ\mathrm C\), so

\[ (170\ \mathrm g)c(T_{\mathrm f}-20.0\,^\circ\mathrm C) =Q_1+Q_2. \]

Therefore,

\[ T_{\mathrm f}-20.0\,^\circ\mathrm C =\frac{2849.2\ \mathrm J} {(170\ \mathrm g)(4.19\ \mathrm{J\,g^{-1}\,K^{-1}})} =4.0\ \mathrm K, \]

\[ \boxed{T_{\mathrm f}=24.0\,^\circ\mathrm C}. \]

4.2

After mixing, the formal salt amounts are

\[ n(\ce{NaCl})=0.0250\ \mathrm{mol}, \qquad n(\ce{NaHSO4})=0.0250\ \mathrm{mol}. \]

During evaporation and sufficiently complete heating, volatile hydrogen chloride can be expelled according to

\[ \ce{NaCl + NaHSO4 -> Na2SO4 + HCl ^}. \]

The two salts are present in exactly the required ratio. Complete hydrogen chloride removal therefore leaves

\[ n(\ce{Na2SO4})=0.0250\ \mathrm{mol}. \]

Using the supplied atomic masses,

\[ M(\ce{Na2SO4}) =2(23)+32+4(16) =142\ \mathrm{g\,mol^{-1}}, \]

and hence

\[ \boxed{ m_{\mathrm{dry}} =(0.0250\ \mathrm{mol})(142\ \mathrm{g\,mol^{-1}}) =3.55\ \mathrm g }. \]

The solution was checked against the official solution.