IChO 2003 · Theory problem 28

Cryoscopy

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Chemists often need a bath in which to carry out a process that has a temperature below the water freezing point (\(0\,^\circ\mathrm{C}\)) and well above the \(\ce{CO2}\) sublimation point (\(-78\,^\circ\mathrm{C}\)) this case they mix water ice prepared at its melting point and \(\ce{NaCl}\). Depending on the quantities used temperatures as low as \(-20\,^\circ\mathrm{C}\) can be reached.

We prepare a cold bath mixing 1 kg of ice at \(0\,^\circ\mathrm{C}\) with 150 g of \(\ce{NaCl}\) in a thermally insulated container. Circle the letters Y or N to indicate if the following statements are correct (Y) or not (N).

a) The mixing process is spontaneous. (Y) (N)

b) The change of entropy during the mixing process is negative. (Y) (N)

c) The following diagram depicts the freezing point of aqueous solutions of \(\ce{NaCl}\) as a function of the composition of the solution (per cent by weight). What is is the freezing point of the bath based on the diagram?

Figure 1

d) If an equal mass of \(\ce{MgCl2}\) were used instead of \(\ce{NaCl}\), would the freezing point be higher? (Y) (N)

Official English text of the problem.

Answers

  • a) Y.
  • b) N. The entropy of the insulated system increases.
  • c) Approximately \(-9\,^\circ\mathrm{C}\), using the overall composition of \(13.0\%\) \(\ce{NaCl}\) by mass.
  • d) Y, under the usual ideal, complete-dissociation cryoscopic model.

Solution

a) Water, \(\ce{H2O}\) (O), is initially present as solid ice at its melting point. Dissolved salt lowers the chemical potential of liquid water, making melting into the salt solution favorable. Salt dissolution and ice melting therefore proceed spontaneously.

Melting absorbs energy. Because the container is thermally insulated, this energy comes from the bath itself, causing its temperature to fall.

b) For the entire insulated system, there is no entropy transfer through heat exchange with the surroundings. Consequently,

\[ \Delta S_{\mathrm{system}}=S_{\mathrm{generated}}>0 \]

for this spontaneous, irreversible process.

The entropy gains associated with mixing and melting outweigh the entropy decrease associated with cooling. Thus the statement that the entropy change is negative is false.

c) Taking the stated quantities as the composition of the resulting solution, the mass percentage of salt is

\[ w(\ce{NaCl}) =\frac{m(\ce{NaCl})}{m(\ce{NaCl})+m(\ce{H2O})}\times100\% =\frac{150\,\mathrm{g}}{150\,\mathrm{g}+1000\,\mathrm{g}}\times100\% =13.0\%. \]

At \(13.0\%\) on the horizontal axis, the descending branch of the graph gives approximately

\[ \boxed{T_{\mathrm f}\approx-9\,^\circ\mathrm{C}}. \]

This is the freezing point associated with the stated overall solution composition.

d) Freezing-point depression depends on the number of dissolved particles per unit mass of solvent:

\[ \Delta T_{\mathrm f}=iK_{\mathrm f}b, \]

where \(i\) is the van ’t Hoff factor and \(b\) is the formula-unit molality.

The ideal dissociation equations are

\[ \ce{NaCl(s) -> Na+(aq) + Cl-(aq)}, \]

\[ \ce{MgCl2(s) -> Mg^2+(aq) + 2Cl-(aq)}. \]

Thus \(i=2\) for sodium chloride and \(i=3\) for magnesium chloride. The dissolved species are sodium ion ([Na+]), magnesium ion ([Mg+2]) and chloride ion ([Cl-]); these SMILES do not represent their hydration shells.

Using the standard approximate molar masses

\[ M(\ce{NaCl})=58.44\,\mathrm{g\,mol^{-1}}, \qquad M(\ce{MgCl2})=95.21\,\mathrm{g\,mol^{-1}}, \]

the amounts of dissolved particles produced by 150 g are

\[ n_{\mathrm{particles}}(\ce{NaCl}) =2\frac{150\,\mathrm{g}}{58.44\,\mathrm{g\,mol^{-1}}} =5.13\,\mathrm{mol}, \]

\[ n_{\mathrm{particles}}(\ce{MgCl2}) =3\frac{150\,\mathrm{g}}{95.21\,\mathrm{g\,mol^{-1}}} =4.73\,\mathrm{mol}. \]

For equal solvent masses,

\[ \frac{\Delta T_{\mathrm f}(\ce{MgCl2})} {\Delta T_{\mathrm f}(\ce{NaCl})} =\frac{4.73}{5.13} \approx0.921. \]

An equal mass of magnesium chloride therefore gives fewer dissolved particles and a smaller freezing-point depression. Its freezing point is higher, so the answer is Y.

The solution was checked against the official solution.