IChO 1985 · Theory problem 5

Thermodynamics of a zinc - silver sulphate galvanic cell

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The equilibrium voltage of the cell,

\[ \ce{Zn}\;/\;\ce{ZnSO4}\;(0.0125\ \mathrm{M}) \;\|\; \ce{Ag2SO4}\;(0.0125\ \mathrm{M})\;/\;\ce{Ag} \]

was measured at several temperatures and the results of the measurements are given in the following table:

\(t\,/\,^\circ\mathrm{C}\) 10 20 30
\(E\,/\,\mathrm{V}\) 1.5784 1.5675 1.5566

Problems:

5.1 Give the equation for the reaction occurring in this galvanic cell.

5.2 Determine the value of the cell voltage at the temperature T = 298 K.

5.3 Determine \(\Delta_{\mathrm r}G_{298}\) of the cell reaction.

5.4 Determine \(\Delta_{\mathrm r}H_{298}\) of the cell reaction.

Official English text of the problem.

Answers

  • 5.1 \(\ce{Zn(s) + 2Ag+(aq) -> Zn^2+(aq) + 2Ag(s)}\), equivalently \(\ce{Zn(s) + Ag2SO4(aq) -> ZnSO4(aq) + 2Ag(s)}\).
  • 5.2 \(E_{298}=1.5622\ \mathrm{V}\), or approximately \(1.562\ \mathrm{V}\).
  • 5.3 \(\Delta_{\mathrm r}G_{298}=-301.5\ \mathrm{kJ\,mol^{-1}}\).
  • 5.4 \(\Delta_{\mathrm r}H_{298}\approx-364.1\ \mathrm{kJ\,mol^{-1}}\).

The molar quantities refer to the reaction as written, consuming one mole of zinc.

Solution

5.1

For the cell written, the positive voltage corresponds to oxidation at the left-hand zinc electrode and reduction at the right-hand silver electrode.

Anode: \[ \ce{Zn(s) -> Zn^2+(aq) + 2e-} \]

Cathode: \[ \ce{2Ag+(aq) + 2e- -> 2Ag(s)} \]

Adding the half-reactions gives \[ \boxed{\ce{Zn(s) + 2Ag+(aq) -> Zn^2+(aq) + 2Ag(s)}}. \]

Thus, the number of electrons transferred per reaction as written is \(n=2\). Sulfate is a spectator ion, so the equivalent salt equation is \[ \boxed{\ce{Zn(s) + Ag2SO4(aq) -> ZnSO4(aq) + 2Ag(s)}}. \]

The ionic species are zinc(II) ion [Zn+2], silver(I) ion [Ag+], and sulfate ion [O-]S(=O)(=O)[O-]. The salts are zinc sulfate [Zn+2].[O-]S(=O)(=O)[O-] and silver(I) sulfate [Ag+].[Ag+].[O-]S(=O)(=O)[O-]. These disconnected ionic SMILES specify composition; they do not represent hydration or ion association in solution.

5.2

The voltage decreases by \(0.0109\ \mathrm{V}\) in each \(10^\circ\mathrm{C}\) interval. We therefore interpolate linearly over the measured range:

\[ \left(\frac{\partial E}{\partial T}\right)_p =\frac{1.5566-1.5784}{303.15-283.15} =-1.090\times10^{-3}\ \mathrm{V\,K^{-1}}. \]

At \(T=298\ \mathrm{K}\), \(t=24.85^\circ\mathrm{C}\). Hence

\[ \begin{aligned} E_{298} &=1.5675+ \left(-1.090\times10^{-3}\right)(298-293.15)\\ &=1.5622135\ \mathrm{V}. \end{aligned} \]

Therefore, \[ \boxed{E_{298}=1.5622\ \mathrm{V}}. \]

Using the usual approximation \(298\ \mathrm{K}\approx25^\circ\mathrm{C}\) instead gives \(1.56205\ \mathrm{V}\); both results are approximately \(1.562\ \mathrm{V}\).

5.3

For a reversible galvanic cell, \[ \Delta_{\mathrm r}G=-nFE. \]

Using the standard Faraday constant \(F=96485\ \mathrm{C\,mol^{-1}}\),

\[ \begin{aligned} \Delta_{\mathrm r}G_{298} &=-2(96485)(1.5622135)\ \mathrm{J\,mol^{-1}}\\ &=-301460\ \mathrm{J\,mol^{-1}}. \end{aligned} \]

Thus, \[ \boxed{\Delta_{\mathrm r}G_{298}=-301.5\ \mathrm{kJ\,mol^{-1}}}. \]

This is the reaction Gibbs energy at the specified solution concentrations.

5.4

At constant pressure and fixed composition, \[ \left(\frac{\partial\Delta_{\mathrm r}G}{\partial T}\right)_p =-\Delta_{\mathrm r}S. \]

Differentiating \(\Delta_{\mathrm r}G=-nFE\) therefore gives \[ \Delta_{\mathrm r}S =nF\left(\frac{\partial E}{\partial T}\right)_p. \]

Consequently, \[ \Delta_{\mathrm r}S_{298} =2(96485)(-1.090\times10^{-3}) =-210.3373\ \mathrm{J\,mol^{-1}\,K^{-1}}. \]

Using \(\Delta_{\mathrm r}H=\Delta_{\mathrm r}G+T\Delta_{\mathrm r}S\),

\[ \begin{aligned} \Delta_{\mathrm r}H_{298} &=-301.4603+ 298(-0.2103373)\ \mathrm{kJ\,mol^{-1}}\\ &=-364.1409\ \mathrm{kJ\,mol^{-1}}. \end{aligned} \]

Hence, \[ \boxed{\Delta_{\mathrm r}H_{298}\approx-364.1\ \mathrm{kJ\,mol^{-1}}}. \]

The negative sign shows that the cell reaction is exothermic.

The solution was checked against the official solution. Where this solution differs from the official solution, the official solution contains an error or a misprint.