Question

Consider a concentration cell that has both electrodes made of some metal M. Solution A in one compartment of the cell contains 1.0 M M²⁺ Solution B in the other cell compartment has a volume of 1.00 L. At the beginning of the experiment, 0.0100 mole of M(NO₃)₂ and 0.0100 mole of Na₂SO₄, are dissolved in solution B (ignore volume changes), where the reaction
${\mathbf{M}}^{\mathbf{2}\mathbf{+}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}\mathbf{+}\mathbf{S}{{\mathbf{O}}_{\mathbf{4}}}^{\mathbf{2}\mathbf{-}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}\mathbf{\rightleftharpoons }{\mathbf{MSO}}_{\mathbf{4}}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$
occurs. For this reaction equilibrium is rapidly established, whereupon the cell potential is found to be +0.44 V at 25°C. Assume that the process
${\mathbf{M}}^{\mathbf{2}\mathbf{+}}\mathbf{+}{\mathbf{e}}^{\mathbf{-}}\mathbf{\to }\mathbf{M}$
has a standard reduction potential of +0.80 V and that no other redox process occurs in the cell. Calculate the value of K_sp for MSO₄(s) at 25°C.

Short answer

Thevalue of K_sp is found to be $1.63\times {10}^{-30}.$

Step-by-step solution

Oxidation and reduction half reaction

The reduction and oxidation half-reaction can be denoted as,

$$\begin{gathered} {{\mathrm{M}}_{\mathrm{A}}}^{+2}\left(\mathrm{aq}\right)+2{\mathrm{e}}^{-}\to \mathrm{M} \\ \mathrm{M}\to {{\mathrm{M}}_{\mathrm{B}}}^{+2}\left(\mathrm{aq}\right)+2{\mathrm{e}}^{-} \end{gathered}$$

The overall reaction can be expressed as,

${{\mathrm{M}}_{\mathrm{A}}}^{+2}\left(\mathrm{aq}\right)\to {{\mathrm{M}}_{\mathrm{B}}}^{+2}\left(\mathrm{aq}\right)$

Expression of Nernst equation

The Nernst equation expression can be denoted as,

$\begin{array}{rcl}{\mathrm{E}}_{\mathrm{cell}}& =& {{\mathrm{E}}^{\mathrm{o}}}_{\mathrm{cell}}-\frac{0.0591}{\mathrm{n}}\log \frac{\left[{{\mathrm{M}}_{\mathrm{B}}}^{+2}\right]}{\left[{{\mathrm{M}}_{\mathrm{A}}}^{+2}\right]}\\ \log \frac{\left[{{\mathrm{M}}_{\mathrm{B}}}^{+2}\right]}{\left[{{\mathrm{M}}_{\mathrm{A}}}^{+2}\right]}& =& ({{\mathrm{E}}^{\mathrm{o}}}_{\mathrm{cell}}-{\mathrm{E}}_{\mathrm{cell}})\frac{\mathrm{n}}{0.0591}\\ \log \frac{\left[{{\mathrm{M}}_{\mathrm{B}}}^{+2}\right]}{1}& =& (0-0.44)\left(\frac{2}{0.0591}\right)\\ \left[{{\mathrm{M}}_{\mathrm{B}}}^{+2}\right]& =& 1.28\times {10}^{-15}\mathrm{M}\end{array}$

$\begin{array}{rcl}{\mathrm{K}}_{\mathrm{sp}}& =& \left[{{\mathrm{M}}_{\mathrm{B}}}^{+2}\right]\left[\mathrm{S}{{\mathrm{O}}_4}^{2+}\right]\\ {\mathrm{K}}_{\mathrm{sp}}& =& (1.28\times {10}^{-15})(1.28\times {10}^{-15})\\ {\mathrm{K}}_{\mathrm{sp}}& =& 1.63\times {10}^{-30}.\end{array}$