Question

For the reaction \[MX\left( s \right) \leftrightarrow {M^ + }\left( {aq} \right) + {X^ - }\left( {aq} \right),\Delta {G^\circ } = 57kJ\]. \[{K_{sp}}\]
A. \[{10^{ - 5}}\]
B. \[{10^{ - 6}}\]
C. \[{10^{ - 10}}\]
D. \[{10^{ - 16}}\]

Answer 1

Hint: \[\Delta {G^\circ }\] stands for Gibbs free energy, and \[{K_{sp}}\] stands for the solubility product of the given compound MX . There is a direct relationship between the solubility product and Gibbs free energy of a compound, which can be used to determine the value of \[{K_{sp}}\] for the compound MX . We must keep in mind that since no particular temperature condition is mentioned at which the reaction is taking place, the reaction is assumed to be occurring at the standard temperature of \[{25^\circ }C\] itself.

Complete step by step answer:
The relationship between the solubility product and Gibbs free energy is given by the formula:
\[\Delta {G^\circ } = - 2.303 \times R \times T \times log{K_{sp}}\] …\[(i)\]
Where R represents the ideal gas constant of reaction;
T represents the temperature of the reaction in Kelvin.

The given value of \[\Delta {G^\circ }\] for the dissociation reaction of \[MX\] is \[57kJ\].
All the values must be in the S.I. unit. So, we have to convert \[kJ\] into \[J\] (joules).
We know that, 1kJ = 1000J
\[ \Rightarrow 57kJ = 57 \times 1000J = 57000J\]
The value of \[R\] in S.I. unit is the constant value of \[8.314J/mol/K\].
The S.I. value of \[{25^\circ }C\] in Kelvin \[ = 25{{ }} + {{ }}273{{ }}K{{ }} = {{ }}298{{ }}K\]

Substituting all the values in the equation \[(i)\] , we get;

\[57000\;J/mol = - 2.303 \times 8.314J/mol/K \times 298K \times log{K_{sp}}\]

\[ \Rightarrow 57000J/mol = - 5705.84J/mol \times log{K_{sp}}\]

\[ \Rightarrow log{K_{sp}} = \dfrac{{57000J/mol}}{{ - 5705.84J/mol}}\]

\[ \Rightarrow log{K_{sp}} = - 10\]

\[ \Rightarrow {K_{sp}} = {10^{ - 10}}\]
Therefore, the value of \[{K_{sp}}\] for the dissociation of \[MX\] using Gibbs free energy of \[57kJ\] is \[{10^{ - 10}}\].

So, the correct answer is Option C.

Additional Information:
Gibbs free energy is the energy available in a system that determines whether a reaction will take place in the system.
Solubility product is a measure or the extent to which a solute dissolves in a solvent after dissociation. Its value increases with the increase in solubility of a compound in a solvent.

Note: A reaction can take place only if Gibbs free energy for the reaction is negative. So, the negative sign in the equation must not be omitted.
To convert degree Celsius into Kelvin, we must add \[273\] to it, as \[{1^\circ }C = 273K\].
It must be kept in mind that the natural base of a logarithm is \[10\]. So, \[log{K_{sp}} = - 10\] means \[lo{g_{10}}{K_{sp}} = - 10\].