Sure, let's solve this problem systematically by examining the given concentrations and the chemical equilibrium reaction:
[tex]\[ M^{2+}(aq) + 4 A^{-}(aq) \longrightarrow M(A)_4^{2-}(aq) \][/tex]
We start with the initial concentrations of [tex]\( M^{2+} \)[/tex] and [tex]\( A^- \)[/tex].
Initial Concentrations:
1. Molar concentration of [tex]\( M^{2+} \)[/tex] = 0.0250 M
2. Molar concentration of [tex]\( A^- \)[/tex] = 0.700 M
Equilibrium Concentrations:
During the reaction, a small amount [tex]\( x = 1.9 \times 10^{-5} \)[/tex] M of [tex]\( M^{2+} \)[/tex] complexes to form [tex]\( M(A)_4^{2-} \)[/tex].
Let's analyze the changes in concentrations that happen because of the reaction:
- [tex]\( M^{2+} \)[/tex] decreases by [tex]\( 1.9 \times 10^{-5} \)[/tex] M.
- [tex]\( A^- \)[/tex] decreases by [tex]\( 4 \times 1.9 \times 10^{-5} \)[/tex] M because four moles of [tex]\( A^- \)[/tex] react for each mole of [tex]\( M^{2+} \)[/tex].
Equilibrium concentration of [tex]\( M^{2+} \)[/tex]:
[tex]\[ [M^{2+}]_{eq} = 0.0250 \, \text{M} - 1.9 \times 10^{-5} \, \text{M} \][/tex]
[tex]\[ [M^{2+}]_{eq} = 0.024981 \, \text{M} \][/tex]
Equilibrium concentration of [tex]\( A^- \)[/tex]:
Change in concentration of [tex]\( A^- \)[/tex]:
[tex]\[ \Delta [A^-] = 4 \times 1.9 \times 10^{-5} \][/tex]
[tex]\[ \Delta [A^-] = 7.6 \times 10^{-5} \, \text{M} \][/tex]
Equilibrium concentration of [tex]\( A^- \)[/tex]:
[tex]\[ [A^-]_{eq} = 0.700 \, \text{M} - 7.6 \times 10^{-5} \, \text{M} \][/tex]
[tex]\[ [A^-]_{eq} = 0.699924 \, \text{M} \][/tex]
Equilibrium concentration of the complex [tex]\( M(A)_4^{2-} \)[/tex]:
The concentration of the complex formed is the same as the amount by which [tex]\( M^{2+} \)[/tex] decreased:
[tex]\[ [M(A)_4^{2-}] = 1.9 \times 10^{-5} \, \text{M} \][/tex]
Formation constant ([tex]\( K_f \)[/tex]) expression:
The formation constant [tex]\( K_f \)[/tex] is expressed in terms of the equilibrium concentrations of the complex, the metal ion, and the ligand:
[tex]\[ K_f = \frac{[M(A)_4^{2-}]}{[M^{2+}]_{eq} [A^-]_{eq}^4} \][/tex]
Substitute the equilibrium concentrations we found:
[tex]\[ K_f = \frac{1.9 \times 10^{-5}}{(0.024981) \times (0.699924)^4} \][/tex]
Plugging in these values:
[tex]\[ K_f \approx 0.0031691313501103525 \][/tex]
Therefore, the formation constant [tex]\( K_f \)[/tex] is:
[tex]\[ K_f \approx 0.003169 \][/tex]
This is the step-by-step solution for finding the value of the formation constant for the given reaction.
