What is the value of the equilibrium constant K for the overall reaction Cr(OH)3(s) + OH−(aq) ⇌ [Cr(OH)4]−(aq) given Ksp = 1.6 × 10^−30 and Kf = 8.0 × 10^29?

• A. −8 × 10^29
• B. 2.0 × 10^−60
• C. 1.3 × 10^0
• D. 5.0 × 10^59

Answer: (C)

Think of the overall reaction as the combination of two processes: the dissolution of the solid and the subsequent formation of the complex. The dissolution of Cr(OH)3(s) is Cr(OH)3(s) ⇌ Cr3+ + 3 OH− with Ksp = 1.6 × 10^−30. The complex formation is Cr3+ + 4 OH− ⇌ [Cr(OH)4]− with Kf = 8.0 × 10^29.

If you add these two equilibria, the Cr3+ cancels, leaving the net reaction Cr(OH)3(s) + OH− ⇌ [Cr(OH)4]−. The overall equilibrium constant is the product of the two constants: K_overall = Ksp × Kf.

Compute: (1.6 × 10^−30) × (8.0 × 10^29) = 1.28 × 10^0 ≈ 1.3.

So the equilibrium constant for the overall reaction is about 1.3, meaning the reaction is not strongly to either side; there is a moderate tendency to form the complex.