Question

What is the silver ion concentration in a solution prepared by mixing 379 mL of 0.369 M silver nitrate with 413 mL of 0.567 M sodium carbonate? The Ksp of silver carbonate is 8.1 × 10^-12. Please help. Thank you!

Answer 1

2Ag⁺(aq) + CO3^2-(aq) --> Ag₂CO₃(s)
ksp Ag₂CO₃ = 8.1 x10^-12 = [Ag⁺]² [CO₃^2-]

0.379 L x0.369 M AgNO₃ = 0.1398 mol AgNO3
0.413 L x 0.567 M Ag₂CO₃ = 0.2341 mol Ag₂CO₃^2-
0.379 L + 0.413 L = 0.792 L
0.1398 mol AgNO3 / 0.792L = 0.1765M
0.2341 mol Ag2CO3 / 0.792L =0.2955M

Think of the problem in two steps: (1) complete precipitation of Ag2CO3 because of the very small value of Ksp, and (2) dissolution of a very small amount of Ag2CO3 to satisfy Ksp

initially [Ag+] = 0.176 M and [CO3 2-] = 0.295 M. In a quantitative reaction,

ksp   = [Ag⁺]² [CO₃^2-]

=> 8.1 x10^-12 = (0.1765 - 2x)² (0.2955 - x)

Instead of solving this cubic equation, let's assume that most of the Ag+ goes into the precipitate:
2x ≈ 0.1765
x ≈ 0.0882

But not all of the Ag+ precipitates, so

8.1 x 10^-12 = [Ag⁺]² (0.2955 - 0.0882 )

=> 39.07 x 10^-12 = [Ag⁺]²

=> [Ag⁺] = 6.25 x 10^-6 M