Silver Ion Concentration

Question

What is the silver ion concentration in a solution prepared by mixing 457 mL 0.378 M silver nitrate with 379 mL 0.495 M sodium carbonate? The Ksp of silver carbonate is 8.1×10^-12

Answer 1

Let's work out the original concentrations of Ag⁺ and CO₃^2-, assuming the salts dissolve completely and the volumes add:

.457 liters (.378 moles AgNO₃/liter)(1Ag⁺/1AgNO₃) = .172746 moles Ag⁺

Concentration of Ag⁺ is .172746 moles/(.457+.379 liters of solutions) = .2066340 M

.379 liters(.495 mole/liter Na₂CO₃)(1CO₃^2-/Na₂CO₃) = .187605 moles CO₃^2-

Concentration of Carbonate is .187605 moles/.836 liters = .2244079 M

Now for the Solubility Product: [Ag⁺]²[CO₃^2-] = K_SP at equilibrium

from the equation, Ag₂CO₃(s) --> 2Ag⁺(aq) + CO₃^2-(aq)

ICE Diagram (Using molarities - V constant) x will be the "molarity" of salt produced (xV to get moles)

Species Ag⁺ CO₃^2-

Initial .2066 .2244

Change -2x -x

Equil .2066 -2x .2244 -x

Solve for x using K expression at equilibrium: (.206634 -2x)^2(.2244079-x) = 8.1e-12

Cubic equation... x = .103313 [Wolfram alpha]

[Ag⁺]_eq = .206634 - 2(.103313) = 8e-6 M

[CO₃^2-]_eq = .2244079 - .103313 = ..1211 M

check: (8e-6)^2(.1211) = 7.8e-12 (A lot of % error from subtraction of Ag⁺)

Hope that helps.