Dissociation of a Complete Ion

Question

Calculate the silver ion concentration, [Ag⁺], of a solution prepared by dissolving 1.00 g of AgNO₃ and 10.0 g of KCN in sufficient water to make 1.00 L of solution. (Hint: Because K_f is very large, assume the reaction goes to completion then calculate the [Ag⁺] produced by dissociation of the complex.)

Answer 1

[Ag⁺] = 5.2 × 10^-20 M

A comment here notes that the formation constant is missing, and that is fair - it is a table value rather than something you can derive. I have used K_f[Ag(CN)₂^-] = 5.6 × 10¹⁸, the value in most general chemistry appendices. Check it against yours, since the final number scales directly with it. Everything else in the problem is fully determined.

Step 1: moles of each reagent. In 1.00 L, molarity equals moles.

AgNO₃, M = 169.87 g/mol: 1.00 / 169.87 = 5.887 × 10^-3 M Ag⁺

KCN, M = 65.12 g/mol: 10.0 / 65.12 = 0.1536 M CN^-

Step 2: react to completion, as the hint directs. Ag⁺(aq) + 2 CN^-(aq) gives Ag(CN)₂^-(aq), so silver is limiting and each silver consumes two cyanides.

CN^- consumed = 2(5.887 × 10^-3) = 1.177 × 10^-2 M

CN^- remaining = 0.1536 - 0.0118 = 0.1418 M

Ag(CN)₂^- formed = 5.887 × 10^-3 M

Note how large the leftover cyanide is. You dissolved a big excess of KCN, and that excess is what drives the free silver so far down in the next step.

Step 3: back-dissociate. Rearranging K_f = [Ag(CN)₂^-] / ([Ag⁺][CN^-]²):

[Ag⁺] = (5.887 × 10^-3) / [(5.6 × 10¹⁸)(0.1418)²] = 5.2 × 10^-20 M

Square the free cyanide term. The 2 in the formula puts it there, and leaving it out inflates the answer sevenfold here without making it look wrong.

Now read what that number says. 5.2 × 10^-20 M in a litre works out to roughly thirty thousand free silver ions in the whole beaker, from the 3.5 × 10²¹ you dissolved. Practically none of the silver exists as a free ion.

This is not a contrived result. It is the basis of the cyanide process for extracting gold and silver from ore: cyanide binds the metal so tightly that it pulls it into solution from solid rock, and the metal is later recovered by breaking the complex. It is also why cyanide is dangerous for the mirror-image reason, since it binds the iron in cytochrome c oxidase with the same avidity and shuts down cellular respiration.