Question

A solution contains \(1.0 \times 10^{-5} \mathrm{M} \mathrm{Ag}^{+}\) and \(2.0 \times 10^{-6} \mathrm{M} \mathrm{CN}^{-}\). Will \(\mathrm{AgCN}(s)\) precipitate? \(\left(K_{\mathrm{sp}}\right.\) for \(\mathrm{AgCN}(s)\) is \(2.2 \times 10^{-12}\).)

Short answer

Yes, precipitation of \(\mathrm{AgCN}(s)\) will occur, as the value of the reaction quotient (\(Q = 2.0 \times 10^{-11}\)) is greater than the solubility product constant (\(K_{\mathrm{sp}} = 2.2 \times 10^{-12}\)).

Step-by-step solution

Write the balanced equation for dissolution

Write the balanced chemical equation for the dissolution of \(\mathrm{AgCN}(s)\): \[\mathrm{AgCN}(s) \rightleftharpoons \mathrm{Ag}^{+}(aq) + \mathrm{CN}^{-}(aq)\]

Calculate the reaction quotient Q

Write the expression for the reaction quotient (\(Q\)) for the dissolution of \(\mathrm{AgCN}(s)\): \[Q = [\mathrm{Ag}^{+}][\mathrm{CN}^{-}]\] Substitute the given concentrations of \(\mathrm{Ag}^{+}\) and \(\mathrm{CN}^{-}\) ions into the expression for \(Q\): \[Q = (1.0 \times 10^{-5} \,\mathrm{M})(2.0 \times 10^{-6}\, \mathrm{M}) = 2.0 \times 10^{-11}\]

Compare Q with Ksp

Now compare \(Q\) with the given \(K_{\mathrm{sp}}\): \[Q = 2.0 \times 10^{-11} > K_{\mathrm{sp}} = 2.2 \times 10^{-12}\] Since \(Q > K_{\mathrm{sp}}\), precipitation of \(\mathrm{AgCN}(s)\) will occur.

Key concepts

Solubility Product Constant (Ksp)

The concept of solubility product constant, or Ksp, is crucial for understanding when a compound will precipitate from a solution. Think of Ksp as a numerical value that measures how much of a solid substance can dissolve in water to form a saturated solution. Specifically, it represents the maximum product of the ion concentrations of a salt that can exist in solution without forming a precipitate.

For any dissociation reaction like

AgCN(s) ⇄ Ag+(aq) + CN−(aq)

the Ksp value would be calculated as: \[Ksp = [Ag^+][CN^-]\]where the square brackets represent the molar concentrations of the ions at equilibrium. A key point to remember is that a low Ksp implies low solubility, and vice versa. If the product of the ion concentrations in the solution exceeds Ksp, this indicates the solution is supersaturated, and precipitation is likely to occur, as seen in our exercise example.

Reaction Quotient (Q)

To predict whether a precipitate will form, we turn to the reaction quotient, or Q. This helpful number is calculated in the same way as the Ksp, but it's for a system that isn't necessarily at equilibrium. It's an 'instant snapshot' of a solution's tendency to form a precipitate or dissolve a solid at any given moment.

Using the reaction:

AgCN(s) ⇄ Ag+(aq) + CN−(aq)

we get the following expression for Q: \[Q = [Ag^+][CN^-]\]You'll plug in the current ion concentrations into this expression, just as you would for Ksp. If Q is equal to Ksp, the system is at equilibrium, and no precipitation or dissolution will occur. If Q is less than Ksp, the solution can dissolve more of the solid (it's unsaturated). However, if Q exceeds Ksp, as in our original exercise, the solution is supersaturated and precipitation happens.

Chemical Equilibrium

Chemical equilibrium is a balance in a reversible chemical reaction where the rates of the forward (dissolution) and reverse (precipitation or re-formation of a solid) reactions are equal. At this point, the concentrations of reactants and products remain constant over time because the system has found a stable mix.

A reaction like

AgCN(s) ⇄ Ag+(aq) + CN−(aq)

can reach equilibrium when a certain amount of solid has dissolved in the solution to produce ions, and those ions are reforming the solid at the same rate they are being produced. This state is numerically defined by Ksp. When changes are made to the system, such as adding more ions or a solid, the equilibrium will shift to accommodate this change—a concept known as Le Chatelier's Principle.

Ionic Product

An ionic product is similar to the reaction quotient, Q, but it is strictly relevant to the ions present in a solution. It's the product of the concentrations of the cations and anions that come from the dissociation of an ionic compound in water. It is critical to understanding the behavior of ionic substances in solution.

For any solution containing ions, you can calculate the ionic product by multiplying the concentration of the cations by the concentration of the anions. In the context of the given exercise:

AgCN(s) ⇄ Ag+(aq) + CN−(aq)

the ionic product is calculated just like the Q: \[Ionic\text{ }Product = [Ag^+][CN^-]\]This ionic product helps predict whether a precipitate will form under given conditions. If the ionic product is greater than Ksp, then the ions exceed their solubility limit and begin to come together to form a solid, resulting in precipitation.