Question

Silver ions are added to a solution with \(\left[\mathrm{Br}^{-}\right]=\left[\mathrm{Cl}^{-}\right]=\left[\mathrm{CO}_{3}^{2-}\right]=\left[\mathrm{AsO}_{4}^{3-}\right]=0.1 M .\) Which compound will precipitate with lowest \(\left[\mathrm{Ag}^{+}\right] ?\) (a) \(\mathrm{AgBr}\left(K_{s p}=5 \times 10^{-13}\right)\) (b) \(\mathrm{AgCl}\left(K_{\mathrm{sp}}=1.8 \times 10^{-10}\right)\) (c) \(\mathrm{Ag}_{2} \mathrm{CO}_{3}\left(K_{s p}=8.1 \times 10^{-12}\right)\) (d) \(\mathrm{Ag}_{3} \mathrm{AsO}_{4}\left(K_{\text {sp }}=1 \times 10^{-22}\right.\) )

Short answer

The compound that will precipitate with the lowest [Ag+] is Ag3AsO4 with a required concentration of 4.64 x 10^-8 M.

Step-by-step solution

Understand Precipitation and Solubility Product

Precipitation occurs when the product of the concentrations of the ions in a solution exceeds the solubility product constant, Ksp, for that particular compound. To find the compound which precipitates at the lowest concentration of Ag+, we need to calculate the concentration of Ag+ that will be in equilibrium with each anion given the Ksp for each compound.

Calculate the Concentration of Ag+ for AgBr Precipitation

For AgBr, Ksp = 5 x 10^-13. Since the stoichiometry of AgBr is 1:1, the equilibrium expression is Ksp = [Ag+][Br-]. Given [Br-] = 0.1 M and solving for [Ag+] gives [Ag+] = Ksp / [Br-] = (5 x 10^-13) / (0.1) = 5 x 10^-12 M.

Calculate the Concentration of Ag+ for AgCl Precipitation

For AgCl, Ksp = 1.8 x 10^-10. The stoichiometry is also 1:1, and thus Ksp = [Ag+][Cl-]. With [Cl-] = 0.1 M, [Ag+] = Ksp / [Cl-] = (1.8 x 10^-10) / (0.1) = 1.8 x 10^-9 M.

Calculate the Concentration of Ag+ for Ag2CO3 Precipitation

For Ag2CO3, Ksp = 8.1 x 10^-12. The stoichiometry is 2Ag+:1CO3^2-, so the equilibrium expression includes a square for the silver ion concentration, Ksp = [Ag+]^2[CO3^2-]. Substituting [CO3^2-] = 0.1 M, we need to solve for [Ag+] in the equation Ksp = [Ag+]^2(0.1). Therefore, [Ag+] = sqrt(Ksp / 0.1) = sqrt(8.1 x 10^-12 / 0.1) = 9 x 10^-6 M.

Calculate the Concentration of Ag+ for Ag3AsO4 Precipitation

For Ag3AsO4, Ksp = 1 x 10^-22. The stoichiometry here is 3Ag+:1AsO4^3-, so we have Ksp = [Ag+]^3[AsO4^3-]. With [AsO4^3-] = 0.1 M, solving for [Ag+] gives [Ag+] = cuberoot(Ksp / 0.1) = cuberoot(1 x 10^-22 / 0.1) = 4.64 x 10^-8 M.

Identify the Compound that Precipitates First

The compound with the smallest [Ag+] required for precipitation is the one that will precipitate first. Comparing the calculated [Ag+] for each compound, Ag3AsO4 requires the lowest [Ag+] concentration to precipitate.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in which the rate of the forward reaction equals the rate of the reverse reaction, meaning that the concentrations of reactants and products remain constant over time. Imagine a dance where two steps forward are matched with two steps back; no net movement is observed, and the dance continues at a steady pace.

This concept is crucial when discussing solubility product (Ksp), especially in the context of predicting whether a salt will precipitate out of solution. In a saturated solution, where no more solute can dissolve, the ionic product (concentration of cations multiplied by the concentration of anions) can provide a snapshot of this equilibrium stage.

At equilibrium, the ionic product equals the Ksp value. If the ionic product is less than the Ksp, more solute can dissolve, whereas if it's greater, the excess solute tends to precipitate. It's like our dance becoming too crowded; eventually, dancers will be pushed out to maintain balance, similar to precipitate forming to maintain equilibrium in a saturated solution.

Precipitation Reactions

Precipitation reactions involve the formation of a solid from a solution. When two soluble salts are mixed, if the resultant combination of anions and cations can form an insoluble compound, a precipitate is born. Think of it as a social event where two guests form an unexpected, exclusive pair, separating themselves from the mix.

In our specific case, we're adding silver ions to a solution that contains several types of anions. Precipitation is a game of match-making between the cations and anions, where the least soluble combination—holding the lowest solubility product (Ksp)—will make a match (precipitate) first. It's a race to see which pair will first say 'enough' and settle down as a solid. Each unique ionic pair has its own threshold (Ksp) that defines when this will happen. By calculating which requires the lowest concentration of silver ions to initiate precipitation, we can predict the winner of this race.

Ksp Calculations

Ksp calculations are crucial when predicting the solubility of ionic compounds in solution. The solubility product (Ksp) is a type of chemical equilibrium constant specific to the dissolution of solids. It's a bit like a recipe that tells you exactly how much of each ingredient (ion) you need to make just the right amount of product (solid) without any leftovers.

For each given ionic compound, the Ksp is calculated by taking the product of the molar concentrations of the ions, each raised to the power of its coefficient in the balanced dissolution equation. For example, for a compound like AgCl, the dissolution is represented as AgCl(s) ↔ Ag+(aq) + Cl-(aq), so Ksp is calculated as [Ag+][Cl-].

The lower the Ksp value, the less soluble the compound, indicating that fewer ions are needed to form a precipitate. By comparing the calculated concentration of silver ions needed for each possible precipitate, we identify the compound with the lowest Ksp value as the one to precipitate first with the lowest concentration of Ag+.