Question

\(\mathrm{AgNO}_{3}(\mathrm{aq})\) is slowly added to a solution that is \(0.250 \mathrm{M}\) \(\mathrm{NaCl}\) and also \(0.0022 \mathrm{M} \mathrm{KBr}\). (a) Which anion will precipitate first, \(\mathrm{Cl}^{-}\) or \(\mathrm{Br}^{-}\) ? (b) What is \(\left[\mathrm{Ag}^{+}\right]\) at the point at which the second anion begins to precipitate? (c) Can the \(\mathrm{Cl}^{-1}\) and \(\mathrm{Br}^{-}\) be separated effectively by this fractional precipitation?

Short answer

(a) Br- precipitates first. (b) The concentration of Ag+ when Cl- begins to precipitate is \(7.2 \times 10^{-10} M\). (c) The separation of Cl- and Br- by fractional precipitation is not effective.

Step-by-step solution

Identifying the anion that precipitates first

The anion that will precipitate first is the one that has the least solubility, i.e., it forms the least soluble salt with Ag+. Refer to the solubility product constants \(K_{sp}\) given in the textbook or a reference table. Standard tables normally list \( K_{sp}(AgCl) = 1.8 \times 10^{-10} \) and \( K_{sp}(AgBr) = 5.0 \times 10^{-13} \). Since \( K_{sp}(AgBr) < K_{sp}(AgCl) \), the Br- ion will precipitate first because AgBr is less soluble than AgCl.

Calculating the concentration of Ag+ when the second anion begins to precipitate

At the point at which the second anion (Cl-) begins to precipitate, the solution is saturated with respect to both salts. Use the \(K_{sp}(AgCl)\) to calculate the concentration of Ag+. The formula for the solubility product constant is \(K_{sp} = [Ag^+][Cl^-]\).\nSince we are given the concentration of Cl- as 0.25 M, we can solve for the concentration of Ag+: \( [Ag^+] = K_{sp} / [Cl^-] = (1.8 \times 10^{-10} mol^2/L^2 ) / 0.25 mol/L = 7.2 \times 10^{-10} M \). Hence, the concentration of Ag+ at the point at which Cl- begins to precipitate is \( 7.2 \times 10^{-10} M \).

Assessing the feasibility of separation

The separation is effective if the Ag+ concentration when Cl- starts precipitating (\(7.2 \times 10^{-10} M\)) is still less than the Ag+ concentration required to start precipitating Br-. Using the formula \( [Ag^+] = K_{sp}(AgBr) / [Br^-] \), we find that the Ag+ concentration required to start precipitating Br- is \( 5.0 \times 10^{-13} mol^2/L^2 / 0.0022 mol/L = 2.27 \times 10^{-10} M \). Since \( [Ag^+] (when Cl- starts precipitating) > [Ag^+] (required to start precipitating Br-), \) the separation is not feasible as Br- will still be in the solution when Cl- starts to precipitate.

Key concepts

Fractional Precipitation

Fractional precipitation is a technique employed in chemistry to separate different ions from a solution based on their individual solubility. When dealing with solutions that contain multiple ions, one can take advantage of differing solubility products (K_sp) to selectively precipitate them from the solution.

In the context of the exercise, we focus on separating halide ions using silver nitrate, which reacts with the halides to form insoluble silver halides. The fractionation depends heavily on the solubility product constants of these compounds; silver bromide (AgBr) has a lower K_sp compared to silver chloride (AgCl), indicating that it is less soluble. As such, when Ag⁺ is added to the solution, AgBr will precipitate out before AgCl. The effectiveness of this separation relies on the concentrations reaching a level where one salt precipitates considerably before the other.

Ionic Product

The ionic product (IP) refers to the product of the molar concentrations of the ions involved in a dissolved ionic compound, raised to the power of their respective stoichiometric coefficients in the balanced equation. It plays a significant role in solubility equilibrium.

It is important to note that the ionic product is closely related to the solubility product constant, K_sp. When IP equals K_sp, the solution is saturated, and any additional ions introduced may result in precipitation. If IP is less than K_sp, the solution can still dissolve more ions without precipitation. In contrast, when IP exceeds K_sp, the excess ions will start to precipitate out of the solution until equilibrium is restored. In our textbook solution, we can monitor the IP of AgCl and AgBr to predict at what point either ion will begin to precipitate.

Separation of Ions

Separation of ions by fractional precipitation is a strategic process dependent upon the difference in solubility product constants of the ionic compounds formed. Successful separation is achieved when one type of ion precipitates completely before the second ion starts to precipitate.

In the example provided, we seek to separate chloride and bromide ions. Due to the disparities in K_sp values and initial ion concentrations, complete separation is not possible in the given conditions. This is evidenced by the fact that the concentration of Ag⁺ required to initiate precipitation of AgBr is lower than the concentration at which AgCl starts to precipitate. This implies that both ions will precipitate simultaneously to some extent, preventing a clean separation. Thus, for effective ion separation, the conditions must be adjusted to widen the gap between the ionic product at which each ion precipitates.

Solubility Equilibrium

Solubility equilibrium pertains to the dynamic balance between a dissolved ionic compound and its undissolved form in a solution. The equilibrium is characterized by a constant value, the solubility product constant K_sp, which signifies the maximum product of ion concentrations that the solution can hold before it becomes supersaturated.

The exercise provided illustrates a scenario where silver ions react with chloride and bromide ions to potentially form precipitates of AgCl and AgBr. When K_sp for each compound is known, one can predict the point at which the solution becomes saturated with either silver chloride or silver bromide — that is when the ionic product equals K_sp. Calculating the concentrations of Ag⁺ at which each ion begins to precipitate allows us to understand the conditions under which solubility equilibrium is established. For the ions in the exercise, solubility equilibrium is reached at the concentration point given, beyond which precipitation occurs.