Question

A \(0.1\) mole of \(\mathrm{AgNO}_{3}\) is dissolved in \(1 \mathrm{~L}\) of \(1 \mathrm{M}-\mathrm{NH}_{3} .\) If \(0.01\) mole of \(\mathrm{NaCl}\) is added to this solution, will \(\mathrm{AgCl}(\mathrm{s})\) precipitate? \(K_{\mathrm{sp}}\) for \(\mathrm{AgCl}=1.8 \times 10^{-10}\) and \(K_{\text {stab }}\) for \(\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}=1.6 \times 10^{7} .\) (a) Yes (b) No (c) Addition of \(\mathrm{NaCl}\) in any amount can never result precipitation. (d) Addition of even smaller amount of \(\mathrm{NaCl}\) may result precipitation.

Short answer

Yes, \(\mathrm{AgCl}\) will precipitate upon addition of \(0.01\) mole of \(\mathrm{NaCl}\).

Step-by-step solution

Determine the initial conditions

Identify the initial concentrations of the species involved. We have 0.1 mole of \(\mathrm{AgNO}_3\) in 1 L, which gives us a concentration of \(0.1\ M\) of \(\mathrm{Ag}^{+}\). Also, \(\mathrm{NH}_3\) concentration is \(1\ M\).

Calculate the complexation reaction quotient

Write down the complexation reaction, which is \(\mathrm{Ag}^{+} + 2\mathrm{NH}_3 \leftrightarrow \mathrm{Ag}(\text{NH}_3)_2^{+}\). The reaction quotient is given by \(Q_c = \frac{[\mathrm{Ag}(\text{NH}_3)_2^{+}]}{[\mathrm{Ag}^{+}][\mathrm{NH}_3]^2}\). Since \(Q_c\) should be equal to the stability constant \(K_{\text{stab}}\), we can calculate the concentration of \(\mathrm{Ag}(\text{NH}_3)_2^{+}\) using \(K_{\text{stab}} = 1.6 \times 10^{7}\) and the known concentrations of \(\mathrm{Ag}^{+}\) and \(\mathrm{NH}_3\).

Deduce the concentration of \(\mathrm{Ag}^{+}\) remaining in the solution

Calculate the concentration of free \(\mathrm{Ag}^{+}\) by applying the stoichiometry of the reaction from step 2. Use the equilibrium expression to find the concentration of uncomplexed \(\mathrm{Ag}^{+}\).

Assert whether \(\mathrm{AgCl}\) precipitates with the \(\mathrm{NaCl}\) addition

With the addition of \(0.01\) mole of \(\mathrm{NaCl}\), we have \(0.01\ M\) of \(\mathrm{Cl}^{-}\). Now evaluate if the product of the concentrations of \(\mathrm{Ag}^{+}\) and \(\mathrm{Cl}^{-}\) exceeds the \(K_{\text{sp}}\) for \(\mathrm{AgCl}\) which is \(1.8 \times 10^{-10}\). If \(\text{[Ag}^{+}\text{][Cl}^{-}] > K_{\text{sp}}\), then \(\mathrm{AgCl}\) will precipitate.

Conclude if precipitation is possible

Using the calculated concentration of \(\mathrm{Ag}^{+}\) and the provided \(\mathrm{Cl}^{-}\) concentration, determine if the ionic product is greater than the solubility product to confirm precipitation. Depending on this calculation, you can choose the appropriate answer among the options provided.

Key concepts

Solubility Product Constant (Ksp)

The Solubility Product Constant, abbreviated as Ksp, is a term in chemistry that indicates how far a compound can dissolve in water. It is specific for each substance and varies with temperature. For a salt like silver chloride (AgCl), which dissociates into silver ions (Ag+) and chloride ions (Cl-), the Ksp is given by the equation \[K_{sp} = [Ag^+][Cl^-]\]In simpler terms, Ksp helps us understand whether a precipitate will form in a solution. If the product of the concentrations of the ions in the solution exceeds the Ksp, it implies that the solution is supersaturated, and precipitation is likely to occur. On the other hand, if the ionic product is less than the Ksp, the solution remains unsaturated and no precipitate forms.

Complex Formation

Complex formation refers to the combination of two or more species to form a coordination compound, which can affect the solubility of certain salts. In the presence of ammonia (NH3), silver ions (Ag+) can form a complex ion with ammonia, such as \[\text{Ag(NH3)_2^+}\].The formation of this complex is governed by the stability constant (Kstab), a special type of equilibrium constant for complex formation reactions. Complex formation can impact the availability of free ions like Ag+ in the solution, thus affecting the saturation level and the potential for precipitation to occur. The ability of a complexing agent like NH3 to 'hold onto' the metal ion and prevent it from reacting with other ions is an important concept in understanding solubility and precipitation dynamics.

Reaction Quotient (Qc)

The Reaction Quotient, represented by Qc, is an expression that allows chemists to determine the direction in which a reaction will proceed to reach equilibrium. For a general reaction \[aA + bB \leftrightarrow cC + dD\],the reaction quotient is calculated as \[Q_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}\].If we compare Qc to the equilibrium constant (K), we can predict whether a reaction will move forward, move in reverse, or is at equilibrium. In the context of our exercise, calculating Qc informed us about the complex formation and helped to determine the concentration of free Ag+ ions in the solution, which is crucial for understanding whether AgCl precipitation will occur.

Silver Chloride (AgCl) Precipitation

Silver Chloride (AgCl) precipitation occurs when the concentrations of Ag+ and Cl- in a solution reach a level where their product exceeds the solubility product Ksp of AgCl. In a solution that is already complexed with NH3, the concentration of free Ag+ ions can be low, which would normally lead to no precipitation. However, the addition of a salt like NaCl increases the Cl- concentration, potentially pushing the ionic product past the Ksp and causing AgCl to precipitate. This process is influenced by several factors, including the total ionic strength of the solution, the presence of complexing agents, and the competing reactions.

Stability Constant (Kstab)

The Stability Constant, often referred to as Kstab, is associated with the formation of a complex ion from a metal ion and one or more ligands. This constant quantifies the affinity of the metal ion for the ligands, with a higher Kstab indicating a more stable and tightly-bound complex. For the complex ion \[\text{Ag(NH3)_2^+}\],the Kstab helps us understand how silver ions will behave in the presence of ammonia in the solution. A high Kstab implies that most Ag+ ions will tend to remain complexed with NH3, making them less available to form precipitates like AgCl, thus decreasing the overall tendency for AgCl to precipitate from the solution.