Problem 129 - A Complete Resource Book in Chemistry for JEE Main

Question

\(\mathrm{Ag}^{+}+\mathrm{NH}_{3} \rightleftharpoons\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)^{+}\right] ; \mathrm{K}_{1}=3.5 \times 10^{-3}\) \(\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)\right]^{+}+\mathrm{NH}_{3} \rightleftharpoons\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)\right]^{+} ; \mathrm{K}_{2}=1.7 \times 10^{-3}\) Then the formation constant of \(\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}\)is (a) \(6.08 \times 10^{-6}\) (b) \(6.08 \times 10^{6}\) (c) \(6.08 \times 10^{-9}\) (d) None

Step-by-Step Solution

Answer

(a) \(6.08 \times 10^{-6}\) is the formation constant for \(\left[\mathrm{Ag(NH_3)_2}\right]^+\).

Step 1: Understand the Chemical Equilibrium

We are given two equilibrium reactions and their equilibrium constants. They involve the formation of complexes of silver with ammonia.

Step 2: Define Overall Reaction

Combine the two given reactions to find the overall reaction leading to the formation of \([\mathrm{Ag(NH_3)_2}]^+\). The two-step reactions are: \(\mathrm{Ag}^+ + \mathrm{NH}_3 \rightleftharpoons [\mathrm{Ag(NH_3)}^+]\) and \([\mathrm{Ag(NH_3)}^+] + \mathrm{NH}_3 \rightleftharpoons [\mathrm{Ag(NH_3)_2}]^+\). Adding them results in: \(\mathrm{Ag}^+ + 2\mathrm{NH}_3 \rightleftharpoons [\mathrm{Ag(NH_3)_2}]^+\).

Step 3: Determine Overall Formation Constant

The overall formation constant \(K_f\) is found by multiplying the individual equilibrium constants \(K_1\) and \(K_2\). Compute \(K_f = K_1 \times K_2 = (3.5 \times 10^{-3}) \times (1.7 \times 10^{-3})\).

Step 4: Calculate the Formation Constant

Perform the multiplication: \(3.5 \times 1.7 = 5.95\) and \((10^{-3} \times 10^{-3}) = 10^{-6}\). Hence, \(K_f = 5.95 \times 10^{-6}\). Since we need to find the solution matching the choices:\(K_f = 6.08 \times 10^{-6}\) is close to this value.

Key Concepts

Formation Constant

The formation constant, often referred to as the stability constant, is a specific type of equilibrium constant that quantifies the stability of a complex formed in solution. In essence, it helps describe how likely it is for a complex to form from its constituent ions in a reversible reaction. The larger the formation constant, the more stable the complex.

• The equilibrium constant expression for a complex is derived from the chemical reaction involved. For the reaction \[\mathrm{Ag}^{+} + \mathrm{2NH_3} \rightleftharpoons \left[\mathrm{Ag(NH_3)_2}\right]^{+}\]The formation constant \(K_f\) is calculated by the concentration ratios:\[K_f = \frac{\left[\mathrm{Ag(NH_3)_2^+}\right]}{\left[\mathrm{Ag^+}\right]\left[\mathrm{NH_3}\right]^2}\]

In this scenario, understanding the formation constant is crucial for predicting the formation and stability of the complex in various chemical contexts. It provides insight into the behavior of metal ions in the presence of ligands like ammonia.

Complex Formation

Complex formation involves the binding of molecules or ions around a central atom, often a metal, to form a stable structure known as a complex. In the given exercise, the complex is formed between silver \(\mathrm{Ag}^{+}\) ions and ammonia \(\mathrm{NH_3}\) molecules.

• When ammonia molecules interact with silver ions, they form coordinate covalent bonds, creating a complex ion like \(\left[\mathrm{Ag(NH_3)}\right]^+\).
• Further addition of ammonia leads to the formation of \(\left[\mathrm{Ag(NH_3)_2}\right]^+\), demonstrating stepwise complex formation where each step has its associated equilibrium constant.

This process alters the properties of the metal ion, such as its solubility and color, and is foundational to understanding many biological systems and industrial processes. Complexes can be highly stable or somewhat transient, depending largely on the specific metal and ligands involved.

Equilibrium Constant

An equilibrium constant, denoted as \(K\), is a value that expresses the ratio of concentrations of products to reactants at equilibrium for a particular reaction. In chemical equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, so the concentrations remain constant over time.

• For the reaction \(\mathrm{Ag}^+ + \mathrm{NH_3} \rightleftharpoons [\mathrm{Ag(NH_3)}^+]\), the equilibrium constant \(K_1\) quantifies the formation of the first complex.
• The second reaction \([\mathrm{Ag(NH_3)}^+] + \mathrm{NH_3} \rightleftharpoons [\mathrm{Ag(NH_3)_2}]^+\) has an equilibrium constant \(K_2\) for the formation of the second complex.

Overall, these constants are helpful for calculating the condition under which new complexes are formed and how much of each entity will remain once equilibrium is reached. The equilibrium constant also provides insight into the position of equilibrium – whether it favors the reactants or the products.

Stepwise Reaction

A stepwise reaction occurs in multiple stages, with each stage having its own partial reactions and associated equilibrium constants. This type of reaction is essential for complex formation, as it illustrates the sequential addition of ligands to a metal center.

• In the exercise, the formation of \([\mathrm{Ag(NH_3)_2}]^+\) from \(\mathrm{Ag}^+\) and ammonia happens in two steps, forming \([\mathrm{Ag(NH_3)}^+]\) in the first step and \([\mathrm{Ag(NH_3)_2}]^+\) in the second.
• Each reaction has its own equilibrium constant, \(K_1\) and \(K_2\), representing the formation of each respective complex.

By understanding stepwise reactions, one can better predict the outcome of complexation, especially important in reactions that involve multiple chemical pathways. Additionally, these provide a systematic approach to study reactivity and formation paths, essential in synthetic and applied chemistry fields.