Question

The \(K_{\mathrm{f}}\) for the formation of the complex ion between $$ \begin{array}{l} \mathrm{Pb}^{2+} \text { and } \mathrm{EDTA}^{4-}: \\ \mathrm{Pb}^{2+}+\mathrm{EDTA}^{4-} \rightleftarrows[\mathrm{Pb}(\mathrm{EDTA})]^{2-} \end{array} $$ is \(1.0 \times 10^{18}\) at \(25^{\circ} \mathrm{C}\). Calculate \(\left[\mathrm{Pb}^{2+}\right]\) at equilibrium in a solution containing \(1.0 \times 10^{-3} \mathrm{M} \mathrm{Pb}^{2+}\) and \(2.0 \times 10^{-3}\) \(M \mathrm{EDTA}^{4-}\)

Short answer

The equilibrium concentration of \([\text{Pb}^{2+}]\) is approximately \(1.0 \times 10^{-3} \) M.

Step-by-step solution

Write the Equilibrium Expression

Begin by writing the equilibrium expression for the formation of the complex ion. The expression for the equilibrium constant \( K_{\mathrm{f}} \) is given by: \[ K_{\mathrm{f}} = \frac{ [\text{Pb(EDTA)}]^{2-} }{ [\text{Pb}^{2+}][\text{EDTA}^{4-}] } \]

Set Up Initial Concentrations

The initial concentration of \( \text{Pb}^{2+} \) is \( 1.0 \times 10^{-3} \text{ M} \) and that of \( \text{EDTA}^{4-} \) is \( 2.0 \times 10^{-3} \text{ M} \). Initially, the concentration of \( [\text{Pb(EDTA)}]^{2-} \) is 0 M.

Define the Change in Concentration

Define \( x \) as the amount of \( \text{Pb}^{2+} \) and \( \text{EDTA}^{4-} \) that react to form \( [\text{Pb(EDTA)}]^{2-} \). The change in concentrations at equilibrium will be: \[ [\text{Pb}^{2+}] = 1.0 \times 10^{-3} - x \] \[ [\text{EDTA}^{4-}] = 2.0 \times 10^{-3} - x \] \[ [\text{Pb(EDTA)}]^{2-} = x \]

Substitute into Equilibrium Expression

Insert the equilibrium concentrations into the expression for \( K_{\mathrm{f}} \): \[ K_{\mathrm{f}} = \frac{x}{(1.0 \times 10^{-3} - x)(2.0 \times 10^{-3} - x)} \]

Solve for x

Assume \( x \) is very small compared to the initial concentrations since \( K_{\mathrm{f}} \) is large, leading to: \[ 1.0 \times 10^{18} = \frac{x}{(1.0 \times 10^{-3})(2.0 \times 10^{-3})} \] Simplifying yields: \[ x = 2.0 \times 10^{-18} \]

Calculate Equilibrium \([\text{Pb}^{2+}]\)

The equilibrium concentration of \( [\text{Pb}^{2+}] \) is approximately: \( 1.0 \times 10^{-3} - 2.0 \times 10^{-18} \approx 1.0 \times 10^{-3} \text{ M} \), as \( x \) is negligible.

Key concepts

Complex Ion Formation

Complex ion formation is a fascinating process that occurs when a metal ion combines with a ligand to form a complex. In the context of chemical equilibrium, this refers to the reversible reaction between a metal ion and a coordinating molecule, such as EDTA, to form a new species.
In our example, the metal ion is lead, \(\text{Pb}^{2+}\), and the ligand is ethylenediaminetetraacetate (EDTA), which is represented as \(\text{EDTA}^{4-}\).
When these two reactants come together, they form a complex ion such as \([\text{Pb(EDTA)}]^{2-}\).
Understanding complex ion formation is crucial, as it directly impacts the behavior of the ions in solutions and the equilibrium position of the formation reaction.

• Complex ions often have unique chemical properties distinct from their component ions.
• Formation of complex ions can significantly reduce the concentration of free metal ions in solution.

This process is utilized in various fields, including analytical chemistry, where it's used to detect and quantify metals in a solution.

Equilibrium Constant (Kf)

The equilibrium constant, often denoted as \(K_{\mathrm{f}}\), describes the ratio of product concentrations to reactant concentrations at chemical equilibrium for a specific reaction.
For complex ion formation, \(K_{\mathrm{f}}\) is specifically known as the "formation constant," indicating the extent to which the complex is formed.

In our exercise, \(K_{\mathrm{f}}\) for the formation of the lead-EDTA complex ion \([\text{Pb(EDTA)}]^{2-}\) is given as \(1.0 \times 10^{18}\).
This exceptionally large value indicates that the formation of the complex ion is highly favorable, resulting in the significant consumption of the parent ions during the process.

• A larger \(K_{\mathrm{f}}\) value suggests a greater tendency for the complex to form, correlating to a more stable complex.
• As \(K_{\mathrm{f}}\) is dependent on temperature, our problem specifies the value at \(25^{\circ}\text{C}\).

In summary, \(K_{\mathrm{f}}\) encapsulates the affinity of the metal ion for binding with the ligand in solution, providing insight into the stability and concentration of the resultant complex.

EDTA (ethylenediaminetetraacetate)

EDTA, or ethylenediaminetetraacetate, is a highly versatile and powerful chelating agent. It effectively binds to metal ions through multiple coordination sites.
In chemistry, it's renowned for its ability to sequester metal ions, forming stable complex ions.
This efficiency arises from its structure, with EDTA having four carboxylate and two amine donor sites, making it hexadentate.

When interacting with \(\text{Pb}^{2+}\), EDTA forms a very stable complex \([\text{Pb(EDTA)}]^{2-}\), which is a key aspect of many chemical processes involving heavy metals.
Some advantages of using EDTA include:

• Increased stability of metal complexes it forms, reducing the concentration of free metal ions in solution.
• Versatility in various applications such as water treatment, analytical chemistry, and medicine.

As seen in our exercise, EDTA, in its \(\text{EDTA}^{4-}\) form, plays a significant role in determining the equilibrium dynamics of metal ion solutions.

Lead Ion Concentration

The concentration of lead ions, \([\text{Pb}^{2+}]\), in solution is a critical factor in understanding the equilibrium state of reactions involving complex ions.
In such systems, the initial and equilibrium concentrations can vary significantly depending on the stability and formation constants like \(K_{\mathrm{f}}\).

For our particular reaction, the initial concentration of lead ions is \(1.0 \times 10^{-3} \text{ M}\).
After accounting for the formation of the lead-EDTA complex, the equilibrium concentration remains almost unchanged at this value.
This is due to the massive \(K_{\mathrm{f}}\) of the reaction, which suggests a negligible change in the initial concentration required to reach equilibrium.
Key points concerning lead ion concentration:

• The greater the stability of the complex ion, the smaller the concentration of free lead ions at equilibrium.
• Accurate calculation of this concentration is essential in assessing potential toxicological and environmental impacts.

Thus, understanding \([\text{Pb}^{2+}]\) and its interactions is essential for the effective management of lead in chemical solutions.