Problem 86 - Chemistry An Atoms First Approach

Question

The equilibrium constant for the following reaction is \(1.0 \times 10^{23}:\) $$\mathrm{Cr}^{3+}(a q)+\mathrm{H}_{2} \mathrm{EDTA}^{2-}(a q) \rightleftharpoons \mathrm{CrEDTA}^{-}(a q)+2 \mathrm{H}^{+}(a q)$$ EDTA is used as a complexing agent in chemical analysis. Solutions of EDTA, usually containing the disodium salt \(\mathrm{Na}_{2} \mathrm{H}_{2} \mathrm{EDTA},\) are used to treat heavy metal poisoning. Calculate \(\left[\mathrm{Cr}^{3+}\right]\) at equilibrium in a solution originally \(0.0010 \mathrm{M}\) in \(\mathrm{Cr}^{3+}\) and \(0.050 M\) in \(\mathrm{H}_{2} \mathrm{EDTA}^{2-}\) and buffered at \(\mathrm{pH}=6.00\).

Step-by-Step Solution

Answer

The equilibrium concentration of Cr³⁺ in the given solution is approximately 0.0010 M.

Step 1: Write down the balanced chemical equation and the expression for the equilibrium constant

First, we need to write down the balanced chemical equation given in the exercise: \[\mathrm{Cr}^{3+}(a q)+\mathrm{H}_{2} \mathrm{EDTA}^{2-}(a q) \rightleftharpoons \mathrm{CrEDTA}^{-}(a q)+2 \mathrm{H}^{+}(a q)\] Then, we write the expression for the equilibrium constant, \(K_{eq}\): \[K_{eq} = \frac{[\mathrm{CrEDTA}^{-}][\mathrm{H^{+}}]^{2}}{[\mathrm{Cr}^{3+}][\mathrm{H}_{2}\mathrm{EDTA}^{2-}]}\] The equilibrium constant value is given as \(1.0 \times 10^{23}\).

Step 2: Set up a RICE (Reaction, Initial, Change, Equilibrium) table to keep track of the concentrations

In order to keep track of the concentrations of each species during the reaction, we can set up a RICE table: | | Cr³⁺ | H₂EDTA²⁻ | CrEDTA⁻ | H⁺ | |------|------|--------|--------|----| | R | | | | | | I | 0.0010 | 0.050 | 0 | \(10^{-6}\) | | C | -x | -x | +x | +2x | | E |0.0010 - x | 0.050 - x | x | \(10^{-6}\) + 2x | Here, RICE stands for Reaction, Initial, Change, and Equilibrium. The initial concentrations of Cr³⁺ and H₂EDTA²⁻ are given as 0.0010 M and 0.050 M, respectively. The initial concentration of CrEDTA⁻ is 0 M, as it has not formed yet. Since the pH is given as 6.00, we know that the initial concentration of H⁺ = \(10^{-6}\) M. During the reaction, the concentrations of the reactants Cr³⁺ and H₂EDTA²⁻ will decrease by an amount x and the concentrations of the products CrEDTA⁻ and H⁺ will increase by x and 2x, respectively.

Step 3: Substitute the equilibrium concentrations into the equilibrium constant expression

Now we can substitute the equilibrium concentrations given in the RICE table into the expression for the equilibrium constant: \begin{align*} 1.0 \times 10^{23} &= \frac{(x)[10^{-6} + 2x]^{2}}{(0.0010 - x)(0.050 - x)} \end{align*} Since \(K_{eq}\) is very large, we can assume that the reaction will go almost to completion, so x is close to the initial concentration of Cr³⁺ (0.0010 M). Therefore, we can simplify the expression by assuming that \(0.0010 - x \approx 0.0010\) and \(0.050 - x \approx 0.050\): \begin{align*} 1.0 \times 10^{23} &\approx \frac{(x)[10^{-6} + 2x]^{2}}{(0.0010)(0.050)} \end{align*}

Step 4: Solve for x (the concentration of CrEDTA⁻ at equilibrium)

Now we need to solve the simplified equation for x: \begin{align*} x[10^{-6} + 2x]^{2} &= 1.0 \times 10^{23} \times 0.0010 \times 0.050 \\ x(10^{-6} + 2x)^{2}& \approx 5.0 \times 10^{20} \end{align*} To simplify the equation, we can also assume that \(2x \ll 10^{-6}\) since the value of x will be very small compared to \(10^{-6}\). This results in: \begin{align*} x(10^{-6})^2 &\approx 5.0 \times 10^{20} \\ x &\approx \frac{5.0 \times 10^{20}}{(10^{-6})^{2}} \\ x &\approx 5.0 \times 10^{-8} \end{align*}

Step 5: Determine the equilibrium concentration of Cr³⁺

We know that the equilibrium concentration of Cr³⁺ is given by \(0.0010 - x\), so: \[[\mathrm{Cr}^{3+}]_{eq} = 0.0010 - 5.0 \times 10^{-8} \approx 0.0010\ \mathrm{M}\] Thus, the equilibrium concentration of Cr³⁺ is approximately 0.0010 M.

Key Concepts

RICE Table

When solving chemical equilibrium problems, the RICE table is a helpful tool for organizing and handling the changes in concentrations. The acronym RICE stands for:

• R: Reaction - The balanced chemical equation for the reaction.
• I: Initial - The initial concentrations of reactants and products before the reaction starts.
• C: Change - The change in concentrations as the reaction progresses towards equilibrium.
• E: Equilibrium - The concentrations of reactants and products once equilibrium is reached.

For the reaction, \[ \mathrm{Cr}^{3+}(aq) + \mathrm{H}_{2} \mathrm{EDTA}^{2-}(aq) \rightleftharpoons \mathrm{CrEDTA}^{-}(aq) + 2 \mathrm{H}^{+}(aq) \]we start by listing the initial concentrations. Let \( x \) be the change in concentration for each reactant and product. At equilibrium, the changes are applied to the initial concentrations. Using the RICE table ensures organized steps and provides a pictorial reference that guides the calculation of equilibrium concentrations.

Complexing Agent

A complexing agent, often known as a chelating agent, is a substance capable of forming multiple bonds with a single metal ion. In our exercise, \(\mathrm{EDTA}^{4-}\) (ethylenediaminetetraacetic acid) is used as a complexing agent. It can tightly bind to metal ions like \(\mathrm{Cr}^{3+}\), forming stable complexes such as \(\mathrm{CrEDTA}^{-}\).

Complexing agents function by:

• Reducing the concentration of free metal ions in solution, which can help in analyses or purify solutions.
• Preventing metal ions from disrupting reactions by stabilizing them in solution.

EDTA is particularly effective because it has multiple binding sites and forms what is called a multi-dentate or polydentate complex. It wraps around the metal ion in a claw-like fashion, effectively sequestering it and rendering it inert to interference in reactions or analyses.

Heavy Metal Poisoning

Heavy metal poisoning occurs when metals like lead, mercury, or chromium accumulate in the body, causing harm. These metals interfere with biological processes and can lead to various health issues such as organ damage and developmental disorders.

This is where chelation therapy comes into play. Chelation therapy involves the administration of chelating agents like EDTA, which bind to heavy metals in the bloodstream and form a stable complex. This complex is then more easily excreted from the body.

• EDTA is particularly effective in treating certain cases of heavy metal poisoning due to its ability to form stable complexes with metal ions.
• Through chelation therapy, harmful metal ions are safely removed from the body, reducing their toxic effects.

Understanding how complexing agents work is essential to grasp how chemical solutions can be applied in medical treatments and environmental cleanups of heavy metal contamination.