Problem 30 - General Chemistry: Principles and Modern Applications

Question

A tabulation of formation constant data lists the following log \(K\) values for the formation of \(\left[\mathrm{CuCl}_{4}\right]^{2-}\): \(\log K_{1}=2.80, \quad \log K_{2}=1.60, \quad \log K_{3}=0.49, \quad\) and \(\log K_{4}=0.73 .\) What is the overall formation constant \(\beta_{4}=K_{\mathrm{f}}\) for \(\left[\mathrm{CuCl}_{4}\right]^{2-} ?\)

Step-by-Step Solution

Answer

To find the short answer, follow the steps above: Convert the log \(K\) values to \(K\) values, calculate the overall formation constant, and compute using the provided constants. The resulting value for \(\beta_{4}\) is your short answer.

Step 1: Convert log \(K\) values to \(K\) values

As the logarithm base 10 of a number \(x\) is given by \(\log x\), to invert this operation and get \(x\), you would use the ten to the power operation. So the formation constants are: \(K_1 = 10^{log K_1}\), \(K_2 = 10^{log K_2}\), \(K_3 = 10^{log K_3}\), \(K_4 = 10^{log K_4}\)

Step 2: Calculation of overall formation constant, \(\beta_{4}\)

Given that the overall formation constant is the product of the formation constants, \(\beta_{4}= K_{f}= K_1\times K_2\times K_3\times K_4\).

Step 3: Substitute and Compute

Substitute the \(K\) values from step 1 into the formula from step 2 and calculate \(\beta_{4}\).

Key Concepts

Complex Formation

Complex formation occurs when a central metal ion such as copper combines with surrounding molecules or ions, called ligands, to form a stable entity known as a complex. This process is typical in coordination chemistry where metals interact with non-metal elements. The ligands are usually capable of donating electron pairs to the metal center. In our example, chloride ions \(\text{Cl}^{-}\) act as ligands bonding with copper. The complexes formed can have various structures:

• The geometry of complexes can be square planar, tetrahedral, or octahedral, depending on the number of ligands involved.
• Common examples include tetrahedral environments for four ligands, like \[\mathrm{CuCl}_{4}]^{2-}\].

Complex formation is essential in various applications, including catalysis and medicine.

Stability Constants

Stability constants are vital in understanding how strongly a metal ion binds with its ligands in solution. These constants, often denoted as \(K\), measure the extent of complexation reactions.When dealing with logs of these constants, such as \(\log K_1\), \(\log K_2\), it is essential to remember that each constant relates to a step in the complex formation process. The higher the value of \(K\), the stronger the interaction:

• A higher \(\log K\) value indicates a more stable complex.
• It determines the likelihood of the metal ion and ligand staying bound together in a given state.

These constants help chemists predict reaction outcomes and are crucial for applications in chemical synthesis and environmental chemistry.

Copper Chloride Complexes

Copper chloride complexes, like \([\mathrm{CuCl}_4]^{2-}\), are examples of metal halide complexes where chloride ions bond with a copper cation. These complexes can range in color and solubility.In forming such complexes:

• Copper typically forms +2 oxidation state complexes.
• The chloride ions serve as the ligands, coordinating with the copper to create a stable tetrahedral geometry.

The formation constants associated with copper chloride complexes are studied to understand the full-circuit complex formation.It's these very constants (composition from \(K_1\) to \(K_4\)), together, helping determine the stability and strength of \([\mathrm{CuCl}_4]^{2-}\) in solution. Each step in complex formation contributes incrementally to how we calculate the overall formation constant \(\beta_4\). Understanding these interactions is foundational in fields such as coordination chemistry and bioinorganic chemistry.