Question

In a dilute nitric acid solution, \(\mathrm{Fe}^{3+}\) reacts with thiocyanate ion \(\left(\mathrm{SCN}^{-}\right)\) to form a dark-red complex: \(\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}+\mathrm{SCN}^{-} \rightleftarrows \mathrm{H}_{2} \mathrm{O}+\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5} \mathrm{NCS}\right]^{2+}\) The equilibrium concentration of \(\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5} \mathrm{NCS}\right]^{2+}\) may be determined by how darkly colored the solution is (measured by a spectrometer). In one such experiment, \(1.0 \mathrm{~mL}\) of \(0.20 \mathrm{M} \mathrm{Fe}\left(\mathrm{NO}_{3}\right)_{3}\) was mixed with \(1.0 \mathrm{~mL}\) of \(1.0 \times 10^{-3} M \mathrm{KSCN}\) and \(8.0 \mathrm{~mL}\) of dilute \(\mathrm{HNO}_{3}\). The color of the solution quantitatively indicated that the \(\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5} \mathrm{NCS}\right]^{2+}\) concentration was \(7.3 \times 10^{-5} \mathrm{M}\) Calculate the formation constant for \(\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5} \mathrm{NCS}\right]^{2+}\)

Short answer

The formation constant \( K_f \) is approximately 13560.

Step-by-step solution

Identify Initial Conditions

Given the initial concentrations, calculate the moles of each reactant before mixing. The volume added is 10 mL when the reactants are combined.

Calculate Initial Concentrations After Mixing

Determine the initial concentrations of Fe(NO3)3 and KSCN in the 10 mL solution. The initial concentration of Fe(NO3)3 is calculated as \( \frac{0.20 \text{ M} \times 1 \text{ mL}}{10 \text{ mL}} = 0.020 \text{ M} \), and for KSCN, it is \( \frac{1.0 \times 10^{-3} \text{ M} \times 1 \text{ mL}}{10 \text{ mL}} = 1.0 \times 10^{-4} \text{ M} \).

Determine Equilibrium Concentrations

At equilibrium, the concentration of the complex \( \left[ \text{Fe(H}_2 \text{O)}_5\text{NCS} \right]^{2+} \) is given as \( 7.3 \times 10^{-5} \text{ M} \). Calculate the remaining \( \text{SCN}^- \) and \( \text{Fe}^{3+} \) concentrations. The initial concentration of \( \text{SCN}^- \) is reduced by \( 7.3 \times 10^{-5} \text{ M} \), giving \( 2.7 \times 10^{-5} \text{ M} \), and for \( \text{Fe}^{3+} \), it's \( 0.020 \text{ M} - 7.3 \times 10^{-5} \text{ M} = 0.019927 \text{ M} \).

Calculate the Formation Constant \( K_f \)

The expression for the formation constant is \( K_f = \frac{[\text{Fe(H}_2 \text{O)}_5\text{NCS}]^{2+}}{[\text{Fe}^{3+}][\text{SCN}^-]} \). Substitute the equilibrium concentrations: \( K_f = \frac{7.3 \times 10^{-5}}{0.019927 \times 2.7 \times 10^{-5}} \).

Compute the Numerical Value of \( K_f \)

Calculate \( K_f \) as follows: \[ K_f = \frac{7.3 \times 10^{-5}}{0.019927 \times 2.7 \times 10^{-5}} \approx \frac{7.3}{0.000538029} \approx 13560 \]. Thus, the formation constant \( K_f \) is approximately \( 13560 \).

Key concepts

Equilibrium Concentration

Understanding equilibrium concentration is crucial in chemical reactions, especially when complexes form in solutions. In our example, we have a reaction between \( \mathrm{Fe}^{3+} \) and thiocyanate \( \mathrm{SCN}^{-} \) forming a complex. The equilibrium concentration of the complex helps determine how much of the reactants remain unreacted.

To find this, we need initial concentrations post-mixing. It's all about subtracting the concentration of the formed complex from the initial reactants. This shows how much reactant turns into the product. By knowing the equilibrium concentration, we can easily find the equilibrium state of the reaction.

Think of it as a balance point where the formation of the complex and the remaining reactants are steady. Understanding this balance is key to determining how a reaction proceeds.

Spectrometry in Chemistry

Spectrometry plays a vital role in chemistry, especially for detecting concentrations of complexes in solutions. In the example provided, we measure the solution's color to deduce the concentration of \( \left[ \mathrm{Fe(H}_2 \mathrm{O)}_5 \mathrm{NCS} \right]^{2+} \).

This method uses light absorption to gauge how much of a complex is present in a sample. A spectrometer helps in quantifying this by measuring the darkness or intensity of the color.

- The more intense the color, the higher the concentration of the complex.- This technique allows for precise concentration measurements without direct sampling.

In essence, spectrometry provides a non-invasive way to understand chemical solutions' makeup through light interactions.

Chemistry Calculation

Chemistry calculations can be daunting, but they become manageable with practice. The goal is to determine how the initial concentrations of reactants change to form the complex. In our case:

• First, calculate initial concentrations by dividing the moles by the total solution volume.
• Then, adjust these concentrations to find the equilibrium values.

By understanding these calculations, you can predict how a reaction will behave under given conditions. Keep in mind:

• It's important to set up your equations correctly from the start.
• Pay attention to units of measure and consistency as they keep calculations accurate.

The calculations allow you to grasp how different components in a reaction influence each other’s final state.

Complex Formation in Solutions

Complex formation in solutions involves reactants like \( \mathrm{Fe}^{3+} \) and \( \mathrm{SCN}^{-} \) combining to form a more stable product. This results in chemical species with varied properties such as color and solubility.

The formation constant, \( K_f \), quantifies this stabilization by indicating the tendency for a complex to form. A higher \( K_f \) suggests a stronger, more stable complex.

In this reaction:- The formation of \( \left[ \mathrm{Fe(H}_2 \mathrm{O)}_5 \mathrm{NCS} \right]^{2+} \) changes the solution's characteristics, often visible as a color change.- Complexes alter the dynamic equilibrium, requiring precise measurements to determine their concentration.

Understanding complex formation is essential as it affects reaction pathways and outcomes, both in laboratory and industrial processes.