Question

To determine the formation constant of a 1:1 complex, the following absorbances were measured at 470 nm in a 2.50-cm cell for the ligand concentrations shown. The total metal concentration was c_M = 7.50 x 10^-4M.

(a) Use linear regression and the Benesi-Hildebrand equation (Equation 14-11) to determine the formation constant and the difference in molar absorptivities at 470 nm.

(b) Use nonlinear regression and Equation 14-10 to find the values of K_f and $∆\epsilon$. Start with initial estimates

of Kf= 50 and$∆\epsilon =50$

Short answer

a) $$\begin{gathered} ∆\epsilon =349.9{\mathrm{cm}}^{-1}{\mathrm{M}}^{-1} \\ {K}_f=250.7 \end{gathered}$$

b)$$\begin{gathered} ∆\epsilon =350 \\ {K}_f=250 \end{gathered}$$

Step-by-step solution

Part (a) Step 1: Given Information

Total metal concentration = c_M = 7.50 x 10^-4M

Part (a) Step 2: Explanation

The tabulated data is:

Plotting the graph as:

Intercept = $\frac{1}{∆\epsilon C_M}$

$$\begin{gathered} 3.8109=\frac{1}{∆\epsilon \times 7.5\times {10}^{-4}} \\ ∆\mathrm{\epsilon }=\frac{1}{3.8109\times 7.5\times {10}^{-4}} \\ ∆\mathrm{\epsilon }=349.9{\mathrm{cm}}^{-1}{\mathrm{M}}^{-1} \\ \mathrm{Slope}=\frac{1}{∆{\mathrm{\epsilon K}}_{\mathrm{f}}{\mathrm{C}}_{\mathrm{M}}} \\ 0.0152=\frac{1}{∆{\mathrm{\epsilon K}}_{\mathrm{f}}{\mathrm{C}}_{\mathrm{M}}} \\ 0.0152=\frac{1}{349.9\times {\mathrm{K}}_{\mathrm{f}}\times 7.5\times {10}^{-4}} \\ {\mathrm{K}}_{\mathrm{f}}=\frac{1}{349.9\times 0.0152\times 7.5\times {10}^{-4}} \\ {\mathrm{K}}_{\mathrm{f}}=250.7 \end{gathered}$$

Part (b) Step 1: Given Information

$$\begin{gathered} ∆\epsilon =50 \\ {K}_f=50 \end{gathered}$$

Part (b) Step 2: Explanation

The tabulated data is:

Initial estimate of K_f = 249.8159

Initial estimate of ∆ε = 350.0477

There is no difference between the data values and the model

Therefore,

$$\begin{gathered} ∆\epsilon =350 \\ {K}_f=250 \end{gathered}$$