8.1: UV/Vis Analysis of Two-Component Mixtures

In Figure 10.1.9 we examined Nessler’s original method for matching the color of a sample to the color of a standard. Matching colors is a labor intensive process for the analyst and, not surprisingly, spectroscopic methods of analysis were slow to find favor. The 1930s and 1940s saw the introduction of photoelectric transducers for ultraviolet and visible radiation, and thermocouples for infrared radiation. As a result, modern instrumentation for absorption spectroscopy routinely became available in the 1940s—further progress has been rapid ever since.

Quantitative Analysis of Mixtures

Suppose we need to determine the concentration of two analytes, X and Y, in a sample. If each analyte has a wavelength where the other analyte does not absorb, then we can proceed using the approach in Example 8.1.5 . Unfortunately, UV/Vis absorption bands are so broad that frequently it is not possible to find suitable wavelengths. Because Beer’s law is additive the mixture’s absorbance, A_mix, is

\[\left(A_{m i x}\right)_{\lambda_{1}}=\left(\varepsilon_{x}\right)_{\lambda_{1}} b C_{X}+\left(\varepsilon_{Y}\right)_{\lambda_{1}} b C_{Y} \label{10.1}\]

where \(\lambda_1\) is the wavelength at which we measure the absorbance. Because Equation \ref{10.1} includes terms for the concentration of both X and Y, the absorbance at one wavelength does not provide enough information to determine either C_X or C_Y. If we measure the absorbance at a second wavelength

\[\left(A_{m i x}\right)_{\lambda_{2}}=\left(\varepsilon_{x}\right)_{\lambda_{2}} b C_{X}+\left(\varepsilon_{Y}\right)_{\lambda_{2}} b C_{Y} \label{10.2}\]

then we can determine C_X and C_Y by solving simultaneously Equation \ref{10.1} and Equation \ref{10.2}. Of course, we also must determine the value for \(\varepsilon_X\) and \(\varepsilon_Y\) at each wavelength. For a mixture of n components, we must measure the absorbance at n different wavelengths.

Example 8.1.2

The concentrations of Fe³⁺ and Cu²⁺ in a mixture are determined following their reaction with hexacyanoruthenate (II), \(\text{Ru(CN)}_6^{4-}\), which forms a purple-blue complex with Fe³⁺ (\(\lambda_\text{max}\) = 550 nm) and a pale-green complex with Cu²⁺ (\(\lambda_\text{max}\) = 396 nm) [DiTusa, M. R.; Schlit, A. A. J. Chem. Educ. 1985, 62, 541–542]. The molar absorptivities (M^–1 cm^–1) for the metal complexes at the two wavelengths are summarized in the following table.

analyte	\(\varepsilon_{550}\)	\(\varepsilon_{396}\)
Fe^3^+	9970	84
Cu^2^+	34	856

When a sample that contains Fe³⁺ and Cu²⁺ is analyzed in a cell with a pathlength of 1.00 cm, the absorbance at 550 nm is 0.183 and the absorbance at 396 nm is 0.109. What are the molar concentrations of Fe³⁺ and Cu²⁺ in the sample?

Solution

Substituting known values into Equation \ref{10.1} and Equation \ref{10.2} gives

\[\begin{aligned} A_{550} &=0.183=9970 C_{\mathrm{Fe}}+34 C_{\mathrm{Cu}} \\ A_{396} &=0.109=84 C_{\mathrm{Fe}}+856 C_{\mathrm{Cu}} \end{aligned} \nonumber\]

To determine C_Fe and C_Cu we solve the first equation for C_Cu

\[C_{\mathrm{Cu}}=\frac{0.183-9970 C_{\mathrm{Fe}}}{34} \nonumber\]

and substitute the result into the second equation.

\[\begin{aligned} 0.109 &=84 C_{\mathrm{Fe}}+856 \times \frac{0.183-9970 C_{\mathrm{Fe}}}{34} \\ &=4.607-\left(2.51 \times 10^{5}\right) C_{\mathrm{Fe}} \end{aligned} \nonumber\]

Solving for C_Fe gives the concentration of Fe³⁺ as \(1.8 \times 10^{-5}\) M. Substituting this concentration back into the equation for the mixture’s absorbance at 396 nm gives the concentration of Cu²⁺ as \(1.3 \times 10^{-4}\) M.

Another approach to solving Example 8.1.2 is to multiply the first equation by 856/34 giving

\[4.607=251009 C_{\mathrm{Fe}}+856 C_\mathrm{Cu} \nonumber\]

Subtracting the second equation from this equation

\[\begin{aligned} 4.607 &=251009 C_{\mathrm{Fe}}+856 C_{\mathrm{Cu}} \\-0.109 &=84 C_{\mathrm{Fe}}+856 C_{\mathrm{Cu}} \end{aligned} \nonumber\]

gives

\[4.498=250925 C_{\mathrm{Fe}} \nonumber\]

and we find that C_Fe is \(1.8 \times 10^{-5}\). Having determined C_Fe we can substitute back into one of the other equations to solve for C_Cu, which is \(1.3 \times 10^{-5}\).

Exercise 8.1.2

The absorbance spectra for Cr³⁺ and Co²⁺ overlap significantly. To determine the concentration of these analytes in a mixture, its absorbance is measured at 400 nm and at 505 nm, yielding values of 0.336 and 0.187, respectively. The individual molar absorptivities (M^–1 cm^–1) for Cr3⁺ are 15.2 at 400 nm and 0.533 at 505 nm; the values for Co²⁺ are 5.60 at 400 nm and 5.07 at 505 nm.

Answer

Substituting into Equation \ref{10.1} and Equation \ref{10.2} gives

\[A_{400} = 0.336 = 15.2C_\text{Cr} + 5.60C_\text{Co} \nonumber\]

\[A_{400} = 0187 = 0.533C_\text{Cr} + 5.07C_\text{Co} \nonumber\]

To determine C_Cr and C_Co we solve the first equation for C_Co

\[C_{\mathrm{Co}}=\frac{0.336-15.2 \mathrm{C}_{\mathrm{Co}}}{5.60} \nonumber\]

and substitute the result into the second equation.

\[0.187=0.533 C_{\mathrm{Cr}}+5.07 \times \frac{0.336-15.2 C_{\mathrm{Co}}}{5.60} \nonumber\]

\[0.187=0.3042-13.23 C_{\mathrm{Cr}} \nonumber\]

Solving for C_Cr gives the concentration of Cr³⁺ as \(8.86 \times 10^{-3}\) M. Substituting this concentration back into the equation for the mixture’s absorbance at 400 nm gives the concentration of Co²⁺ as \(3.60 \times 10^{-2}\) M.

To obtain results with good accuracy and precision the two wavelengths should be selected so that \(\varepsilon_X > \varepsilon_Y\) at one wavelength and \(\varepsilon_X < \varepsilon_Y\) at the other wavelength. It is easy to appreciate why this is true. Because the absorbance at each wavelength is dominated by one analyte, any uncertainty in the concentration of the other analyte has less of an impact. Figure 8.1.11 shows that the choice of wavelengths for Practice Exercise 8.1.2 are reasonable. When the choice of wavelengths is not obvious, one method for locating the optimum wavelengths is to plot \(\varepsilon_X / \varepsilon_y\) as function of wavelength, and determine the wavelengths where \(\varepsilon_X / \varepsilon_y\) reaches maximum and minimum values [Mehra, M. C.; Rioux, J. J. Chem. Educ. 1982, 59, 688–689].

Figure 8.1.11 . Visible absorption spectra for 0.0250 M Cr³⁺, 0.0750 M Co²⁺, and for a mixture of Cr³⁺ and Co²⁺. The two wavelengths used to analyze the mixture of Cr³⁺ and Co²⁺ are shown by the dashed lines. The data for the two standards are from Brewer, S. Solving Problems in Analytical Chemistry, John Wiley & Sons: New York, 1980.

When the analyte’s spectra overlap severely, such that \(\varepsilon_X \approx \varepsilon_Y\) at all wavelengths, other computational methods may provide better accuracy and precision. In a multiwavelength linear regression analysis, for example, a mixture’s absorbance is compared to that for a set of standard solutions at several wavelengths [Blanco, M.; Iturriaga, H.; Maspoch, S.; Tarin, P. J. Chem. Educ. 1989, 66, 178–180]. If A_SX and A_SY are the absorbance values for standard solutions of components X and Y at any wavelength, then

\[A_{SX}=\varepsilon_{X} b C_{SX} \label{10.3}\]

\[A_{SY}=\varepsilon_{Y} b C_{SY} \label{10.4}\]

where C_SX and C_SY are the known concentrations of X and Y in the standard solutions. Solving Equation \ref{10.3} and Equation \ref{10.4} for \(\varepsilon_X\) and for \(\varepsilon_Y\), substituting into Equation \ref{10.1}, and rearranging, gives

\[\frac{A_{\operatorname{mix}}}{A_{S X}}=\frac{C_{X}}{C_{S X}}+\frac{C_{Y}}{C_{S Y}} \times \frac{A_{S Y}}{A_{S X}} \nonumber\]

To determine C_X and C_Y the mixture’s absorbance and the absorbances of the standard solutions are measured at several wavelengths. Graphing A_mix/A_SX versus A_SY/A_SX gives a straight line with a slope of C_Y/C_SY and a y-intercept of C_X/C_SX. This approach is particularly helpful when it is not possible to find wavelengths where \(\varepsilon_X > \varepsilon_Y\) and \(\varepsilon_X < \varepsilon_Y\).

The approach outlined here for a multiwavelength linear regression uses a single standard solution for each analyte. A more rigorous approach uses multiple standards for each analyte. The math behind the analysis of such data—which we call a multiple linear regression—is beyond the level of this text. For more details about multiple linear regression see Brereton, R. G. Chemometrics: Data Analysis for the Laboratory and Chemical Plant, Wiley: Chichester, England, 2003.

Example 8.1.3

Figure \(PageIndex{10.11}\) shows visible absorbance spectra for a standard solution of 0.0250 M Cr³⁺, a standard solution of 0.0750 M Co²⁺, and a mixture that contains unknown concentrations of each ion. The data for these spectra are shown here.

\(\lambda\) (nm)	ACr	ACu	Amix	\(\lambda\) (nm)	ACr	ACu	Amix
375	0.26	0.01	0.53	520	0.19	0.38	0.63
400	0.43	0.03	0.88	530	0.24	0.33	0.70
425	0.39	0.07	0.83	540	0.28	0.26	0.73
440	0.29	0.13	0.67	550	0.32	0.18	0.76
455	0.20	0.21	0.54	570	0.38	0.08	0.81
470	0.14	0.28	0.47	575	0.39	0.06	0.82
480	0.12	0.30	0.44	580	0.38	0.05	0.79
490	0.11	0.34	0.45	600	0.34	0.03	0.70
500	0.13	0.38	0.51	625	0.24	0.02	0.49

Use a multiwavelength regression analysis to determine the composition of the unknown.

Solution

First we need to calculate values for A_mix/A_SX and for A_SY/A_SX. Let’s define X as Co²⁺ and Y as Cr³⁺. For example, at a wavelength of 375 nm A_mix/A_SX is 0.53/0.01, or 53 and A_SY/A_SX is 0.26/0.01, or 26. Completing the calculation for all wavelengths and graphing A_mix/A_SX versus A_SY/A_SX gives the calibration curve shown in Figure 8.1.12 . Fitting a straight-line to the data gives a regression model of

\[\frac{A_{\operatorname{mix}}}{A_{S X}}=0.636+2.01 \times \frac{A_{S Y}}{A_{S X}} \nonumber\]

Using the y-intercept, the concentration of Co²⁺ is

\[\frac{C_{X}}{C_{S X}}=\frac{\left[\mathrm{Co}^{2+}\right]}{0.0750 \mathrm{M}}=0.636 \nonumber\]

or [Co²⁺] = 0.048 M; using the slope the concentration of Cr³⁺ is

\[\frac{C_{Y}}{C_{S Y}}=\frac{\left[\mathrm{Cr}^{3+}\right]}{0.0250 \mathrm{M}}=2.01 \nonumber\]

or [Cr³⁺] = 0.050 M.

Figure 8.1.12 . Multiwavelength linear regression analysis for the data in Example 8.1.3 .

Exercise 8.1.3

A mixture of \(\text{MnO}_4^{-}\) and \(\text{Cr}_2\text{O}_7^{2-}\), and standards of 0.10 mM KMnO₄ and of 0.10 mM K₂Cr₂O₇ give the results shown in the following table. Determine the composition of the mixture. The data for this problem is from Blanco, M. C.; Iturriaga, H.; Maspoch, S.; Tarin, P. J. Chem. Educ. 1989, 66, 178–180.

\(\lambda\) (nm)	AMn	ACr	Amix
266	0.042	0.410	0.766
288	0.082	0.283	0.571
320	0.168	0.158	0.422
350	0.125	0.318	0.672
360	0.036	0.181	0.366

Answer

Letting X represent \(\text{MnO}_4^{-}\) and letting Y represent \(\text{Cr}_2\text{O}_7^{2-}\), we plot the equation

\[\frac{A_{\operatorname{mix}}}{A_{SX}}=\frac{C_{X}}{C_{SX}}+\frac{C_{Y}}{C_{S Y}} \times \frac{A_{S Y}}{A_{SX}} \nonumber\]

placing A_mix/A_SX on the y-axis and A_SY/A_SX on the x-axis. For example, at a wavelength of 266 nm the value A_mix/A_SX of is 0.766/0.042, or 18.2, and the value of A_SY/A_SX is 0.410/0.042, or 9.76. Completing the calculations for all wavelengths and plotting the data gives the result shown here

Fitting a straight-line to the data gives a regression model of

\[\frac{A_{\text { mix }}}{A_{\text { SX }}}=0.8147+1.7839 \times \frac{A_{SY}}{A_{SX}} \nonumber\]

Using the y-intercept, the concentration of \(\text{MnO}_4^{-}\) is

\[\frac{C_{X}}{C_{S X}}=0.8147=\frac{\left[\mathrm{MnO}_{4}^{-}\right]}{1.0 \times 10^{-4} \ \mathrm{M} \ \mathrm{MnO}_{4}^{-}} \nonumber\]

or \(8.15 \times 10^{-5}\) M \(\text{MnO}_4^{-}\), and using the slope, the concentration of \(\text{Cr}_2\text{O}_7^{2-}\) is

\[\frac{C_{Y}}{C_{S Y}}=1.7839=\frac{\left[\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\right]}{1.00 \times 10^{-4} \ \mathrm{M} \ \text{Cr}_{2} \mathrm{O}_{7}^{2-}} \nonumber\]

or \(1.78 \times 10^{-4}\) M \(\text{Cr}_2\text{O}_7^{2-}\).