1.9: Problems

1. Provide the missing information in the following table.

wavelength (m)	frequency (s^–1)	wavenumber (cm^–1)	energy (J)
\(4.50 \times 10^{-9}\)
	\(1.33 \times 10^{15}\)
		3215
			\(7.20 \times 10^{-19}\)

2. Provide the missing information in the following table.

[analyte] (M)	absorbance	%T	molar absorptivity (M^–1 cm^–1)	pathlength (cm)
\(1.40 \times 10^{-4}\)			1120	1.00
	0.563		750	1.00
\(2.56 \times 10^{-4}\)	0.225		456540
\(1.55 \times 10^{-3}\)	0.167		1550	5.00
		33.3		1.00
\(4.35 \times 10^{-3}\)		21.2
\(1.20 \times 10^{-4}\)		81.3		10.00

3. A solution’s transmittance is 35.0%. What is the transmittance if you dilute 25.0 mL of the solution to 50.0 mL?

4. A solution’s transmittance is 85.0% when measured in a cell with a pathlength of 1.00 cm. What is the %T if you increase the pathlength to 10.00 cm?

5. The accuracy of a spectrophotometer is evaluated by preparing a solution of 60.06 ppm K₂Cr₂O₇ in 0.0050 M H₂SO₄, and measuring its absorbance at a wavelength of 350 nm in a cell with a pathlength of 1.00 cm. The expected absorbance is 0.640. What is the expected molar absorptivity of K₂Cr₂O₇ at this wavelength?

6. A chemical deviation to Beer’s law may occur if the concentration of an absorbing species is affected by the position of an equilibrium reaction. Consider a weak acid, HA, for which K_a is \(2 \times 10^{-5}\). Construct Beer’s law calibration curves of absorbance versus the total concentration of weak acid (C_total = [HA] + [A^–]), using values for C_total of \(1.0 \times 10^{-5}\), \(3.0 \times 10^{-5}\), \(5.0 \times 10^{-5}\), \(7.0 \times 10^{-5}\), \(9.0 \times 10^{-5}\), \(11 \times 10^{-5}\), and \(13 \times 10^{-5}\) M for the following sets of conditions and comment on your results:

(a) \(\varepsilon_{HA} = \varepsilon_{A^-} = 2000\) M^–1 cm^–1; unbuffered solution.

(b) \(\varepsilon_{HA} = 2000\) M^–1 cm^–1; \(\varepsilon_{A^-} = 500\) M^–1 cm^–1; unbuffered solution.

(c) \(\epsilon_{HA} = 2000\) M^–1 cm^–1; \(\epsilon_{A^-} = 500\) M^–1 cm^–1; solution buffered to a pH of 4.5.

Assume a constant pathlength of 1.00 cm for all samples.

7. One instrumental limitation to Beer’s law is the effect of polychromatic radiation. Consider a line source that emits radiation at two wavelengths, \(\lambda^{\prime}\) and \(\lambda^{\prime \prime}\). When treated separately, the absorbances at these wavelengths, A′ and A′′, are

\[A^{\prime}=-\log \frac{P_{\mathrm{T}}^{\prime}}{P_{0}^{\prime}}=\varepsilon^{\prime} b C \quad \quad A^{\prime \prime}=-\log \frac{P_{\mathrm{T}}^{\prime \prime}}{P_{0}^{\prime \prime}}=\varepsilon^{\prime \prime} b C \nonumber\]

If both wavelengths are measured simultaneously the absorbance is

\[A=-\log \frac{\left(P_{\mathrm{T}}^{\prime}+P_{\mathrm{T}}^{\prime \prime}\right)}{\left(P_{0}^{\prime}+P_{0}^{\prime \prime}\right)} \nonumber\]

(a) Show that if the molar absorptivities at \(\lambda^{\prime}\) and \(\lambda^{\prime \prime}\) are the same (\(\varepsilon^{\prime} = \varepsilon^{\prime \prime} = \varepsilon\)), then the absorbance is equivalent to

\[A=\varepsilon b C \nonumber\]

(b) Construct Beer’s law calibration curves over the concentration range of zero to \(1 \times 10^{-4}\) M using \(\varepsilon^{\prime} = 1000\) M^–1 cm^–1 and \(\varepsilon^{\prime \prime} = 1000\) M^–1 cm^–1, and \(\varepsilon^{\prime} = 1000\) M^–1 cm^–1 and \(\varepsilon^{\prime \prime} = 100\) M^–1 cm^–1. Assume a value of 1.00 cm for the pathlength and that \(P_0^{\prime} = P_0^{\prime \prime} = 1\). Explain the difference between the two curves.

8. A second instrumental limitation to Beer’s law is stray radiation. The following data were obtained using a cell with a pathlength of 1.00 cm when stray light is insignificant (P_stray = 0).

[analyte] (mM)	absorbance
0.00	0.00
2.00	0.40
4.00	0.80
6.00	1.20
8.00	1.60
10.00	3.00

Calculate the absorbance of each solution when P_stray is 5% of P₀, and plot Beer’s law calibration curves for both sets of data. Explain any differences between the two curves. (Hint: Assume P₀ is 100).

9. In the process of performing a spectrophotometric determination of iron, an analyst prepares a calibration curve using a single-beam spectrophotometer similar to that shown in Figure 10.3.2. After preparing the calibration curve, the analyst drops and breaks the cuvette. The analyst acquires a new cuvette, measures the absorbance of the sample, and determines the %w/w Fe in the sample. Does the change in cuvette lead to a determinate error in the analysis? Explain.

10. The spectrophotometric methods for determining Mn in steel and for determining glucose use a chemical reaction to produce a colored spe- cies whose absorbance we can monitor. In the analysis of Mn in steel, colorless Mn²⁺ is oxidized to give the purple \(\text{MnO}_4^{-}\) ion. To analyze for glucose, which is also colorless, we react it with a yellow colored solution of the \(\text{Fe(CN)}_6^{3-}\), forming the colorless \(\text{Fe(CN)}_6^{4-}\) ion. The directions for the analysis of Mn do not specify precise reaction conditions, and samples and standards are treated separately. The conditions for the analysis of glucose, however, require that the samples and standards are treated simultaneously at exactly the same temperature and for exactly the same length of time. Explain why these two experimental procedures are so different.

11. One method for the analysis of Fe³⁺, which is used with a variety of sample matrices, is to form the highly colored Fe³⁺–thioglycolic acid complex. The complex absorbs strongly at 535 nm. Standardizing the method is accomplished using external standards. A 10.00-ppm Fe³⁺ working standard is prepared by transferring a 10-mL aliquot of a 100.0 ppm stock solution of Fe³⁺ to a 100-mL volumetric flask and diluting to volume. Calibration standards of 1.00, 2.00, 3.00, 4.00, and 5.00 ppm are prepared by transferring appropriate amounts of the 10.0 ppm working solution into separate 50-mL volumetric flasks, each of which contains 5 mL of thioglycolic acid, 2 mL of 20% w/v ammonium citrate, and 5 mL of 0.22 M NH₃. After diluting to volume and mixing, the absorbances of the external standards are measured against an appropriate blank. Samples are prepared for analysis by taking a portion known to contain approximately 0.1 g of Fe³⁺, dissolving it in a minimum amount of HNO₃, and diluting to volume in a 1-L volumetric flask. A 1.00-mL aliquot of this solution is transferred to a 50-mL volumetric flask, along with 5 mL of thioglycolic acid, 2 mL of 20% w/v ammonium citrate, and 5 mL of 0.22 M NH₃ and diluted to volume. The absorbance of this solution is used to determine the concentration of Fe3⁺ in the sample.

(a) What is an appropriate blank for this procedure?

(b) Ammonium citrate is added to prevent the precipitation of Al³⁺. What is the effect on the reported concentration of iron in the sample if there is a trace impurity of Fe³⁺ in the ammonium citrate?

(c) Why does the procedure specify that the sample contain approximately 0.1 g of Fe³⁺?

(d) Unbeknownst to the analyst, the 100-mL volumetric flask used to prepare the 10.00 ppm working standard of Fe³⁺ has a volume that is significantly smaller than 100.0 mL. What effect will this have on the reported concentration of iron in the sample?

12. A spectrophotometric method for the analysis of iron has a linear calibration curve for standards of 0.00, 5.00, 10.00, 15.00, and 20.00 mg Fe/L. An iron ore sample that is 40–60% w/w is analyzed by this method. An approximately 0.5-g sample is taken, dissolved in a minimum of concentrated HCl, and diluted to 1 L in a volumetric flask using distilled water. A 5.00 mL aliquot is removed with a pipet. To what volume—10, 25, 50, 100, 250, 500, or 1000 mL—should it be diluted to minimize the uncertainty in the analysis? Explain.

13. Lozano-Calero and colleagues developed a method for the quantitative analysis of phosphorous in cola beverages based on the formation of the blue-colored phosphomolybdate complex, (NH₄)₃[PO₄(MoO₃)₁₂] [Lozano-Calero, D.; Martín-Palomeque, P.; Madueño-Loriguillo, S. J. Chem. Educ. 1996, 73, 1173–1174]. The complex is formed by adding (NH₄)₆Mo₇O₂₄ to the sample in the presence of a reducing agent, such as ascorbic acid. The concentration of the complex is determined spectrophotometrically at a wavelength of 830 nm, using an external standards calibration curve.

In a typical analysis, a set of standard solutions that contain known amounts of phosphorous is prepared by placing appropriate volumes of a 4.00 ppm solution of P₂O₅ in a 5-mL volumetric flask, adding 2 mL of an ascorbic acid reducing solution, and diluting to volume with distilled water. Cola beverages are prepared for analysis by pouring a sample into a beaker and allowing it to stand for 24 h to expel the dissolved CO₂. A 2.50-mL sample of the degassed sample is transferred to a 50-mL volumetric flask and diluted to volume. A 250-μL aliquot of the diluted sample is then transferred to a 5-mL volumetric flask, treated with 2 mL of the ascorbic acid reducing solution, and diluted to volume with distilled water.

(a) The authors note that this method can be applied only to noncol- ored cola beverages. Explain why this is true.

(b) How might you modify this method so that you can apply it to any cola beverage?

(c) Why is it necessary to remove the dissolved gases?

(d) Suggest an appropriate blank for this method?

(e) The author’s report a calibration curve of

\[A=-0.02+\left(0.72 \ \mathrm{ppm}^{-1}\right) \times C_{\mathrm{P}_{2} \mathrm{O}_{5}} \nonumber\]

A sample of Crystal Pepsi, analyzed as described above, yields an absorbance of 0.565. What is the concentration of phosphorous, reported as ppm P, in the original sample of Crystal Pepsi?

Crystal Pepsi was a colorless, caffeine-free soda produced by PepsiCo. It was available in the United States from 1992 to 1993.

14. EDTA forms colored complexes with a variety of metal ions that may serve as the basis for a quantitative spectrophotometric method of analysis. The molar absorptivities of the EDTA complexes of Cu²⁺, Co²⁺, and Ni²⁺ at three wavelengths are summarized in the following table (all values of \(\varepsilon\) are in M^–1 cm^–1).

metal ion	\(\varepsilon_{462.9}\)	\(\varepsilon_{732.0}\)	\(\varepsilon_{378.7}\)
Co^2^+	15.8	2.11	3.11
Cu^2^+	2.32	95.2	7.73
Ni^2^+	1.79	3.03	13.5

Using this information determine the following, assuming a pathlength, b, of 1.00 cm for all measurements:

(a) The concentration of Cu²⁺ in a solution that has an absorbance of 0.338 at a wavelength of 732.0 nm.

(b) The concentrations of Cu²⁺ and Co²⁺ in a solution that has an absorbance of 0.453 at a wavelength of 732.0 nm and 0.107 at a wavelength of 462.9 nm.

(c) The concentrations of Cu²⁺, Co²⁺, and Ni²⁺ in a sample that has an absorbance of 0.423 at a wavelength of 732.0 nm, 0.184 at a wavelength of 462.9 nm, and 0.291 at a wavelength of 378.7 nm.

15. The concentration of phenol in a water sample is determined by using steam distillation to separate the phenol from non-volatile impurities, followed by reacting the phenol in the distillate with 4-aminoantipyrine and K₃Fe(CN)₆ at pH 7.9 to form a colored antipyrine dye. A phenol standard with a concentration of 4.00 ppm has an absorbance of 0.424 at a wavelength of 460 nm using a 1.00 cm cell. A water sample is steam distilled and a 50.00-mL aliquot of the distillate is placed in a 100-mL volumetric flask and diluted to volume with distilled water. The absorbance of this solution is 0.394. What is the concentration of phenol (in parts per million) in the water sample?

16. Saito describes a quantitative spectrophotometric procedure for iron based on a solid-phase extraction using bathophenanthroline in a poly(vinyl chloride) membrane [Saito, T. Anal. Chim. Acta 1992, 268, 351–355]. In the absence of Fe²⁺ the membrane is colorless, but when immersed in a solution of Fe²⁺ and I^–, the membrane develops a red color as a result of the formation of an Fe²⁺–bathophenanthroline complex. A calibration curve determined using a set of external standards with known concentrations of Fe²⁺ gave a standardization relationship of

\[A=\left(8.60 \times 10^{3} \ \mathrm{M}^{-1}\right) \times\left[\mathrm{Fe}^{2+}\right] \nonumber\]

What is the concentration of iron, in mg Fe/L, for a sample with an absorbance of 0.100?

17. In the DPD colorimetric method for the free chlorine residual, which is reported as mg Cl₂/L, the oxidizing power of free chlorine converts the colorless amine N,N-diethyl-p-phenylenediamine to a colored dye that absorbs strongly over the wavelength range of 440–580 nm. Analysis of a set of calibration standards gave the following results.

mg Cl2/L	absorbance
0.00	0.000
0.50	0.270
1.00	0.543
1.50	0.813
2.00	1.084

A sample from a public water supply is analyzed to determine the free chlorine residual, giving an absorbance of 0.113. What is the free chlorine residual for the sample in mg Cl₂/L?

18. Lin and Brown described a quantitative method for methanol based on its effect on the visible spectrum of methylene blue [Lin, J.; Brown, C. W. Spectroscopy 1995, 10(5), 48–51]. In the absence of methanol, methylene blue has two prominent absorption bands at 610 nm and 663 nm, which correspond to the monomer and the dimer, respectively. In the presence of methanol, the intensity of the dimer’s absorption band decreases, while that for the monomer increases. For concentrations of methanol between 0 and 30% v/v, the ratio of the two absorbance, A₆₆₃/A₆₁₀, is a linear function of the amount of methanol. Use the following standardization data to determine the %v/v methanol in a sample if A₆₁₀ is 0.75 and A₆₆₃ is 1.07.

%v/v methanol	A663/A610	%v/v methanol	A663/A610
0.0	1.21	20.0	1.62
5.0	1.29	25.0	1.74
10.0	1.42	30.0	1.84
15.0	1.52

19. The concentration of the barbiturate barbital in a blood sample is determined by extracting 3.00 mL of blood with 15 mL of CHCl₃. The chloroform, which now contains the barbital, is extracted with 10.0 mL of 0.45 M NaOH (pH ≈ 13). A 3.00-mL sample of the aqueous extract is placed in a 1.00-cm cell and an absorbance of 0.115 is measured. The pH of the sample in the absorption cell is then adjusted to approximately 10 by adding 0.50 mL of 16% w/v NH₄Cl, giving an absorbance of 0.023. When 3.00 mL of a standard barbital solution with a concentration of 3 mg/100 mL is taken through the same procedure, the absorbance at pH 13 is 0.295 and the absorbance at a pH of 10 is 0.002. Report the mg barbital/100 mL in the sample.

20. Jones and Thatcher developed a spectrophotometric method for analyzing analgesic tablets that contain aspirin, phenacetin, and caffeine [Jones, M.; Thatcher, R. L. Anal. Chem. 1951, 23, 957–960]. The sample is dissolved in CHCl₃ and extracted with an aqueous solution of NaHCO₃ to remove the aspirin. After the extraction is complete, the chloroform is transferred to a 250-mL volumetric flask and diluted to volume with CHCl₃. A 2.00-mL portion of this solution is then diluted to volume in a 200-mL volumetric flask with CHCl₃. The absorbance of the final solution is measured at wavelengths of 250 nm and 275 nm, at which the absorptivities, in ppm^–1 cm^–1, for caffeine and phenacetin are

analyte	\(\varepsilon_{250}\)	\(\varepsilon_{275}\)
caffeine	0.0131	0.0485
phenacetin	0.0702	0.0159

Aspirin is determined by neutralizing the NaHCO₃ in the aqueous solution and extracting the aspirin into CHCl₃. The combined extracts are diluted to 500 mL in a volumetric flask. A 20.00-mL portion of the solution is placed in a 100-mL volumetric flask and diluted to volume with CHCl₃. The absorbance of this solution is measured at 277 nm, where the absorptivity of aspirin is 0.00682 ppm^–1 cm^–1. An analgesic tablet treated by this procedure is found to have absorbances of 0.466 at 250 nm, 0.164 at 275 nm, and 0.600 at 277 nm when using a cell with a 1.00 cm pathlength. Report the milligrams of aspirin, caffeine, and phenacetin in the analgesic tablet.

21. The concentration of SO₂ in a sample of air is determined by the p-rosaniline method. The SO₂ is collected in a 10.00-mL solution of \(\text{HgCl}_4^{2-}\), where it reacts to form \(\text{Hg(SO}_3 )_2\), by pulling air through the solution for 75 min at a rate of 1.6 L/min. After adding p-rosaniline and formaldehyde, the colored solution is diluted to 25 mL in a volumetric flask. The absorbance is measured at 569 nm in a 1-cm cell, yielding a value of 0.485. A standard sample is prepared by substituting a 1.00-mL sample of a standard solution that contains the equivalent of 15.00 ppm SO₂ for the air sample. The absorbance of the standard is found to be 0.181. Report the concentration of SO₂ in the air in mg SO₂/L. The density of air is 1.18 g/liter.

22. Seaholtz and colleagues described a method for the quantitative analysis of CO in automobile exhaust based on the measurement of infrared radiation at 2170 cm^–1 [Seaholtz, M. B.; Pence, L. E.; Moe, O. A. Jr. J. Chem. Educ. 1988, 65, 820–823]. A calibration curve is prepared by filling a 10-cm IR gas cell with a known pressure of CO and measuring the absorbance using an FT-IR, giving a calibration equation of

\[A=-1.1 \times 10^{-4}+\left(9.9 \times 10^{-4}\right) \times P_{\mathrm{CO}} \nonumber\]

Samples are prepared by using a vacuum manifold to fill the gas cell. After measuring the total pressure, the absorbance at 2170 cm^–1 is measured. Results are reported as %CO (P_CO/P_total). The analysis of five exhaust samples from a 1973 coupe gives the following results.

Ptotal (torr)	absorbance
595	0.1146
354	0.0642
332	0.0591
233	0.0412
143	0.0254

Determine the %CO for each sample, and report the mean and the 95% confidence interval.

23. Figure 10.3.8 shows an example of a disposable IR sample card made using a thin sheet of polyethylene. To prepare an analyte for analysis, it is dissolved in a suitable solvent and a portion of the sample placed on the IR card. After the solvent evaporates, leaving the analyte behind as a thin film, the sample’s IR spectrum is obtained. Because the thickness of the polyethylene film is not uniform, the primary application of IR cards is for a qualitative analysis. Zhao and Malinowski reported how an internal standardization with KSCN can be used for a quantitative IR analysis of polystyrene [Zhao, Z.; Malinowski, E. R. Spectroscopy 1996, 11(7), 44–49]. Polystyrene is monitored at 1494 cm^–1 and KSCN at 2064 cm^–1. Standard solutions are prepared by placing weighed portions of polystyrene in a 10-mL volumetric flask and diluting to volume with a solution of 10 g/L KSCN in methyl isobutyl ketone. A typical set of results is shown here.

g polystyrene	0.1609	0.3290	0.4842	0.6402	0.8006
A1494	0.0452	0.1138	0.1820	0.3275	0.3195
A2064	0.1948	0.2274	0.2525	0.3580	0.2703

When a 0.8006-g sample of a poly(styrene/maleic anhydride) copolymer is analyzed, the following results are obtained.

replicate	A1494	A2064
1	0.2729	0.3582
2	0.2074	0.2820
3	0.2785	0.3642

What is the %w/w polystyrene in the copolymer? Given that the reported %w/w polystyrene is 67%, is there any evidence for a determinate error at \(\alpha\) = 0.05?

24. The following table lists molar absorptivities for the Arsenazo complexes of copper and barium [Grossman, O.; Turanov, A. N. Anal. Chim. Acta 1992, 257, 195–202]. Suggest appropriate wavelengths for analyzing mixtures of copper and barium using their Arsenzao complexes.

wavelength (nm)	\(\varepsilon_\text{Cu}\) (M^–1 cm^–1)	\(\varepsilon_\text{Ba}\) (M^–1 cm^–1)
595	11900	7100
600	15500	7200
607	18300	7400
611	19300	6900
614	19300	7000
620	17800	7100
626	16300	8400
635	10900	9900
641	7500	10500
645	5300	10000
650	3500	8600
655	2200	6600
658	1900	6500
665	1500	3900
670	1500	2800
680	1800	1500

25. Blanco and colleagues report several applications of multiwavelength linear regression analysis for the simultaneous determination of two-component mixtures [Blanco, M.; Iturriaga, H.; Maspoch, S.; Tarin, P. J. Chem. Educ. 1989, 66, 178–180]. For each of the following, determine the molar concentration of each analyte in the mixture.

(a) Titanium and vanadium are determined by forming complexes with H₂O₂. Results for a mixture of Ti(IV) and V(V) and for stan-dards of 63.1 ppm Ti(IV) and 96.4 ppm V(V) are listed in the following table.

wavelength (nm)	ATi(V) Std	AV(V) Std	Amix
390	0.895	0.326	0.651
430	0.884	0.497	0.743
450	0.694	0.528	0.665
470	0.481	0.512	0.547
510	0.173	0.374	0.314

(b) Copper and zinc are determined by forming colored complexes with 2-pyridyl-azo-resorcinol (PAR). The absorbances for PAR, a mixture of Cu²⁺ and Zn²⁺, and standards of 1.00 ppm Cu²⁺ and 1.00 ppm Zn²⁺ are listed in the following table. Note that you must correct the absorbances for the each metal for the contribution from PAR.

wavelength (nm)	APAR	ACu Std	AZn Std	Amix
480	0.211	0.698	0.971	0.656
496	0.137	0.732	1.018	0.668
510	0.100	0.732	0.891	0.627
526	0.072	0.602	0.672	0.498
540	0.056	0.387	0.306	0.290

26. The stoichiometry of a metal–ligand complex, ML_n, is determined by the method of continuous variations. A series of solutions is prepared in which the combined concentrations of M and L are held constant at \(5.15 \times 10^{-4}\) M. The absorbances of these solutions are measured at a wavelength where only the metal–ligand complex absorbs. Using the following data, determine the formula of the metal–ligand complex.

mole fraction of M	mole fraction of L	absorbance
1.0	0.0	0.001
0.9	0.1	0.126
0.8	0.2	0.260
0.7	0.3	0.389
0.6	0.4	0.515
0.5	0.5	0.642
0.4	0.6	0.775
0.3	0.7	0.771
0.2	0.8	0.513
0.1	0.9	0.253
0.0	1.0	0.000

27. The stoichiometry of a metal–ligand complex, ML_n, is determined by the mole-ratio method. A series of solutions are prepared in which the metal’s concentration is held constant at \(3.65 \times 10^{-4}\) M and the ligand’s concentration is varied from \(1 \times 10^{-4}\) M to \(1 \times 10^{-3}\) M. Using the following data, determine the stoichiometry of the metal-ligand complex.

[ligand] (M)	absorbance	[ligand] (M)	absorbance
\(1.0 \times 10^{-4}\)	0.122	\(6.0 \times 10^{-4}\)	0.752
\(2.0 \times 10^{-4}\)	0.251	\(7.0 \times 10^{-4}\)	0.873
\(3.0 \times 10^{-4}\)	0.376	\(8.0 \times 10^{-4}\)	0.937
\(4.0 \times 10^{-4}\)	0.496	\(9.0 \times 10^{-4}\)	0.962
\(5.0 \times 10^{-4}\)	0.625	\(1.0 \times 10^{-3}\)	1.002

28. The stoichiometry of a metal–ligand complex, ML_n, is determined by the slope-ratio method. Two sets of solutions are prepared. For the first set of solutions the metal’s concentration is held constant at 0.010 M and the ligand’s concentration is varied. The following data are obtained at a wavelength where only the metal–ligand complex absorbs.

[ligand] (M)	absorbance
\(1.0 \times 10^{-5}\)	0.012
\(2.0 \times 10^{-5}\)	0.029
\(3.0 \times 10^{-5}\)	0.042
\(4.0 \times 10^{-5}\)	0.055
\(5.0 \times 10^{-5}\)	0.069

For the second set of solutions the concentration of the ligand is held constant at 0.010 M, and the concentration of the metal is varied, yielding the following absorbances.

[metal] (M)	absorbance
\(1.0 \times 10^{-5}\)	0.040
\(2.0 \times 10^{-5}\)	0.085
\(3.0 \times 10^{-5}\)	0.125
\(4.0 \times 10^{-5}\)	0.162
\(5.0 \times 10^{-5}\)	0.206

Using this data, determine the stoichiometry of the metal-ligand complex.

29. Kawakami and Igarashi developed a spectrophotometric method for nitrite based on its reaction with 5, 10, 15, 20-tetrakis(4-aminophenyl) porphrine (TAPP). As part of their study they investigated the stoichiometry of the reaction between TAPP and \(\text{NO}_2^-\). The following data are derived from a figure in their paper [Kawakami, T.; Igarashi, S. Anal. Chim. Acta 1996, 333, 175–180].

[TAPP] (M)	[\(\text{NO}_2^-\)] (M)	absorbance
\(8.0 \times 10^{-7}\)	0	0.227
\(8.0 \times 10^{-7}\)	\(4.0 \times 10^{-8}\)	0.223
\(8.0 \times 10^{-7}\)	\(8.0 \times 10^{-8}\)	0.211
\(8.0 \times 10^{-7}\)	\(1.6 \times 10^{-7}\)	0.191
\(8.0 \times 10^{-7}\)	\(3.2 \times 10^{-7}\)	0.152
\(8.0 \times 10^{-7}\)	\(4.8 \times 10^{-7}\)	0.127
\(8.0 \times 10^{-7}\)	\(6.4 \times 10^{-7}\)	0.107
\(8.0 \times 10^{-7}\)	\(8.0 \times 10^{-7}\)	0.092
\(8.0 \times 10^{-7}\)	\(1.6 \times 10^{-6}\)	0.058
\(8.0 \times 10^{-7}\)	\(2.4 \times 10^{-6}\)	0.045
\(8.0 \times 10^{-7}\)	\(3.2 \times 10^{-6}\)	0.037
\(8.0 \times 10^{-7}\)	\(4.0 \times 10^{-6}\)	0.034

What is the stoichiometry of the reaction?

30. The equilibrium constant for an acid–base indicator is determined by preparing three solutions, each of which has a total indicator concentration of \(1.35 \times 10^{-5}\) M. The pH of the first solution is adjusted until it is acidic enough to ensure that only the acid form of the indicator is present, yielding an absorbance of 0.673. The absorbance of the second solution, whose pH is adjusted to give only the base form of the indicator, is 0.118. The pH of the third solution is adjusted to 4.17 and has an absorbance of 0.439. What is the acidity constant for the acid–base indicator?

31. The acidity constant for an organic weak acid is determined by measuring its absorbance as a function of pH while maintaining a constant total concentration of the acid. Using the data in the following table, determine the acidity constant for the organic weak acid.

pH	absorbance	pH	absorbance
1.53	0.010	4.88	0.193
2.20	0.010	5.09	0.227
3.66	0.035	5.69	0.288
4.11	0.072	7.20	0.317
4.35	0.103	7.78	0.317
4.75	0.169

32. Suppose you need to prepare a set of calibration standards for the spectrophotometric analysis of an analyte that has a molar absorptivity of 1138 M^–1 cm^–1 at a wavelength of 625 nm. To maintain an acceptable precision for the analysis, the %T for the standards should be between 15% and 85%.

(a) What is the concentration for the most concentrated and for the least concentrated standard you should prepare, assuming a 1.00-cm sample cell.

(b) Explain how you will analyze samples with concentrations that are 10 μM, 0.1 mM, and 1.0 mM in the analyte.

33. When using a spectrophotometer whose precision is limited by the uncertainty of reading %T, the analysis of highly absorbing solutions can lead to an unacceptable level of indeterminate errors. Consider the analysis of a sample for which the molar absorptivity is \(1.0 \times 10^4\) M^–1 cm^–1 and for which the pathlength is 1.00 cm.

(a) What is the relative uncertainty in concentration for an analyte whose concentration is \(2.0 \times 10^{-4}\) M if s_T is ±0.002?

(b) What is the relative uncertainty in the concentration if the spectrophotometer is calibrated using a blank that consists of a \(1.0 \times 10^{-4}\) M solution of the analyte?

34. Hobbins reported the following calibration data for the flame atomic absorption analysis for phosphorous [Hobbins, W. B. “Direct Determination of Phosphorous in Aqueous Matricies by Atomic Absorption,” Varian Instruments at Work, Number AA-19, February 1982].

mg P/L	absorbance
2130	0.048
4260	0.110
6400	0.173
8530	0.230

To determine the purity of a sample of Na₂HPO₄, a 2.469-g sample is dissolved and diluted to volume in a 100-mL volumetric flask. Analysis of the resulting solution gives an absorbance of 0.135. What is the purity of the Na₂HPO₄?

35. Bonert and Pohl reported results for the atomic absorption analysis of several metals in the caustic suspensions produced during the manufacture of soda by the ammonia-soda process [Bonert, K.; Pohl, B. “The Determination of Cd, Cr, Cu, Ni, and Pb in Concentrated CaCl₂/NaCl solutions by AAS,” AA Instruments at Work (Varian) Number 98, November, 1990].

(a) The concentration of Cu is determined by acidifying a 200.0-mL sample of the caustic solution with 20 mL of concentrated HNO₃, adding 1 mL of 27% w/v H₂O₂, and boiling for 30 min. The resulting solution is diluted to 500 mL in a volumetric flask, filtered, and analyzed by flame atomic absorption using matrix matched standards. The results for a typical analysis are shown in the following table.

solution	mg Cu/L	absorbance
blank	0.000	0.007
standard 1	0.200	0.014
standard 2	0.500	0.036
standard 3	1.000	0.072
standard 4	2.000	0.146
sample		0.027

Determine the concentration of Cu in the caustic suspension.

(b) The determination of Cr is accomplished by acidifying a 200.0-mL sample of the caustic solution with 20 mL of concentrated HNO₃, adding 0.2 g of Na₂SO₃ and boiling for 30 min. The Cr is isolated from the sample by adding 20 mL of NH₃, producing a precipitate that includes the chromium as well as other oxides. The precipitate is isolated by filtration, washed, and transferred to a beaker. After acidifying with 10 mL of HNO₃, the solution is evaporated to dryness. The residue is redissolved in a combination of HNO₃ and HCl and evaporated to dryness. Finally, the residue is dissolved in 5 mL of HCl, filtered, diluted to volume in a 50-mL volumetric flask, and analyzed by atomic absorption using the method of standard additions. The atomic absorption results are summarized in the following table.

sample	mg Cradded/L	absorbance
blank		0.001
sample		0.045
standard addition 1	0.200	0.083
standard addition 2	0.500	0.118
standard addition 3	1.000	0.192

Report the concentration of Cr in the caustic suspension.

36. Quigley and Vernon report results for the determination of trace metals in seawater using a graphite furnace atomic absorption spectrophotometer and the method of standard additions [Quigley, M. N.; Vernon, F. J. Chem. Educ. 1996, 73, 671–673]. The trace metals are first separated from their complex, high-salt matrix by coprecipitating with Fe³⁺. In a typical analysis a 5.00-mL portion of 2000 ppm Fe³⁺ is added to 1.00 L of seawater. The pH is adjusted to 9 using NH₄OH, and the precipitate of Fe(OH)₃ allowed to stand overnight. After isolating and rinsing the precipitate, the Fe(OH)₃ and coprecipitated metals are dissolved in 2 mL of concentrated HNO₃ and diluted to volume in a 50-mL volumetric flask. To analyze for Mn²⁺, a 1.00-mL sample of this solution is diluted to 100 mL in a volumetric flask. The following samples are injected into the graphite furnace and analyzed.

sample	absorbance
2.5-µL sample + 2.5 µL of 0 ppb Mn^2^+	0.223
2.5-µL sample + 2.5 µL of 2.5 ppb Mn^2^+	0.294
2.5-µL sample + 2.5 µL of 5.0 ppb Mn^2^+	0.361

Report the ppb Mn²⁺ in the sample of seawater.

37. The concentration of Na in plant materials are determined by flame atomic emission. The material to be analyzed is prepared by grinding, homogenizing, and drying at 103^oC. A sample of approximately 4 g is transferred to a quartz crucible and heated on a hot plate to char the organic material. The sample is heated in a muffle furnace at 550^oC for several hours. After cooling to room temperature the residue is dissolved by adding 2 mL of 1:1 HNO₃ and evaporated to dryness. The residue is redissolved in 10 mL of 1:9 HNO₃, filtered and diluted to 50 mL in a volumetric flask. The following data are obtained during a typical analysis for the concentration of Na in a 4.0264-g sample of oat bran.

sample	mg Na/L	emission (arbitrary units)
blank	0.00	0.0
standard 1	2.00	90.3
standard 2	4.00	181
standard 3	6.00	272
standard 4	8.00	363
standard 5	10.00	448
sample		238

Report the concentration of μg Na/g sample.

38. Yan and colleagues developed a method for the analysis of iron based its formation of a fluorescent metal–ligand complex with the ligand 5-(4-methylphenylazo)-8-aminoquinoline [Yan, G.; Shi, G.; Liu, Y. Anal. Chim. Acta 1992, 264, 121–124]. In the presence of the surfactant cetyltrimethyl ammonium bromide the analysis is carried out using an excitation wavelength of 316 nm with emission monitored at 528 nm. Standardization with external standards gives the following calibration curve.

\[I_{f}=-0.03+\left(1.594 \ \mathrm{mg}^{-1} \ \mathrm{L}\right) \times \frac{\mathrm{mg} \ \mathrm{Fe}^{3+}}{\mathrm{L}} \nonumber\]

A 0.5113-g sample of dry dog food is ashed to remove organic materials, and the residue dissolved in a small amount of HCl and diluted to volume in a 50-mL volumetric flask. Analysis of the resulting solution gives a fluorescent emission intensity of 5.72. Determine the mg Fe/L in the sample of dog food.

39. A solution of \(5.00 \times 10^{-5}\) M 1,3-dihydroxynaphthelene in 2 M NaOH has a fluorescence intensity of 4.85 at a wavelength of 459 nm. What is the concentration of 1,3-dihydroxynaphthelene in a solution that has a fluorescence intensity of 3.74 under identical conditions?

40. The following data is recorded for the phosphorescent intensity of several standard solutions of benzo[a]pyrene.

[benzo[a]pyrene] (M)	emission intensity
0	0.00
\(1.00 \times 10^{-5}\)	0.98
\(3.00 \times 10^{-5}\)	3.22
\(6.00 \times 10^{-5}\)	6.25
\(1.00 \times 10^{-4}\)	10.21

What is the concentration of benzo[a]pyrene in a sample that yields a phosphorescent emission intensity of 4.97?

41. The concentration of acetylsalicylic acid, C₉H₈O₄, in aspirin tablets is determined by hydrolyzing it to the salicylate ion, \(\text{C}_7 \text{H}_5 \text{O}_2^-\), and determining its concentration spectrofluorometrically. A stock standard solution is prepared by weighing 0.0774 g of salicylic acid, C₇H₆O₂, into a 1-L volumetric flask and diluting to volume. A set of calibration standards is prepared by pipeting 0, 2.00, 4.00, 6.00, 8.00, and 10.00 mL of the stock solution into separate 100-mL volumetric flasks that contain 2.00 mL of 4 M NaOH and diluting to volume. Fluorescence is measured at an emission wavelength of 400 nm using an excitation wavelength of 310 nm with results shown in the following table.

mL of stock solution	emission intensity
0.00	0.00
2.00	3.02
4.00	5.98
6.00	9.18
8.00	12.13
10.00	14.96

Several aspirin tablets are ground to a fine powder in a mortar and pestle. A 0.1013-g portion of the powder is placed in a 1-L volumetric flask and diluted to volume with distilled water. A portion of this solution is filtered to remove insoluble binders and a 10.00-mL aliquot transferred to a 100-mL volumetric flask that contains 2.00 mL of 4 M NaOH. After diluting to volume the fluorescence of the resulting solution is 8.69. What is the %w/w acetylsalicylic acid in the aspirin tablets?

42. Selenium (IV) in natural waters is determined by complexing with ammonium pyrrolidine dithiocarbamate and extracting into CHCl₃. This step serves to concentrate the Se(IV) and to separate it from Se(VI). The Se(IV) is then extracted back into an aqueous matrix using HNO₃. After complexing with 2,3-diaminonaphthalene, the complex is extracted into cyclohexane. Fluorescence is measured at 520 nm following its excitation at 380 nm. Calibration is achieved by adding known amounts of Se(IV) to the water sample before beginning the analysis. Given the following results what is the concentration of Se(IV) in the sample.

[Se(IV)] added (nM)	emission intensity
0.00	323
2.00	597
4.00	862
6.00	1123

43. Fibrinogen is a protein that is produced by the liver and found in human plasma. Its concentration in plasma is clinically important. Many of the analytical methods used to determine the concentration of fibrinogen in plasma are based on light scattering following its precipitation. For example, da Silva and colleagues describe a method in which fibrino- gen precipitates in the presence of ammonium sulfate in a guanidine hydrochloride buffer [da Silva, M. P.; Fernandez-Romero, J. M.; Luque de Castro, M. D. Anal. Chim. Acta 1996, 327, 101–106]. Light scattering is measured nephelometrically at a wavelength of 340 nm. Analysis of a set of external calibration standards gives the following calibration equation

\[I_{\mathrm{s}}=-4.66+9907.63 C \nonumber\]

where I_s is the intensity of scattered light and C is the concentration of fibrinogen in g/L. A 9.00-mL sample of plasma is collected from a patient and mixed with 1.00 mL of an anticoagulating agent. A 1.00-mL aliquot of this solution is diluted to 250 mL in a volumetric flask and is found to have a scattering intensity of 44.70. What is the concentration of fibrinogen, in gram per liter, in the plasma sample?