14.7: Problems

1. The following data were obtained for four compounds separated on a 20-m capillary column.

compound	tr (min)	w (min)
A	8.04	0.15
B	8.26	0.15
C	8.43	0.16

(a) Calculate the number of theoretical plates for each compound and the average number of theoretical plates for the column, in mm.

(b) Calculate the average height of a theoretical plate.

(c) Explain why it is possible for each compound to have a different number of theoretical plates.

2. Using the data from Problem 1, calculate the resolution and the selectivity factors for each pair of adjacent compounds. For resolution, use both equation 12.2.1 and equation 12.3.3, and compare your results. Discuss how you might improve the resolution between compounds B and C. The retention time for an nonretained solute is 1.19 min.

3. Use the chromatogram in Figure 12.8.1 , obtained using a 2-m column, to determine values for t_r, w, \(t_r^{\prime}\), k, N, and H.

Figure 12.8.1 . Chromatogram for Problem 3.

4. Use the partial chromatogram in Figure 12.8.2 to determine the resolution between the two solute bands.

Figure 12.8.2 . Chromatogram for Problem 4.

5. The chromatogram in Problem 4 was obtained on a 2-m column with a column dead time of 50 s. Suppose you want to increase the resolution between the two components to 1.5. Without changing the height of a theoretical plate, what length column do you need? What height of a theoretical plate do you need to achieve a resolution of 1.5 without increasing the column’s length?

6. Complete the following table.

NB	\(\alpha\)	kB	R
100000	1.05	0.50
10000	1.10		1.50
10000		4.0	1.00
	1.05	3.0	1.75

7. Moody studied the efficiency of a GC separation of 2-butanone on a dinonyl phthalate packed column [Moody, H. W. J. Chem. Educ. 1982, 59, 218–219]. Evaluating plate height as a function of flow rate gave a van Deemter equation for which A is 1.65 mm, B is 25.8 mm•mL min^–1, and C is 0.0236 mm•min mL^–1.

(a) Prepare a graph of H versus u for flow rates between 5 –120 mL/min.

(b) For what range of flow rates does each term in the Van Deemter equation have the greatest effect?

(c) What is the optimum flow rate and the corresponding height of a theoretical plate?

(d) For open-tubular columns the A term no longer is needed. If the B and C terms remain unchanged, what is the optimum flow rate and the corresponding height of a theoretical plate?

(e) Compared to the packed column, how many more theoretical plates are in the open-tubular column?

8. Hsieh and Jorgenson prepared 12–33 μm inner diameter HPLC columns packed with 5.44-μm spherical stationary phase particles [Hsieh, S.; Jorgenson, J. W. Anal. Chem. 1996, 68, 1212–1217]. To evaluate these columns they measured reduced plate height, h, as a function of reduced flow rate, v,

\[b=\frac{H}{d_{p}} \quad v=\frac{u d_{p}}{D_{m}} \nonumber\]

where d_p is the particle diameter and D_m is the solute’s diffusion coefficient in the mobile phase. The data were analyzed using van Deemter plots. The following table contains a portion of their results for norepinephrine.

internal diameter (µm)	A	B	C
33	0.63	1.32	0.10
33	0.67	1.30	0.08
23	0.40	1.34	0.09
23	0.58	1.11	0.09
17	0.31	1.47	0.11
17	0.40	1.41	0.11
12	0.22	1.53	0.11
12	0.19	1.27	0.12

(a) Construct separate van Deemter plots using the data in the first row and in the last row for reduced flow rates in the range 0.7–15. Determine the optimum flow rate and plate height for each case given d_p = 5.44 μm and D_m = \(6.23 \times 10^{-6}\) cm² s^–1.

(b) The A term in the van Deemter equation is strongly correlated with the column’s inner diameter, with smaller diameter columns providing smaller values of A. Offer an explanation for this observation. Hint: consider how many particles can fit across a capillary of each diameter.

When comparing columns, chromatographers often use dimensionless, reduced parameters. By including particle size and the solute’s diffusion coefficient, the reduced plate height and reduced flow rate correct for differences between the packing material, the solute, and the mobile phase.

9. A mixture of n-heptane, tetrahydrofuran, 2-butanone, and n-propanol elutes in this order when using a polar stationary phase such as Carbowax. The elution order is exactly the opposite when using a nonpolar stationary phase such as polydimethyl siloxane. Explain the order of elution in each case.

10. The analysis of trihalomethanes in drinking water is described in Representative Method 12.4.1. A single standard that contains all four trihalomethanes gives the following results.

compound	concentration (ppb)	peak area
CHCl3	1.30	\(1.35 \times 10^4\)
CHCl2Br	0.90	\(6.12 \times 10^4\)
CHClBr2	4.00	\(1.71 \times 10^4\)
CHBr3	1.20	\(1.52 \times 10^4\)

Analysis of water collected from a drinking fountain gives areas of \(1.56 \times 10^4\), \(5.13 \times10^4\), \(1.49 \times 10^4\), and \(1.76 \times 10^4\) for, respectively, CHCl₃, CHCl₂Br, CHClBr₂, and CHBr₃. All peak areas were corrected for variations in injection volumes using an internal standard of 1,2-dibromopentane. Determine the concentration of each of the trihalomethanes in the sample of water.

11. Zhou and colleagues determined the %w/w H₂O in methanol by capillary column GC using a polar stationary phase and a thermal conductivity detector [Zhou, X.; Hines, P. A.; White, K. C.; Borer, M. W. Anal. Chem. 1998, 70, 390–394]. A series of calibration standards gave the following results.

%w/w H2O	peak height (arb. units)
0.00	1.15
0.0145	2.74
0.0472	6.33
0.0951	11.58
0.1757	20.43
0.2901	32.97

(a) What is the %w/w H₂O in a sample that has a peak height of 8.63?

(b) The %w/w H₂O in a freeze-dried antibiotic is determined in the following manner. A 0.175-g sample is placed in a vial along with 4.489 g of methanol. Water in the vial extracts into the methanol. Analysis of the sample gave a peak height of 13.66. What is the %w/w H₂O in the antibiotic?

12. Loconto and co-workers describe a method for determining trace levels of water in soil [Loconto, P. R.; Pan, Y. L.; Voice, T. C. LC•GC 1996, 14, 128–132]. The method takes advantage of the reaction of water with calcium carbide, CaC₂, to produce acetylene gas, C₂H₂. By carrying out the reaction in a sealed vial, the amount of acetylene produced is determined by sampling the headspace. In a typical analysis a sample of soil is placed in a sealed vial with CaC₂. Analysis of the headspace gives a blank corrected signal of \(2.70 \times 10^5\). A second sample is prepared in the same manner except that a standard addition of 5.0 mg H₂O/g soil is added, giving a blank-corrected signal of \(1.06 \times 10^6\). Determine the milligrams H₂O/g soil in the soil sample.

13. Van Atta and Van Atta used gas chromatography to determine the %v/v methyl salicylate in rubbing alcohol [Van Atta, R. E.; Van Atta, R. L. J. Chem. Educ. 1980, 57, 230–231]. A set of standard additions was prepared by transferring 20.00 mL of rubbing alcohol to separate 25-mL volumetric flasks and pipeting 0.00 mL, 0.20 mL, and 0.50 mL of methyl salicylate to the flasks. All three flasks were diluted to volume using isopropanol. Analysis of the three samples gave peak heights for methyl salicylate of 57.00 mm, 88.5 mm, and 132.5 mm, respectively. Determine the %v/v methyl salicylate in the rubbing alcohol.

14. The amount of camphor in an analgesic ointment is determined by GC using the method of internal standards [Pant, S. K.; Gupta, P. N.; Thomas, K. M.; Maitin, B. K.; Jain, C. L. LC•GC 1990, 8, 322–325]. A standard sample is prepared by placing 45.2 mg of camphor and 2.00 mL of a 6.00 mg/mL internal standard solution of terpene hydrate in a 25-mL volumetric flask and diluting to volume with CCl₄. When an approximately 2-μL sample of the standard is injected, the FID signals for the two components are measured (in arbitrary units) as 67.3 for camphor and 19.8 for terpene hydrate. A 53.6-mg sample of an analgesic ointment is prepared for analysis by placing it in a 50-mL Erlenmeyer flask along with 10 mL of CCl₄. After heating to 50^oC in a water bath, the sample is cooled to below room temperature and filtered. The residue is washed with two 5-mL portions of CCl₄ and the combined filtrates are collected in a 25-mL volumetric flask. After adding 2.00 mL of the internal standard solution, the contents of the flask are diluted to volume with CCl₄. Analysis of an approximately 2-μL sample gives FID signals of 13.5 for the terpene hydrate and 24.9 for the camphor. Report the %w/w camphor in the analgesic ointment.

15. The concentration of pesticide residues on agricultural products, such as oranges, is determined by GC-MS [Feigel, C. Varian GC/MS Application Note, Number 52]. Pesticide residues are extracted from the sample using methylene chloride and concentrated by evaporating the methylene chloride to a smaller volume. Calibration is accomplished using anthracene-d₁₀ as an internal standard. In a study to determine the parts per billion heptachlor epoxide on oranges, a 50.0-g sample of orange rinds is chopped and extracted with 50.00 mL of methylene chloride. After removing any insoluble material by filtration, the methylene chloride is reduced in volume, spiked with a known amount of the internal standard and diluted to 10 mL in a volumetric flask. Analysis of the sample gives a peak–area ratio (A_analyte/A_intstd) of 0.108. A series of calibration standards, each containing the same amount of anthracene-d₁₀ as the sample, gives the following results.

ppb heptachlor epoxide	Aanalyte/Aintstd
20.0	0.065
60.0	0.153
200.0	0.637
500.0	1.554
1000.0	3.198

Report the nanograms per gram of heptachlor epoxide residue on the oranges.

16. The adjusted retention times for octane, toluene, and nonane on a particular GC column are 15.98 min, 17.73 min, and 20.42 min, respectively. What is the retention index for each compound?

17. The following data were collected for a series of normal alkanes using a stationary phase of Carbowax 20M.

alkane	\(t_r^{\prime}\) (min)
pentane	0.79
hexane	1.99
heptane	4.47
octane	14.12
nonane	33.11

What is the retention index for a compound whose adjusted retention time is 9.36 min?

18. The following data were reported for the gas chromatographic analysis of p-xylene and methylisobutylketone (MIBK) on a capillary column [Marriott, P. J.; Carpenter, P. D. J. Chem. Educ. 1996, 73, 96–99].

injection mode	compound	tr (min)	peak area (arb. units)	peak width (min)
split	MIBK	1.878	54285	0.028
	p-xylene	5.234	123483	0.044
splitless	MIBK	3.420	2493005	1.057
	p-xylene	5.795	3396656	1.051

Explain the difference in the retention times, the peak areas, and the peak widths when switching from a split injection to a splitless injection.

19. Otto and Wegscheider report the following retention factors for the reversed-phase separation of 2-aminobenzoic acid on a C₁₈ column when using 10% v/v methanol as a mobile phase [Otto, M.; Wegscheider, W. J. Chromatog. 1983, 258, 11–22].

pH	k
2.0	10.5
3.0	16.7
4.0	15.8
5.0	8.0
6.0	2.2
7.0	1.8

Explain the effect of pH on the retention factor for 2-aminobenzene.

20. Haddad and associates report the following retention factors for the reversed-phase separation of salicylamide and caffeine [Haddad, P.; Hutchins, S.; Tuffy, M. J. Chem. Educ. 1983, 60, 166-168].

%v/v methanol	30%	35%	40%	45%	50%	55%
ksal	2.4	1.6	1.6	1.0	0.7	0.7
kcaff	4.3	2.8	2.3	1.4	1.1	0.9

(a) Explain the trends in the retention factors for these compounds.

(b) What is the advantage of using a mobile phase with a smaller %v/v methanol? Are there any disadvantages?

21. Suppose you need to separate a mixture of benzoic acid, aspartame, and caffeine in a diet soda. The following information is available.

	tr in aqueous mobile phase of pH
compound	3.0	3.5	4.0	4.5
benzoic acid	7.4	7.0	6.9	4.4
aspartame	5.9	6.0	7.1	8.1
caffeine	3.6	3.7	4.1	4.4

(a) Explain the change in each compound’s retention time.

(b) Prepare a single graph that shows retention time versus pH for each compound. Using your plot, identify a pH level that will yield an acceptable separation.

22. The composition of a multivitamin tablet is determined using an HPLC with a diode array UV/Vis detector. A 5-μL standard sample that contains 170 ppm vitamin C, 130 ppm niacin, 120 ppm niacinamide, 150 ppm pyridoxine, 60 ppm thiamine, 15 ppm folic acid, and 10 ppm riboflavin is injected into the HPLC, giving signals (in arbitrary units) of, respectively, 0.22, 1.35, 0.90, 1.37, 0.82, 0.36, and 0.29. The multivitamin tablet is prepared for analysis by grinding into a powder and transferring to a 125-mL Erlenmeyer flask that contains 10 mL of 1% v/v NH₃ in dimethyl sulfoxide. After sonicating in an ultrasonic bath for 2 min, 90 mL of 2% acetic acid is added and the mixture is stirred for 1 min and sonicated at 40^oC for 5 min. The extract is then filtered through a 0.45-μm membrane filter. Injection of a 5-μL sample into the HPLC gives signals of 0.87 for vitamin C, 0.00 for niacin, 1.40 for niacinamide, 0.22 for pyridoxine, 0.19 for thiamine, 0.11 for folic acid, and 0.44 for riboflavin. Report the milligrams of each vitamin present in the tablet.

23. The amount of caffeine in an analgesic tablet was determined by HPLC using a normal calibration curve. Standard solutions of caffeine were prepared and analyzed using a 10-μL fixed-volume injection loop. Results for the standards are summarized in the following table.

concentration (ppm)	signal (arb. units)
50.0	8.354
100.0	16925
150.0	25218
200.0	33584
250.0	42002

The sample is prepared by placing a single analgesic tablet in a small beaker and adding 10 mL of methanol. After allowing the sample to dissolve, the contents of the beaker, including the insoluble binder, are quantitatively transferred to a 25-mL volumetric flask and diluted to volume with methanol. The sample is then filtered, and a 1.00-mL aliquot transferred to a 10-mL volumetric flask and diluted to volume with methanol. When analyzed by HPLC, the signal for caffeine is found to be 21 469. Report the milligrams of caffeine in the analgesic tablet.

24. Kagel and Farwell report a reversed-phase HPLC method for determining the concentration of acetylsalicylic acid (ASA) and caffeine (CAF) in analgesic tablets using salicylic acid (SA) as an internal standard [Kagel, R. A.; Farwell, S. O. J. Chem. Educ. 1983, 60, 163–166]. A series of standards was prepared by adding known amounts of ace- tylsalicylic acid and caffeine to 250-mL Erlenmeyer flasks and adding 100 mL of methanol. A 10.00-mL aliquot of a standard solution of salicylic acid was then added to each. The following results were obtained for a typical set of standard solutions.

	milligrams of		peak height ratios for
standard	ASA	CAF	ASA/SA	CAF/SA
1	200.0	20.0	20.5	10.6
2	250.0	40.0	25.1	23.0
3	300.0	60.0	30.9	36.8

A sample of an analgesic tablet was placed in a 250-mL Erlenmeyer flask and dissolved in 100 mL of methanol. After adding a 10.00-mL portion of the internal standard, the solution was filtered. Analysis of the sample gave a peak height ratio of 23.2 for ASA and of 17.9 for CAF.

(a) Determine the milligrams of ASA and CAF in the tablet.

(b) Why is it necessary to filter the sample?

(c) The directions indicate that approximately 100 mL of methanol is used to dissolve the standards and samples. Why is it not necessary to measure this volume more precisely?

(d) In the presence of moisture, ASA decomposes to SA and acetic acid. What complication might this present for this analysis? How might you evaluate whether this is a problem?

25. Bohman and colleagues described a reversed-phase HPLC method for the quantitative analysis of vitamin A in food using the method of standard additions Bohman, O.; Engdahl, K. A.; Johnsson, H. J. Chem. Educ. 1982, 59, 251–252]. In a typical example, a 10.067-g sample of cereal is placed in a 250-mL Erlenmeyer flask along with 1 g of sodium ascorbate, 40 mL of ethanol, and 10 mL of 50% w/v KOH. After refluxing for 30 min, 60 mL of ethanol is added and the solution cooled to room temperature. Vitamin A is extracted using three 100-mL portions of hexane. The combined portions of hexane are evaporated and the residue containing vitamin A transferred to a 5-mL volumetric flask and diluted to volume with methanol. A standard addition is prepared in a similar manner using a 10.093-g sample of the cereal and spiking with 0.0200 mg of vitamin A. Injecting the sample and standard addition into the HPLC gives peak areas of, respectively, \(6.77 \times10^3\) and \(1.32 \times 10^4\). Report the vitamin A content of the sample in milligrams/100 g cereal.

26. Ohta and Tanaka reported on an ion-exchange chromatographic method for the simultaneous analysis of several inorganic anions and the cations Mg²⁺ and Ca²⁺ in water [Ohta, K.; Tanaka, K. Anal. Chim. Acta 1998, 373, 189–195]. The mobile phase includes the ligand 1,2,4-benzenetricarboxylate, which absorbs strongly at 270 nm. Indirect detection of the analytes is possible because its absorbance decreases when complexed with an anion.

(a) The procedure also calls for adding the ligand EDTA to the mobile phase. What role does the EDTA play in this analysis?

(b) A standard solution of 1.0 mM NaHCO₃, 0.20 mM NaNO₂, 0.20 mM MgSO₄, 0.10 mM CaCl₂, and 0.10 mM Ca(NO₃)₂ gives the following peak areas (arbitrary units).

ion	\(\text{HCO}_3^-\)	Cl^–	\(\text{NO}_2^-\)	\(\text{NO}_3^-\)
peak area	373.5	322.5	264.8	262.7
ion	Ca^2^+	Mg^2^+	\(\text{SO}_4^{2-}\)
peak area	458.9	352.0	341.3

Analysis of a river water sample (pH of 7.49) gives the following results.

ion	\(\text{HCO}_3^-\)	Cl^–	\(\text{NO}_2^-\)	\(\text{NO}_3^-\)
peak area	310.0	403.1	3.97	157.6
ion	Ca^2^+	Mg^2^+	\(\text{SO}_4^{2-}\)
peak area	734.3	193.6	324.3

Determine the concentration of each ion in the sample.

(c) The detection of \(\text{HCO}_3^-\) actually gives the total concentration of carbonate in solution ([\(\text{CO}_3^{2-}\)]+[\(\text{HCO}_3^-\)]+[H₂CO₃]). Given that the pH of the water is 7.49, what is the actual concentration of \(\text{HCO}_3^-\)?

(d) An independent analysis gives the following additional concentrations for ions in the sample: [Na⁺] = 0.60 mM; [\(\text{NH}_4^+\)] = 0.014 mM; and [K⁺] = 0.046 mM. A solution’s ion balance is defined as the ratio of the total cation charge to the total anion charge. Determine the charge balance for this sample of water and comment on whether the result is reasonable.

27. The concentrations of Cl^–, \(\text{NO}_2^-\), and \(\text{SO}_4^{2-}\) are determined by ion chromatography. A 50-μL standard sample of 10.0 ppm Cl^–, 2.00 ppm \(\text{NO}_2^-\), and 5.00 ppm \(\text{SO}_4^{2-}\) gave signals (in arbitrary units) of 59.3, 16.1, and 6.08 respectively. A sample of effluent from a wastewater treatment plant is diluted tenfold and a 50-μL portion gives signals of 44.2 for Cl^–, 2.73 for \(\text{NO}_2^-\), and 5.04 for \(\text{SO}_4^{2-}\). Report the parts per million for each anion in the effluent sample.

28. A series of polyvinylpyridine standards of different molecular weight was analyzed by size-exclusion chromatography, yielding the following results.

formula weight	retention volume (mL)
600000	6.42
100000	7.98
30000	9.30
3000	10.94

When a preparation of polyvinylpyridine of unknown formula weight is analyzed, the retention volume is 8.45 mL. Report the average formula weight for the preparation.

29. Diet soft drinks contain appreciable quantities of aspartame, benzoic acid, and caffeine. What is the expected order of elution for these compounds in a capillary zone electrophoresis separation using a pH 9.4 buffer given that aspartame has pK_a values of 2.964 and 7.37, benzoic acid has a pK_a of 4.2, and the pK_a for caffeine is less than 0. Figure 12.8.3 provides the structures of these compounds.

Figure 12.8.3 . Structures for the compounds in Problem 29.

30. Janusa and coworkers describe the determination of chloride by CZE [Janusa, M. A.; Andermann, L. J.; Kliebert, N. M.; Nannie, M. H. J. Chem. Educ. 1998, 75, 1463–1465]. Analysis of a series of external standards gives the following calibration curve.

\[\text { area }=-883+5590 \times \mathrm{ppm} \text{ Cl}^{-} \nonumber\]

A standard sample of 57.22% w/w Cl^– is analyzed by placing 0.1011-g portions in separate 100-mL volumetric flasks and diluting to volume. Three unknowns are prepared by pipeting 0.250 mL, 0.500 mL, an 0.750 mL of the bulk unknown in separate 50-mL volumetric flasks and diluting to volume. Analysis of the three unknowns gives areas of 15 310, 31 546, and 47 582, respectively. Evaluate the accuracy of this analysis.

31. The analysis of \(\text{NO}_3^-\) in aquarium water is carried out by CZE using \(\text{IO}_4^-\) as an internal standard. A standard solution of 15.0 ppm \(\text{NO}_3^-\) and 10.0 ppm \(\text{IO}_4^-\) gives peak heights (arbitrary units) of 95.0 and 100.1, respectively. A sample of water from an aquarium is diluted 1:100 and sufficient internal standard added to make its concentration 10.0 ppm in \(\text{IO}_4^-\). Analysis gives signals of 29.2 and 105.8 for \(\text{NO}_3^-\) and \(\text{IO}_4^-\), respectively. Report the ppm \(\text{NO}_3^-\) in the sample of aquarium water.

32. Suggest conditions to separate a mixture of 2-aminobenzoic acid (pK_a1 = 2.08, pK_a2 = 4.96), benzylamine (pK_a = 9.35), and 4-methylphenol (pK_a2 = 10.26) by capillary zone electrophoresis. Figure \(PageIndex{4}\) provides the structures of these compounds.

Figure 12.8.4 . Structures for the compounds in Problem 32.

33. McKillop and associates examined the electrophoretic separation of some alkylpyridines by CZE [McKillop, A. G.; Smith, R. M.; Rowe, R. C.; Wren, S. A. C. Anal. Chem. 1999, 71, 497–503]. Separations were carried out using either 50-μm or 75-μm inner diameter capillaries, with a total length of 57 cm and a length of 50 cm from the point of injection to the detector. The run buffer was a pH 2.5 lithium phosphate buffer. Separations were achieved using an applied voltage of 15 kV. The electroosmotic mobility, μ_eof, as measured using a neutral marker, was found to be \(6.398 \times 10^{-5}\) cm² V^–1 s^–1. The diffusion coefficient for alkylpyridines is \(1.0 \times 10^{-5}\) cm² s^–1.

(a) Calculate the electrophoretic mobility for 2-ethylpyridine given that its elution time is 8.20 min.

(b) How many theoretical plates are there for 2-ethylpyridine?

(c) The electrophoretic mobilities for 3-ethylpyridine and 4-ethylpyridine are \(3.366 \times 10^{-4}\) cm² V^–1 s^–1 and \(3.397 \times 10^{-4} \text{ cm}^2 \text{ V}^{-1} \text{ s}^{-1}\), respectively. What is the expected resolution between these two alkylpyridines?

(d) Explain the trends in electrophoretic mobility shown in the following table.

alkylpyridine	\(\mu_{ep}\) (cm^2 V^–1 s^–1)
2-methylpyridine	\(3.581 \times 10^{-4}\)
2-ethylpyridine	\(3.222 \times 10^{-4}\)
2-propylpyridine	\(2.923 \times 10^{-4}\)
2-pentylpyridine	\(2.534 \times 10^{-4}\)
2-hexylpyridine	\(2.391 \times 10^{-4}\)

(e) Explain the trends in electrophoretic mobility shown in the following table.

alkylpyridine	\(\mu_{ep}\) (cm^2 V^–1 s^–1)
2-ethylpyridine	\(3.222 \times 10^{-4}\)
3-ethylpyridine	\(3.366 \times 10^{-4}\)
4-ethylpyridine	\(3.397 \times 10^{-4}\)

(f) The pK_a for pyridine is 5.229. At a pH of 2.5 the electrophoretic mobility of pyridine is \(4.176 \times 10^{-4}\) cm² V^–1 s^–1. What is the expected electrophoretic mobility if the run buffer’s pH is 7.5?