In the Classroom
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Sulfuric Acid and Water: Paradoxes of Dilution I. A. Leenson Department of Chemistry, Moscow State University, Moscow, 119899, Russia;
[email protected] The principal goal of this article is to elucidate some intriguing and uncommon phenomena that one finds in the thermochemistry and equilibrium properties of aqueous solutions of sulfuric acid. Every student knows that: The water should be in before You add the H2SO4. Or else you might not feel too well— You might not even live to tell!
Many students also know that the reason lies in the large quantity of heat liberated on mixing of the two liquids. As a result, the thin layer of water being spread on the dense liquid of acid can easily get boiled and sprayed around. However, even professional chemists can be somewhat surprised to learn how many moles of water must be added to one mole of sulfuric acid to complete the evolution of heat. An additional goal of this article is to show that sulfuric acid dissociates only slightly in the second equilibrium and the common belief that in dilute solutions (e.g., 0.1% by weight) it is dissociated completely to H+ and SO42− ions is erroneous. This article also contains interesting information about thermochemical measurements in the 19th century, a discussion concerning the heats of hydration, and a way to obtain unexpected results from properly plotted graphs. The article consists of three parts and Supplemental Material.W Historical and modern experimental data concerning thermochemistry and equilibrium in aqueous solutions of sulfuric acid are presented in the first two parts. Comparison of the historic and modern findings and a brief discussion concerning the composition of the solution at very high dilutions is presented in part 3. Questions and problems for students ranging from simple to more sophisticated are presented in the Supplemental Material.W The answers and possible solutions of the problems are also provided. This article could be useful in general, inorganic, and physical chemistry courses. Heat of Dilution: Historical Accounts The heats of formation of aqueous solutions of sulfuric acid are scientifically significant. For instance, these data are essential in the thermochemistry of both inorganic and organic sulfur compounds (1). In addition, a study of heats of solution can provide interesting insights into the thermodynamics of the process and solute–solvent interaction (2). Thermal phenomena of different magnitudes accompany many processes between solvent and solute. A great quantity of data on the heat effects associated with the solution of H2SO4 in water and with successive dilution of concentrated solutions has been generated from the time of Thomsen and Berthelot to the present, and the methods of measurements have been continuously refined. The accepted values originate from experiments that are relatively old, although accurate (3). Here we examine the total or inwww.JCE.DivCHED.org
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tegral heat of dilution, that is, the difference in enthalpy between a solution and its components. The heats of solution (or dilution), Q, of a mole of sulfuric acid with n moles of water have been measured by many investigators. As far back as 1826 the famous Swedish chemist Jöns Jacob Berzelius and the Russian chemist Herman Henry Hess observed that when sulfuric acid was mixed with one-fourth its weight of water, the temperature rose from 0 ⬚C to 100 ⬚C. Hence, if a bulb of water is immersed in a flask where the acid and water are mixed the water will boil. Danish physicochemist Hans Peter Jørgen Julius Thomsen showed that the maximum rise of temperature (159 ⬚C) occurs when one part by weight of the acid is mixed with 0.338 part of water (4). It appears that Julius Thomsen (1826–1909) was the first person to carry out comprehensive measurements on the dilution of sulfuric acid (5). His work had important consequences for the theory of solutions. While another prominent thermochemist Marcelin Berthelot (1827–1907) argued in favor of the existence of hydrates of definite composition in solution, Thomsen’s conclusions were in direct opposition and gained a ready acceptance among chemists: Thomsen’s results with H2SO4 are of special importance, not only from his having studied this acid more fully than any other substance, but that the seeming regularity of these results is for ever being urged against the hydrate theory of solution. (6)
As Thomsen pointed out, ...the heat evolved on diluting liquid sulfuric acid with water is a continuous function of the water used, and excluded absolutely the acceptance of definite hydrates as existing in the solution. (6)
As P. Muir wrote in his textbook, “the generally adopted ‘hydrate theory’ of solution can scarcely be expected to survive the dissemination of Thomsen’s researches” (7). These conclusions are based on the regular and smooth curves when Q values are plotted against n (Figure 1). Throughout these curves there are no signs of irregularities or special points that would indicate the formation of definite hydrates. The data obtained by Thomsen are of direct interest for our discussion. The results are given in Table 1 and correspond to the 12 circles in Figure 1. At first glance, the dilution results have incredible precision, especially for the highest dilutions: five significant digits when 1599 moles (or about 30 L of H2O) are mixed with 1 mole (less than 54 mL) of sulfuric acid.1 Of course, Thomsen and Berthelot used rather precise devices. The readers can evaluate difficulties involved in every measurement: All the heat evolved in the process must be used in raising the temperature of the mixed liquids. Berthelot employed a fairly large platinum vessel (about 600 cm3 capacity). Berthelot places his calorimeter proper, with its cover and thermometer, in an outer vessel of thin cop-
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Of course, Thomsen knew about this aberration and argued that it might be due the contraction on mixing acid with water; but this explanation is highly improbable because the discrepancies are especially large in dilute solutions.
per, which is immediately surrounded by another vessel of silver with a cover; this vessel is placed in a considerably larger double walled vessel of sheet iron containing from 10 to 14 liters of water between the walls, and furnished with a stirrer and thermometer.... The water is placed in the outer vessel some days before an experiment is to be made. The whole apparatus stands on the wooden bench in a large room protected from the sun’s rays. All the liquids, the thermometers, the stirrer, & c., are placed in this room some days before the experiments begin so that everything may be as nearly as possible at the same temperature. (7)
Modern Data
We can add that Thomsen used an optical tube to measure temperature to 0.001 ⬚C. Nevertheless, his results seem even more surprising if we look at the original figures of Thomsen’s measurements (5). For instance, in the run with n = 1 the temperature rose from 20.785 to 30.049 ⬚C upon mixing. This seems precise enough to calculate Q with the desired precision. However, with dilution the situation worsens. Thomsen honestly published all values obtained even for unsuccessful experiments. For example, for n = 400 temperature increased from 20.930 to only 20.997 ⬚C in one run, did not change in another, and even decreased slightly (from 20.808 to 20.805 ⬚C) in the third run! Thomsen deduced a hyperbolic empirical equation Q =
17860 n cal n + 1.7983
to fit his results (5), which is shown as the dotted line in Figure 1. He showed the near agreement of the found and calculated values for n = 1, 2, 3, 5, 9, 19, and 1599. Unfortunately, as Pickering wrote (somewhat overcritically), ...the portion where this agreement exists is but little more than 1% of his whole curve, and throughout the remaining 99% of it there is no agreement at all, the differences reaching as high a value as 683 cal (at n = 99)... (6)
Table 1. Heat Evolved upon Mixing One Mole of H2SO4 with n Moles of H2O
100
Heat of Dilution / (kJ molⴚ1)
After Thomsen, many investigators measured the Q values for different n with increasing precision. For example, in the glass calorimeter constructed in the National Bureau of Standards in early 1960s the temperature was measured to 0.00005 ⬚C by means of platinum resistance thermometer. This device was used to measure the heat of formation of H2SO4⭈2500H2O (8). Thermoelectric batteries with 1000 couples made it possible to measure temperature difference of the order of 10᎑7 K (9). As the precision of measurements increased, the results achieved by Thomsen and other scientists were improved upon. In addition, R. E. Wilson suggested that Thomsen probably had a small quantity of water in his “pure” H2SO4 (10). (It is noteworthy that Wilson calls the Danish chemist alternately Thomsen, Thomson, and Thompsen in his article!) In 1952 NBS issued Selected Values of Chemical Thermodynamic Properties (11) where standard heats of formation for aqueous solutions of sulfuric acid were tabulated for n varying from 0.5 to 500,000 (Table 2). These values were critically recalculated on the basis of many original experimental data. Later the analogous data (66 values from n = 0.2 to n = 500,000) were reported in the Russian publication that supplied new experimental data compiled by a group of scholars under the supervision of Glushko (12). The difference in values in the two compilations is slight and can be safely discarded in our discussion (for example, for n = 1000, Q = 18.72 kcal in ref 12 versus 18.78 kcal in ref 11, and for n = ∞, Q = 23.13 kcal versus 23.00 kcal; the difference is connected with slightly different values accepted in refs 11 and 12 for the standard heat of formation of pure liquid H2SO4 at 298 K, namely, ᎑193.91 kcal兾mol and ᎑194.6 ± 0.2 kcal兾mol, respectively).
n/mol 80
60
40
20
0 -1
0
1
2
3
4
5
∞
log(n ) Figure 1. Heat evolved upon mixing one mole of H2SO4 with n moles of water: circles—Thomsen’s experimental data (data shown in Table 1), dotted line—values calculated from Thomsen’s empirical formula, solid line—modern results (11).
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0001
06.379
0002
09.418
0003
11.137
0005
13.108
0009
14.952
0019
16.256
0049
16.684
0099
16.858
0199
17.065
0399
17.313
0799
17.641
1599
17.857
NOTE: Data from ref 5.
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The NBS data are also shown in Figure 1 by the solid line. The Q value for the infinite dilution was obtained from the last several values using Debye–Hückel limiting law (13–15): below about 0.001 M concentration the measured heats of dilution are proportional to square root of molality. Incidentally, it is obvious that the quantity of heat evolved in this process is related to the concentration of the acid used. Therefore the moderately large concentration of an acid sample can be determined using even a primitive calorimeter. A simple but fairly accurate procedure was proposed for measuring the heats of interaction between water and sulfuric acid of various concentrations and for applying it to determine the concentration of unknown acid samples within the range of 50–98% (16). Heat of Dilution and Dissociation of Sulfuric Acid in Aqueous Solutions In diluted aqueous solutions, sulfuric acid dissociates in two steps: H2SO4 → H+ + HSO4−
(1)
H+ + SO42−
(2)
HSO4−
In highly diluted solutions nearly all of the sulfuric acid is present as sulfate and hydrogen ions; at higher concentrations most of it is bisulfate and hydrogen ions (17). In moderately dilute solutions the first equilibrium proceeds completely. The conductivity of H2SO4 at various concentrations reflects the composition of a solution (at concentrations above 5.0 M the conductivity decreases). Measurement of the sulfuric acid conductivity is a well-known experiment in electrochemistry (see for example ref 18).
Table 2. Heats of Mixing One Mole of H2SO4 with n Moles of Water n/mol
Q/kcal
n/mol
Q/kcal
0.5
03.76
00 0100
17.68
01
06.71
000 200
17.91
02
10.02
000 300
18.09
03
11.71
00 0500
18.34
04
12.92
00 1000
18.78
05
13.87
00 2000
19.33
06
14.52
00 5000
20.18
07
15.04
010,000
20.81
08
15.44
020,000
21.42
10
16.02
050,000
22.07
15
16.77
070,000
22.24
20
17.09
100,000
22.38
30
17.37
200,000
22.59
40
17.47
500,000
22.78
50
17.53
∞
23.00
75
17.61
NOTE: Data from ref 11.
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Some useful information about degree of ionization can be obtained from a study of the Raman and NMR spectra of the solution (19, 20). The intensity of a selected Raman line is proportional to the concentration of the species giving rise to that line. For example, the 910, 1043, and 980 cm᎑1 lines are produced by H2SO4, HSO4−, and SO42−, respectively (17). Some values one can find in Gmelin’s handbook (21). In one of such experiments, it was shown that diluted acid (0.101 M) had approximately the following ionic composition (expressed in mole per cent): H+ 56.8, HSO4− 29.5, SO42− 13.7 (20). It is, therefore, obvious that even in rather diluted acid (0.1 M corresponds to less than 1 mass %) dissociation in the second equilibrium is far from complete; the present value for K2 is only 0.012 at 25 ⬚C (22).2 The dissociation in the second equilibrium proceeds just in half, that is [HSO4−] = [SO42−], only in a very diluted solution (0.008 M or less than 0.08%). The shape of the curves in Figure 1 deserves a special comment. Thomsen’s experimental data fit rather well into his formula for 1 < n < 20 (so that Pickering’s criticism was too severe). For the more diluted solutions the deviation is fairly large (about 4.3% for n = 100), but the hyperbolic formula gives the limiting heat of dilution as 17,860 cal (74.73 kJ), which is practically identical with the number obtained experimentally when n = 1599. Of course, had Thomsen extended his determinations to more diluted solutions, he would have found still greater deviation from his formula (more than 20% for n = 105). However, Thomsen could not obtain precise data for extremely high dilutions. Besides, his experimental results (plateau in the graph) indicated (for Thomsen) the uselessness of further dilution. These conclusions are in a good agreement with that of Young and Blatz (17). They pointed out that some of the experimental values required the 10᎑6 ⬚C or better precision in temperature measurements. These authors’ estimate of the probable error of a single point at the highest dilutions is 40–100 cal兾mol, and at least one of the two points nearest total dilution, [H2SO4] → 0, must be in error by 100 cal兾mol or more (with the indicated precision in the temperature measurements, namely 10᎑6 ⬚C or better). It should also be pointed out that Thomsen carried out his measurements in the middle of the 19th century and he surely knew nothing about the electrolytic dissociation of H2SO4 molecules in dilute aqueous solutions (Svante Arrhenius put forward his theory in 1884 and this theory accounts for peculiarities of the correct upper curve in Figure 1). Therefore, Thomsen was obviously sure that he determined the limiting Q value corresponding to infinite dilution. The explanation of the shape of the upper curve is rather straightforward. As we can see, the heats of hydration rapidly increase as n → 10. When n > 50 the curve is almost a smooth horizontal line up to n = 100–200 ([H2SO4] = 0.3– 0.5 M). At such concentrations, as it was shown earlier, sulfuric acid dissociates mainly by the first reaction and only at [H2SO4] = 0.008 M (n ≈ 7000) dissociation of HSO4− ions proceeds by 50%. The heat evolved when n < 100–200 is determined by the hydration of only two ions (H+ and HSO4−) obtained from one molecule of H2SO4. Only at very large dilutions does dissociation of HSO4− become important to contribute enough to the Q values. We can see that the curve goes upward again when n > 1000, but much more
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slowly owing to the small value of K2; consequently, an extremely high dilution is needed to shift the equilibrium in eq 2 to the right and complete evolution of heat (three ions from one molecule of H2SO4). It can be seen from Table 2 (or from the appropriate graph) that at n = 500,000 the heat of dilution Q almost reaches its limiting value. One might propose that each ion at such dilution is surrounded with a huge hydrate shell that seems to consist of thousands of layers of water molecules. But this is not the case: fewer than 30 water molecules could be lined up along the radius of a hydrate shell. In conclusion, it is relevant to quote an article published in this Journal (25): The now-accepted model of a hydrated ion in solution is a dynamic one. At any instant, the ion is surrounded by a first shell of suitably oriented water molecules, their being the primary solvation number of the ion. Far away from the ion, there is the unperturbed structure of bulk water. In between there is a secondary solvation shell, a few molecules thick, in which the structure of the solvent is modified to an intermediate extent. This secondary solvation shell is responsible for as much as half the total hydration energy of a singly charged ion.... These generalizations establish beyond doubt the great importance of interactions beyond the primary hydration sphere in determining total hydration energies.... It is clearly impossible to treat ion hydration satisfactorily on the basis of considering only the interaction of the ion and its immediate solvent neighbors. W
Supplemental Material
Problems and assignments for students together with answers and possible solutions are available in this issue of JCE Online. Notes 1. The detailed discussion of precision, accuracy, errors, etc. is not the subject of this paper. As Robert M. Sykes correctly noted in his letter, a “moratorium on papers concerning balancing equations could be profitably extended to papers on error propagation and significant figures” (J. Chem. Educ. 1998, 75, 970). It is also interesting to note in this context that in one of his articles Keith J. Laidler (J. Chem. Educ. 1984, 61, 496) pointed out “the extravagant use of significant figures” in some older scientific works. 2. Dissociation constants and activities for strong inorganic acids have been discussed also by E. L. King (23), and the nature of the species present in sulfuric acid, especially in concentrated solutions, as well as the difference between “activity equilibrium constant” and “concentration equilibrium constant” and dependence of the later on acid molarity can be found in ref 24.
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Literature Cited 1. Mansson, M.; Sunner, S. Acta Chem. Scand. 1967, 17, 723– 727. 2. Mundell, D. W. J. Chem. Educ. 1990, 67, 426–427. 3. Good, W. D.; Lacina, J. L.; McCullough, J. P. J. Am. Chem. Soc. 1960, 82, 5589–5591. 4. Mellor J. W. A Comprehensive Treatise on Inorganic and Theoretical Chemistry, 2nd ed.; Longmans, Green and Co.: London, 1946–1948; Vol. 10, pp 405–406. 5. Thomsen, J. Thermochemische Untersuchungen; J. A. Barth Verlag: Leipzig, Germany, 1883; Bd. III, S. 19, 44–50. 6. Pickering, S. U. J. Chem. Soc. 1890, 57, 64–184. 7. Muir, M. M. P. The Elements of Thermal Chemistry; MacMillan: London, 1885, p 167. 8. Johnson, W. H.; Ambrose, J. K. J. Res. Nat. Bur. Stand. 1963, 67A, 427–430. 9. Hemminger, W.; Hoene, G. Calorimetry, Fundamentals and Practice; Verlag: Weinheim, Germany, 1984. 10. Wilson, R. E. Ind. Eng. Chem. 1921, 13, 326–331. 11. Rossini, F. D.; Wagman, D. D.; Evans, W. H.; Levine, S.; Jaffe, I. Selected Values of Chemical Thermodynamic Properties. Circular of the National Bureau of Standards 500; US Government Printing Office: Washington DC, 1952; pp 41–43. 12. Termicheskiye Konstanty Veshchestv (Thermal Properties of Substances), Glushko, V. P., Ed.; VINITI: Moscow, 1966; vol. 2. 13. Lange, E.; Monheim, J.; Robinson, A. L. J. Am. Chem. Soc. 1933, 55, 4733–4744. 14. Lange, E. Heats of Dilution of Dilute Solutions of Strong and Weak Electrolytes. In The Structure of Electrolytic Solutions; Hamer, W. J., Ed.; Wiley: New York, 1959; Chapter 9, pp 135–151. 15. Glasstone, S. Thermodynamics for Chemists; D.van Nostrand: New York, 1947; p 451. 16. Wolthuis, E.; Leegwater, A.; Ploeg, J. V. J. Chem. Educ. 1961, 38, 472–473. 17. Young, T. F.; Blatz, L. A. Chem. Rev. 1949, 44, 93–115. 18. West, L. E.; Gahler, A. J. Chem. Educ. 1942, 19, 366–388. 19. Comprehensive Inorganic Chemistry; Pergamon Press: New York, 1973; Vol. 1, p 121. 20. Young, T. F.; Maranville, L. F.; Smith, H. M. In The Structure of Electrolytic Solutions; Hamer, W. J., Ed.; Wiley: New York, 1959; Chapter 4, pp 35–63. 21. Gmelin’s Handbuch der anorganischen Chemie, 8 Aufl.; Verlag: Weinheim, Germany, 1960; Sys. No 9, Teil B, Lfr. 2, S. 655, 740. 22. Handbook of Chemistry and Physics, 69th ed.; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 1988. 23. King, E. L. J. Chem. Educ. 1986, 63, 490–491. 24. Brubaker, C. H. J. Chem. Educ. 1957, 34, 325–326. 25. Sharpe, A. G. J. Chem. Educ. 1990, 67, 309–315.
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