In the Classroom
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Dulong and Petit’s Law: We Should Not Ignore Its Importance Mary Laing and Michael Laing* 61 Baines Road, Durban 4001, Republic of South Africa; *
[email protected] Recently while tidying up our library of chemistry books, we came across a first-year laboratory manual of 1953 (50 years old!), and in it, as experiment 4, was the determination of the atomic weight1 of a metal (1). In this experiment it is the atomic weight that is determined directly and not equivalent weight. In a typical first-year laboratory it is the equivalent weight of a metal that is actually determined: either by evolution of hydrogen gas, by reduction of the oxide (with hydrogen or burner gas), by air combustion of the metal, or by electrolysis (2). To calculate the atomic weight the student must know at least one of the following: the valency of the metal, that hydrogen is H2, that oxygen is O2, that the valency of hydrogen is one, or that the valency of oxygen is two. A first-year student should ask “What experiment can I (as a student) carry out in the laboratory in one day, without recourse to this information, to determine the atomic weight of a metal?” The answer is “Measure its specific heat capacity, and then apply the classical law of Dulong and Petit.” This remarkable empirical discovery, first enunciated in 1819, states that “the atoms of all simple bodies have exactly the same capacity for heat” or, in modern terms, the molar heat capacity, Cm, is the same for all solid elements, equal to the product of the molar mass, M, and specific heat capacity, c, of that element (3, 4). Cm = Mc = 6 cal mol1 K1 = 25 J mol1 K1
(1)
We checked the indexes of some well-known first-year textbooks for “Dulong and Petit”. The names were absent in several, although the terms specific heat or heat capacity were present (5–9). Some mentioned specific heat of metals (10, 11). Only one textbook specifically mentions Dulong and Petit’s law, and its importance in estimating (and correcting) atomic weights (12). The textbooks of 30 years ago were better in describing Dulong and Petit’s law and its application (13, 14). Its value in correcting atomic weights is well described in the old books (15–17). Correcting the Periodic Table Two very recent discussions of the historical and philosophical background of the development of the periodic table describe how Mendeleyeff during 1870 used the law of Dulong and Petit to obtain correct atomic weights for indium, cerium, and uranium (wrong in the 1869 table) and thus produced the remarkable table of 1871 (18, 19). The periodic table published by Mendeleyeff in 1869 is shown in Figure 1. The elements In, Ce, and Ur, with atomic weights of 75, 92, and 116, respectively, are positioned according to their atomic weights as accepted at that time. It was evident to Mendeleyeff that these elements were not well-positioned (uranium between cadmium and tin!) and so during 1870 he obtained pure samples of these metals and determined their specific heat capacities. Using the relationship of Dulong www.JCE.DivCHED.org
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and Petit he corrected their atomic weights to the values of 113, 138, and 240, respectively, and these he incorporated in his revised periodic table of 1871, shown in Figure 2. The structure of this table remained essentially unchanged to about 1950 when the medium or long form was adopted in which the d transition elements were spread out and the lanthanides and actinides were removed from the body of the table. Thus Dulong and Petit’s law was critically important for the preparation of the first good periodic table whose structure remains largely the same to this day.2
Figure 1. Mendeleyeff’s 1869 periodic table. Positions of the elements In, Ce, and Ur are indicated. Wrong atomic weights are obvious.
Figure 2. Mendeleyeff’s periodic table of 1871. The newly corrected positions of In, Ce, and U are indicated. The format of this periodic table remained unchanged for about 70 years.
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Law of Pierre Louis Dulong and Alexis Thérèse Petit The specific heat capacities of the elements are interesting because, unlike most other physical properties that are periodic with atomic weight, the specific heat capacities are not, but decrease monotonically with atomic weight. As Dulong and Petit observed, the product of atomic weight and specific heat capacity is approximately constant, 6 cal mol1 C1 or 25 J mol1 C1 for such disparate metals as Na, Al, Zn, Pb, U—quite amazing. Rarely does one see a graph of specific heat capacity versus atomic weight (Figure 3), but there is one in the venerable classic text by Mellor, 1927 (20). This displays the characteristic hyperbolic curve showing the relationship between specific heat capacity and atomic weight of the metallic elements. Not only did Dulong and Petit in 1819 use their law to correct atomic weights, for example, lead (17), but Berzelius also used the relationship to deduce the correct atomic weights that he published in 1827 (Fe = 54) and that had been wrong in his list of 1815 (Fe =111) (21). But it was not clear why this relationship should exist.
ing an atom vibrating in simple harmonic motion in three independent orthogonal directions. Each mode of vibration thus contributes kBT to the energy of the atom (1/2kBT for the kinetic energy of the vibration and 1/2kBT for the potential energy). Thus the total energy of the 3NA oscillators is 3RT for 1 gram atom (or mole). The molar heat capacity of a monatomic solid therefore is 3R per degree, 6 cal per mole per degree, twice that of a monatomic ideal gas (22). A remarkable result. The large variation in the value of c with temperature (Figure 4) was explained for the first time by Einstein in 1907 using the quantum approach of Max Planck. Subsequently Debye, in 1911, put forward a theoretical model whose calculated values almost completely fitted the observed data (23). Thus the genius of three Nobel Prize winners was needed to finally bring understanding. The heat capacity of the solid is dependent on a maximum characteristic vibration frequency of the oscillators, νm,
Problems and their Explanation In the 1840s Regnault showed that the elements Be, B, C, and Si had exceptionally low specific heat capacities at room temperature (20): these elements had low atomic weights and high melting points. Measurements at low temperatures showed that for all elements c decreased toward zero with decreasing temperature. However, c approached the regular value of 6 cal mol1 C1 even for Be, B, C, and Si with substantial increase in temperature. (The values shown on Figure 3 for Be, B, and C are clearly not correct. They should be far smaller. Correct values are in Table 1). In 1871, Boltzmann applied Maxwell’s principle of equipartition of energy to the problem of the heat capacity of solids. He assumed that the ideal crystalline solid was built of monatomic unit cells, one atom at each lattice position, consisting of 3NA oscillators (NA is Avogadro’s number) each be-
Figure 4. The value of the molar heat capacity, Cm, of a solid element changes with temperature. The value of C approaches 3R asymptotically with increasing temperature. That this occurs close to room temperature for most metallic elements is serendipitous. (This figure is from ref 20, p 802; temperatures are in degrees Celsius. The units of atomic heat are cal mol1 °C1.)
Figure 3. The relationship between specific heat capacity and atomic weight of the elements at 0 °C. The specific heat capacities for Be, B, and C are not correct. All should be displaced to lower values: Be 0.43, B 0.24, and C 0.17. There is no value for gadolinium. (This figure is taken directly from ref 20, p 261. The units of specific heat capacity are cal g1 C1.)
Figure 5. The molar heat capacity of all elements follows the Debye equation and depends on the characteristic Debye temperature θ for the element. It is serendipitous that θ for most elements is not much higher than room temperature, so that T/θ is about 0.7. Values of θ are C 1900; Al 389; Ag 214; Pb 88 K. Comparison with Figure 4 shows that lead already fits the Dulong and Petit law at 173 C while aluminum does so only above 27 C. (This figure is from ref 22, p 420. The units of atomic heat capacity are cal mol1 K1, and T is in K.)
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In the Classroom
and the value of hνm兾kB is termed the characteristic or Debye temperature, θ. Cm at temperature T兾θ is the same for all solid elements. θ is different for all solids and is related to the rigidity and melting point of the solid; typical values are: Ag 214, Al 389, Zn 235, and Cu 313 K. Elements with a small atomic weight and a high melting point that are hard (and are covalently bonded and not ductile) have large values of θ: examples are carbon, silicon, and boron whose θ’s all exceed 1000 K. Figure 4 shows the differences in behavior of lead, aluminum, and carbon, and Figure 5 shows how well the Debye equation reproduces the behavior of silver, aluminum, and carbon. One can conclude that the specific heat capacity of a solid metallic element will obey Dulong and Petit’s law so long as it is measured at a temperature T equal to or greater than about 0.7 θ. Table 1 lists the specific heat capacities and molar heat capacities of metals. Plotting the data from Table 1 shows
Figure 6. A plot of specific heat capacity versus molar mass of representative data from Table 1. The curve is hyperbolic. The elements Be and Gd are not obviously misplaced. The position that Be should have if its molar heat capacity were 25 J mol1 K1 is marked by the solid square, which is indicated by the arrowhead.
Figure 7. A plot of the values of 1/specific heat capacity versus molar mass of representative data from the metals in Table 1. The curve is linear, except for the two metals Be and Gd, which are now clearly displaced from the line. The slope yields a value of 26.5 J mol1 K1 for molar mass x specific heat (Mc) (close to 3R). The slope, that is, Cm, for Be is 16.4 and for Gd is 37.0 J mol1 K1. The slope of the line through Au yields a value of 25.4 J mol1 K1 and the value for Hg is 28.0 J mol1 K1. Values for the elements B, C, and Si are included for comparison.
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Table 1. Specific Heat Capacities of Metals (at 25 °C) Metal
Specific Heata / (J g1 K1 )
Molar Heat Capacityb / (J mol1 K1 )
Atomic Weight
(Specific Heat)1 / (g K J1 )
Li
3.582
24.77
006.9
0.279
Na
1.228
28.24
023.0
0.814
K
0.757
29.58
039.1
1.321
Bec
1.825
16.44
009.0
0.548
Mg
1.023
24.89
024.3
0.977
Ca
0.647
25.31
040.1
1.545
Al
0.897
24.35
027.0
1.115
Sc
0.568
25.52
045.0
1.760
Y
0.298
26.53
088.9
3.355
La
0.195
27.11
138.9
5.128
Ced
0.192
26.94
140.1
5.208
Sm
0.197
29.54
150.3
5.076
Eu
0.182
27.66
152.0
5.494
Gd
0.236
37.03
157.2
4.237
Yb
0.155
26.74
173.0
6.452
Lu
0.154
26.86
175.0
6.493
Ti
0.523
25.02
047.9
1.912
Zr
0.278
25.36
091.2
3.597
Hf
0.144
25.73
178.5
6.944
V
0.489
24.89
050.9
2.049
Cr
0.449
23.35
052.0
2.227
Mo
0.251
24.06
095.9
3.984
Mn
0.479
26.32
054.9
2.088
Fe
0.449
25.10
055.8
2.227
Ru
0.238
24.06
101.1
4.202
Co
0.421
24.81
058.9
2.375
Rh
0.243
24.98
102.9
4.115
Ni
0.444
26.07
058.7
2.252
Pd
0.246
25.98
106.4
4.065 7.519
Pt
0.133
25.86
195.1
Cu
0.385
24.43
063.5
2.597
Ag
0.235
25.35
107.9
4.255 7.752
Au
0.129
25.42
197.0
Zn
0.388
25.40
065.4
2.577
Cd
0.232
25.98
112.4
4.310
Hg
0.140
27.98
200.6
7.143
Ga
0.371
25.86
069.7
2.695
Ind
0.233
26.74
114.8
4.292
Sn
0.228
25.77
118.7
4.386
Pb
0.129
26.44
207.2
7.752
Sb
0.207
25.23
121.7
4.831
Bi
0.122
25.52
209.0
8.197
Th
0.113
27.32
232.0
8.849
Ud
0.116
27.66
238.0
8.621
Bc
1.026
11.09
010.8
0.975
Cc
0.709
08.52
012.0
1.410
Sic
0.705
19.79
028.1
1.418
Data from ref 27. Data from ref 28. These four elements have the highest melting points of the first 20 elements in the periodic table: d Be 1551, B 2573, C 3820, Si 1683 K. The atomic weights of these three elements were corrected by Mendeleyeff in 1870, by the method of Dulong and Petit. a
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nicely both the patterns and the anomalies (such as beryllium and gadolinium; Figures 6 and 7). The low values for Be, B, C, and Si are evident and are explained by the Debye model. The curve in Figure 6 is a good hyperbola. The very small value for Be and large value for Gd are not readily apparent but become obvious when 1兾c is plotted versus atomic
Figure 8. A plot of the specific heat capacity, cv, of nickel versus temperature. The specific heat capacity at room temperature is 0.106 cal g1 °C1. The dramatic increase in the specific heat capacity at the Curie temperature of 365 °C is caused by the transition from a ferromagnetic structure at low temperatures to regular paramagnetism above TCurie. (The figure is from ref 23, p 25).
weight. As shown in Figure 7, both Be and Gd are markedly displaced from the line of slope 26.5 J mol1 K1. The Anomaly of Gadolinium The value of Cm for gadolinium, 37 J mol1 K1 at room temperature, is anomalously large: the largest for all elements. This is due to the phase change of the metal from ferromagnetic to paramagnetic occurring at room temperature, that is, its Curie temperature is 26 C (23). The same phenomenon is found in nickel at 365 C (Figure 8). This effect cannot be explained by the classic Debye model that considers only the frequency of lattice vibrations of the atoms and cannot take into account magnetic effects. These anomalously large values for Cm are due to structural changes that are related to the number of unpaired electrons in the atoms, and how the magnetic dipoles of the atoms become aligned below the Curie temperature. Discussion of the magnetic properties of metals is beyond the scope of this article. Similar unexpected large values for the specific heat capacity of a solid are observed when there are order–disorder structural changes within the solid. The alloy beta-brass, 50:50 Cu兾Zn, has an ordered simple cubic CsCl-type structure at room temperature. This changes to a randomly disordered body-centered cubic structure at about 470 C. The value of the specific heat capacity at this temperature is anomalously large (Figure 9) (23). It is thus clear that the value of the specific heat capacity of an element will not necessarily change smoothly with temperature but can have large discontinuities associated with structural phase changes. Also the basic value of c is very dependent on the hardness of material and the temperature at which c is measured. (The specific heat capacity of gadolinium at 50 C is not abnormal.) Figure 4 shows quite clearly that if the standard temperature for measuring specific heat capacity were, say, 100 C, there would be no Dulong and Petit’s law. We thus arrive at the conclusion that the remarkable observation of Dulong and Petit is almost fortuitous, and applies well at room temperature for most elements simply because their Debye temperatures are less than 100 C. Nevertheless it was the very simplicity of Dulong and Petit’s Law that made it so valuable to the chemists of the 19th century, as it enabled them to obtain correct atomic weights. Activities for Students
Figure 9. A plot of specific heat capacity, cp, of the alloy betabrass versus temperature. At room temperature the Zn and Cu atoms are ordered in a simple cubic CsCl structure; cp for this alloy is 0.093 cal g1 C1. The large increase in the specific heat capacity at about 470 °C is caused by the Zn and Cu atoms becoming randomly disordered in the body-centered cubic unit cell. (The figure is from ref 23, p 37. The units of cp are cal g1 °C1).
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There have been a small number of articles in this Journal about specific heat capacity and Dulong and Petit’s law (24–26) but in general they have not emphasized the importance of the measurement of the specific heat capacity of an element in the creation by Mendeleyeff of the important periodic table of 1871. It seems that it would be a worthwhile experience for students to examine the relationship between the atomic weight and specific heat capacity of an element, deduce the law of Dulong and Petit, and observe how the atomic weight of an element can thus be determined. There are two aspects that can be examined:
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In the Classroom • Practical determination of the specific heat capacity of a metal; details of the experimental method (1) are given as a laboratory experiment in the Supplemental Material.W • Processing of data, which is described in detail as a classroom activity in the Supplemental Material.W
Laboratory Experiment
Determining the Specific Heat Capacity of a Metal The student uses a nested pair of Styrofoam coffee cups as a calorimeter. Pellets of the metal are weighed into a test tube, which is then heated in a bath of boiling water for at least 15 min. The temperature of the boiling water is recorded to ±0.1 C. Using some ice chips, about 50 mL of water is cooled to about 10 C below room temperature. Approximately 50 mL of this cold water is weighed into the polystyrene cup, and the temperature of the water is recorded. The hot metal pellets are poured into the cold water, while stirring with the thermometer. The student records the highest temperature attained to ±0.1 C. From these data the student records the energy gained by the water in the coffeecup calorimeter and hence the specific heat capacity of the metal in J兾(g C). It is important here for the student to evaluate the effect of an error of, say, 1 C. Determining an Approximate Atomic Weight Dulong and Petit’s law states that the product of the molar mass, M, of an element and its specific heat capacity, c, is approximately 6 cal mol1 K1 or 25 J mol1 K1. The approximate atomic weight of the metal is easily estimated by M = 25兾c. The student is asked to identify the unknown metal from the following possibilities: Mg, Al, Ti, Cr, Fe, Ni, Cu, Zn, Sn, Pb, and Hg. The student is then asked to explain why it is not possible to be certain. In the determination of specific heat capacity, there should be about 10 different metals in a class of 30 to 100 students. It should become evident to the students that they cannot differentiate among the elements between Cr and Ni because the precision of the measurements is too low. It is here that a density measurement or a simple chemical reaction can identify the metal. Students can get good results for the atomic weight (±2%) with careful work. The exercise is safe to carry out and inexpensive to do because the metals are easily available and can be reused. Specific Heat Capacity of Sodium The student is asked to describe how the specific heat capacity of sodium (a metal that reacts violently with water) could be determined. Determining the specific heat capacity of a metal that reacts with water is not simple. Water cannot be used. The liquid in the calorimeter could be a paraffinic hydrocarbon or even silicone oil. The metal may have to be cooled rather than heated. There is not a specific answer. Hazards The technique is relatively straightforward. None of the metals are hazardous under the conditions of the experiment.
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(In practice, mercury is not used as an unknown). The only potential danger is the boiling water that could cause burns. Students should be cautioned to handle the beaker and test tube with care. Classroom Activity: Exercises for the Student in Processing of Data The student extracts the values of the specific heat capacities (c) of the common group 2 metals, Be, Mg and Zn, Cd, and Hg, from the CRC Handbook of Chemistry and Physics (27) and plots these values versus atomic weights. He or she is asked to decide whether there is a relationship between c and M, and if so, to say what it is. This process is repeated for the lanthanides plus Ba, Al, Sc, Y, and Hf, and then for the 3d metals, Ca and Sc to Cu and Zn, and again to determine whether there is a relationship between c and M. The three plots are now combined and the data for the group 1 metals and the platinum metals are added. (The plotting can be done by hand or by using the spreadsheet option in a program like EXCEL). The student is asked to decide whether there is a relationship between c and M for all metals. Students must plot both c versus M and 1兾c versus M to come to a correct conclusion. If any metal is an exception to the general relationship, the student must compare its values of: density, melting point, compressibility (κ), expansion coefficient (α), electrical conductivity (or resistivity), ∆Hat, IE1, and magnetic susceptibility χ (27, 28) with those of its nearest neighbors across a period, and comment on any discontinuities. Systematically adding data, block-by-block, and seeking a relationship at each stage until a satisfactory result is obtained, is a good example of the application of the scientific method. Students will probably ask why the law of Dulong and Petit holds so well for metallic elements whose molar masses range from 7 to 238 g mol1! This can lead to a discussion of those topics that bridge the boundary between chemistry and physics: Boltzmann statistics, the classical principle of equipartition of energy, and the application of quantum mechanics by Einstein and Debye to explain the behavior of the specific heat capacity of solids with changing temperature (29, 30). Such concepts are beyond the realm of firstyear chemistry, nevertheless a lecturer should review these topics in preparation for the awkward questions. Debye, Einstein, and Planck have unlocked a deep secret of Nature. Conclusion This application of Dulong and Petit’s law is a good example of how science actually occurs: how Mendeleyeff observed that elements were wrongly placed in his first periodic table of 1869, and how he knew that they had to obey the periodic law and therefore deliberately redetermined their atomic weights and thus placed them correctly in the periodic system of 1871. It is quite remarkable how many solid elements obey Dulong and Petit’s law. Gaining an understanding of why this should be requires a synthesis of many physical concepts. Indeed, Dulong and Petit’s law is important and interesting: it involved, separately, three Nobel Prize winners.
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Acknowledgments It was the stimulating discussions at the Second Harry Wiener International Conference; The Periodic Table: Into the 21st Century, held in Canada, 14–20 July, 2003, about the development of the periodic table between 1869 and 1871 that prompted us to write this article. It is a pleasure to thank the reviewers for their valuable and constructive comments. W
Supplemental Material
The laboratory experiment, the classroom activity, and notes for the instructor are available in this issue of JCE Online. Notes 1. Atomic weight is used because this term was used historically and is numerically identical to molar mass (which is in gram mol1). 2. The following comment by one of the reviewers about the elements sums up the situation rather well. “There is a certain irony in using a non-periodic property in order to establish the periodicity of chemical and physical properties.”
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9. Umland, J. B.; Bellama, J. M. General Chemistry, 3rd ed.; Brooks/Cole: Pacific Grove, CA, 1999; pp 198, 199. 10. Moeller, T.; Bailar, J. C.; Kleinberg, J.; Guss, C. O.; Castellion, M. E.; Metz, C. Chemistry, 3rd ed.; Harcourt Brace Jovanovich: New York, 1989; pp 140, 141, 156. 11. Kotz, J. C.; Treichel, P. Chemistry & Chemical Reactivity, 3rd ed.; Saunders: New York, 1996; pp 265–267, 311. 12. Petrucci, R. H.; Harwood, W. S.; Herring, F. G. General Chemistry, 8th ed.; Prentice Hall: Upper Saddle River, NJ, 2002; pp 224–226, 258, 264. 13. Mahan, B. H. University Chemistry; Addison–Wesley: Reading, MA, 1965; pp 10–12. 14. Campbell, J. A. Chemical Systems; W. H. Freeman: San Francisco, 1970; pp 125–127. 15. Partington, J. R. A Text-Book of Inorganic Chemistry, 5th ed.; Macmillan: London, 1937; pp 382, 383, 387, 388, 417, 418. 16. Caven, R. M.; Lander, G. D. Systematic Inorganic Chemistry, 6th ed; Blackie: Glasgow, 1939; pp 13, 14, 35, 36. 17. Ihde, A. J. The Development of Modern Chemistry; Dover: New York, 1984; pp 145–147, 246, 247. 18. Kaji, M. Bull. Hist. Chem. 2002, 27, 4–16. 19. Brooks, N. M. Foundations of Chem. 2002, 4, 127–147. 20. Mellor, J. W. A Comprehensive Treatise on Inorganic and Theoretical Chemistry; Longmans, Green: London, 1927; Vol. I, pp 259–261, Fig 5, p 802. 21. van Spronsen, J. W. The Periodic System of Chemical Elements; Elsevier: Amsterdam, 1969; pp 46–50, 61. 22. Glasstone, S. Textbook of Physical Chemistry, 2nd ed.; van Nostrand: New York, 1946; pp 329–332, 413–422. 23. Seitz, F. The Modern Theory of Solids; McGraw–Hill: New York, 1940; pp 13–18, 23–25, 30, 35–37. 24. Dence, J. B. J. Chem. Educ. 1972, 49, 798–804. 25. (a) Bindel, T. H. J. Chem. Educ. 1990, 67, 165–166. (b) Bindel, T. H.; Fochi, J. C. J. Chem. Educ. 1997, 74, 955– 958. 26. Qiu, L.; White, M. A. J. Chem. Educ. 2001, 78, 1076–1079. 27. CRC Handbook of Chemistry and Physics, 76th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1995; pp 4–120, 12– 1727, 12–173. 28. Emsley, J. The Elements; Clarendon Press: Oxford, 1989. 29. Adamson, A. A. A Textbook of Physical Chemistry; Academic Press: New York, 1973; pp 157–159. 30. Levine, I. N. Physical Chemistry, 3rd ed.; McGraw–Hill: New York, 1988; pp 883–887, 895 (prob. 24.36).
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