In the Classroom
The Constituent Additivity Method to Estimate Heat Capacities of Complex Inorganic Solids Liyan Qiu and Mary Anne White* Department of Chemistry, Dalhousie University, Halifax, NS B3H 4J3, Canada; *
[email protected] Heat capacity is one of the most fundamental properties of matter. For example, the importance of heat capacity in understanding atmospheric temperature gradients associated with El Niño has recently been discussed in This Journal (1). From a material’s heat capacity, other thermodynamic quantities can be obtained, and these can be used to understand the conditions of and driving force for thermodynamic equilibrium, including chemical reactivity and thermodynamic stability (2). Furthermore, knowledge of heat capacity is important for many material applications, from energy storage to heat conduction. For example, in investigations of new materials as potential thermoelectrics, knowledge of heat capacity is required to convert thermal diffusivity (the quantity usually measured) to thermal conductivity (one of the quantities required to assess the thermoelectric figure of merit). Although thousands of new compounds are reported each year, not all their heat capacities are known. Traditional calorimetric methods (3) can provide accurate heat capacity data, but this relies on availability of both suitable apparatus and sample. Thus, a means of estimating heat capacity would be very useful. Background The well-known Dulong–Petit law, first stated in 1819, indicates that all atoms of simple bodies have exactly the same heat capacity (4). This generalization applies to solids at room temperature, and the common value of the molar heat capacity of an elemental solid is 3R, where R is the gas constant. The Dulong–Petit law can be understood in terms of equipartition theory (5). In the middle of the 19th century, the Dulong–Petit law was extended to compounds, leading to the Neumann–Kopp law, which states that the heat capacity of a compound in the solid state is the sum of the atomic heat capacities of the elements from which it is composed (6 ). An example is MoSi2, whose heat capacity over the temperature range 110 to 870 °C deviates from that of the corresponding sum of the elemental heat capacities by only 0.3% or less (7 ). The Dulong–Petit law is useful to estimate the roomtemperature heat capacities of solid elements and the Neumann–Kopp law can be used for some simple solids such as binary compounds or alloys. However, for complex solid compounds, estimates based on the Neumann–Kopp law can deviate from experimental results, as shown in Table 1 for several inorganic compounds at T = 298.15 K. Presumably this is because the chemical properties (determined by the chemical bonds) and hence the physical properties (such as lattice dynamics) of elements are significantly changed upon formation of compounds. The Neumman–Kopp law is based on additivity of the properties of the elements. Another approach—consideration of thermodynamic contributions from the “constituent 1076
groups” of a compound—has been used with considerable success to estimate thermodynamic properties of complex inorganic compounds, especially the standard-state entropies of mineral oxides (8). This method is rather widely used in the mineralogical literature, but is less commonly discussed in physical chemistry classes and textbooks, which are more likely to be limited to the introduction of the Dulong–Petit law and the Einstein and Debye models of heat capacity. In this report, we describe what we call the “constituent additivity” model of the heat capacity of complex inorganic materials, present examples of the use of this model, and discuss its basis and the uncertainty that it introduces in the determination of related thermodynamic quantities. We believe that this could be of interest both for teaching thermodynamics and for comparison with heat capacities of inorganic solids determined in undergraduate laboratory experiments. Some Examples of Constituent Additivity The experimental heat capacities at T = 298.15 K for several inorganic salts are presented in Table 1 in comparison with their heat capacities calculated from additivity of the experimental heat capacities of the appropriate proportions of their simple “constituent” compounds. In all these simple cases, and many more, the additivity of the heat capacity from the constituent compounds gives a rather good representation (within a few percent) of the true heat capacity (9), often much better than the Neumann–Kopp law estimate. However, comparison of the constituent-additive and the experimental heat capacities as a function of temperature would provide a more rigorous test. Table 1. Experimental and Literature Values for Heat Capacity of Some Inorganic Materials at T = 298.15 K C/R a Compound
Ref
Sr3MgSi2O8
10
31.0
Na8Al6Si6O24Cl2 14 (Sodalite)
97.7
Ca3(PO4)2
19
27.4
LaFeO3
22
MgTa2O6
25
NaAlO2
27
aR
Exptl
Constituent Additivity Neumann– Kopp Law b Value "Constituent" Ref 31.5
MgO SrO SiO2
11 12 13
98.0
Na2O Al2O3 SiO2 NaCl
15 16 13 17,18
22.8
27.9
CaO P2O5
20 21
13.0
15
12.7
La2O3 Fe2O3
23 24
21.2
19.5
20.8
MgO Ta2O5
11 26
Na2O Al2O3
15 16
8.86
32.1
107
9.8
is the gas constant. from ref 9.
bData
Journal of Chemical Education • Vol. 78 No. 8 August 2001 • JChemEd.chem.wisc.edu
8.91
In the Classroom
Temperature Dependence
Figure 1. The heat capacity, C p , for sodalite, from (solid line) experimental data and (dashed line) calculated from that of Na2O, Al2O3, α-SiO2, and NaCl.
Figure 2. The relative difference between experimental heat capacities and those calculated by the constituent additivity method for MgTa2O5, Sr3MgSi2O8, LaFeO3, CaZrTi2O7, and CaTiO3.
Figure 3. The relative difference between experimental heat capacities and those calculated by the constituent additivity method for Sr3MgSi2O8 based on different SiO2 polymorphs: α-quartz, cristobalite, and tridymite.
Historically, one of the first successes of quantum mechanics was Einstein’s explanation of the temperature dependence of heat capacity, especially that C→0 as T→0 K, because of thermal depopulation of quantized vibrational energy levels. The exact temperature dependence of C is an indication of the strength of the interactions between constituent atoms. We have considered the temperature dependence of the heat capacities of several complex inorganic solids in comparison with additivity of heat capacity of their “constituents” over the whole temperature range in which accurate experimental data are available. Sodalite, a mineral of idealized formula Na8Al6Si6O24Cl2, is a system for which reliable data are available over a very wide temperature range (100–1000 K). Figure 1 compares its experimental heat capacity (14) with calculated results based on Na2O (15), Al2O3 (16 ), α-SiO2 (28), and NaCl (17, 18) as constituents. The agreement is remarkable despite significant differences in structure, especially considering the open framework structure of sodalite. The differences between the experimental heat capacities and the constituent-additivity prediction for five more complex inorganic solids are quantified in Figure 2. These are MgTa2O6 (25) (considered in terms of MgO [11] + Ta2O5 [26 ]); LaFeO3 (22) (considered as 1⁄2[La2O3 (23) + Fe2O3 (24)]); CaZrTi2O7 (29) (considered as CaO [20] + ZrO2 [30] + 2TiO2 [31]); CaTiO3 (29) (considered as CaO [20] + TiO2 [31]); Sr3MgSi2O8 (10) (considered as MgO [11] + 3SrO [12] + 2SiO2 [13]). The last is an interesting case because the different possible forms of SiO2 allow an investigation of the sensitivity of this method to the type of polymorph; the relevant data are given in Figure 3. For SiO2 polymorphs, it appears that the difference is well within the accuracy of the model, but it would require further exploration to know if the use of different polymorphs generally makes such a small difference. In some cases, direct “constituent” solids that would be required to calculate the heat capacity by constituent-additive methods might not exist. A case would be carbonates, for example MCO3, in which the experimental heat capacity of MO might be known, but CO2 would not be a suitable constituent for this method. However, in this case, data for M′CO3 and M′O (where M′ ≠ M) could be used to calculate the heat capacity of MCO 3 as follows: Cp (MCO3) = C p(MO) + Cp(M′CO3) – Cp(M′O). Nd2O2CO3 (32) and La2O2CO3 (32) were treated in this way (with data for Nd2O3 [33], SrCO3 [34] and SrO [12], and La2O3 [23]); the results are shown in Figure 4. The results in Figures 1 to 4 indicate that the constituent additivity calculation method gives a remarkably good estimate of heat capacity, much better than ±5% above about 130 K, with deviations increasing to about 15% at lower temperatures. The examples above show that the constituent-additivity approach is applicable over a broad temperature range for several inorganic solids. However, the model does not always work with such quantitative success. For example, for the zeolite known as Linde type A, Na96Al96Si96O384, the observed heat capacity (35) exceeds that calculated by constituent additivity from Na2O (15), Al2O3 (16 ), and SiO2 (13) by 30% at T = 100 K and 9% at T = 300 K. The reason for this discrepancy is not clear, especially considering that the constituentadditivity model works so well for sodalite, which has a rather
JChemEd.chem.wisc.edu • Vol. 78 No. 8 August 2001 • Journal of Chemical Education
1077
In the Classroom
Figure 4. The relative difference between experimental heat capacities and those calculated by the constituent additivity method, for Nd2O2CO3 and La2O2CO3.
Figure 5. Propagation of error in heat capacity (Cp) as it gives rise to error in enthalpy (H ), entropy (S) and Gibbs energy (G). Two different types of error in Cp are given in the two sets of curves.
similar structure. However, for a large number of complex inorganic compounds that we have investigated, we have found that the constituent-additivity model usually works within a few percent at and above room temperature; the deviation increases to about 15% at T = 50 K.
modes dominate, because the constituent solids usually are simpler and hence have smaller unit cells and therefore larger (relative) acoustical contributions to the heat capacity. However, this ignores the relationship between the acoustical contribution and the structure and nature of the bonding within the solid (i.e., differences in the Debye temperature), and it would not be realistic to expect these interactions to be precisely the same for both the constituents and the complex solids. As we observe (Figs. 1–4), the low-temperature estimates sometimes exceed and sometimes underrepresent the true heat capacity. Therefore, these few examples show that we should be careful of over-interpretation with respect to the correlation between the mode assignments for the complex solid and its constituent solids, especially since heat capacity is a broad reflection of all thermal excitations, not a means of uniquely assigning thermally excited modes.
Possible Explanations for Additivity From the examples above, especially at ambient and higher temperatures, this approach could provide results nearly comparable with experimental uncertainty. Therefore, it is worth searching for some possible explanations for the basis for constituent additivity of heat capacity of complex inorganic materials. The two major factors that determine the heat capacities of solids are structure and composition. Put more explicitly, the structural contributions refer to those from vibrational modes in solids. It is reasonable to imagine that the optical modes of a complex solid would strongly resemble those of its constituent compounds, especially if the constituent solids have bonding arrangements (i.e., coordination) similar to that of the “complex” solid. In that case, the heat capacity should be primarily a function of composition. The latter comment takes into account that the dominant modes in a polyatomic solid will be optical, with fewer acoustical modes. Specifically, there will be 3 acoustical modes, compared with (3N – 3) optical modes, where N is the number of atoms in a primitive unit cell (21).1 The dominance of optical modes is especially true at reasonably high temperatures and for complex solids (large N ) where the optical modes can provide the main contribution to the heat capacity. At lower temperatures, where the acoustic modes contribute a greater proportion of the heat capacity, the constituent additivity model would be expected to break down, as is observed. Furthermore, this model is known to require similar coordination of the elements in the complex solid and in the constituents considered. One could reasonably expect that the calculated heat capacity of the complex solids would be larger than experimental values at very low temperatures where the acoustical
1078
Utility and Limitations of the Constituent Additivity Method If the heat capacity of a complex solid showed exact constituent additivity, then so would other thermodynamic quantities such as enthalpy, entropy, and Gibbs energy, and this method could be very useful in establishing relationships concerning matters such as relative thermodynamic stability. Of course, if Cp is overestimated or underestimated by a consistent percentage across the entire temperature range, then H (= H0 + ∫C dT ), S (= S0 + ∫C T ᎑1dT ) and G (= H – TS ) would be displaced by the same percentage. However, underestimation of Cp at the lowest temperatures and overestimation at higher temperatures or vice versa would lead to a more complicated error in the other thermodynamic functions, and this error would vary with temperature. Examples that are somewhat typical of the uncertainties in Cp found here with the constituent additivity method are shown in Figure 5, along with propagated uncertainties in H, S, and G. This figure shows that, although the low-temperature uncertainty in Cp can be compensated by a change in its sign as the temperature is increased, the low-temperature error dominates
Journal of Chemical Education • Vol. 78 No. 8 August 2001 • JChemEd.chem.wisc.edu
In the Classroom
the uncertainty in the derived thermodynamic functions until temperatures much higher than the one where the fit is exact, meaning that a reasonable expectation of uncertainty in G at room temperature could be ±10%. Whether this error in G would make the constituent additivity model too unreliable for chemical thermodynamic stability calculations would depend on the exact system. However, it has been shown (36 ) that Gibbs energies of formation of zeolites can be handled by a similar method, and furthermore the entropy of CaZrTi2O7 has been shown (29) to be accurately represented by consideration of its constituent compounds. One certain limitation of the constituent-additivity heat capacity approach is that it cannot accurately predict structural, magnetic, or other phase transitions in complex solids. This requires much more information than this model could provide. Despite these drawbacks, a reasonably accurate heat capacity at or above about 150 K could be expected from the constituent-additivity method, provided that (i) the experimental data for the constituent solids are reliable, (ii) the coordination within the constituent solids is similar to the complex solids, and (iii) there are no solid–solid phase transitions. This method could be especially useful for the rapid assessment of a material’s properties (e.g. to calculate heat storage capabilities or heat flux under a temperature gradient) and under favorable conditions, the constituent-additivity model can be used to estimate thermodynamic stability. Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council of Canada and the Killam Trusts. Note 1. Acoustical modes are lattice vibrations that include the motions induced by sound waves. For a molecular solid, these would be vibrational motions of the whole molecule on the lattice site. Optical modes correspond to lattice vibrations that can interact with light through changes in dipole moments or polarizability as the atoms vibrate. Note that the primitive unit cell might not be the same as the chemical unit cell. For example, the primitive unit cell could include two chemical units related by a symmetry operation.
Literature Cited 1. Blanck, H. F. J. Chem. Educ. 1999, 76, 1635. 2. For an example of recent studies of thermal stability, see White, M. A.; Harnish, R. S. Chem. Mater. 1998, 10, 833. 3. White; M. A. In Comprehensive Supramolecular Chemistry, Vol. 8; Davies, J. E. D.; Ripmeester, J. A., Eds.; Pergamon: Oxford, 1996; Chapter 4. 4. Laidler, K. J. The World of Physical Chemistry; Oxford University Press: Oxford, 1993; p 322. 5. See, for example, Dence, J. B. J. Chem. Educ. 1972, 49, 798. 6. Gopal, E. S. R. Specific Heats at Low Temperatures, Plenum: New York, 1966. 7. Walker, B. E.; Grand, J. A.; Miller, R. R. J. Phys. Chem. 1956, 60, 231.
8. Helgeson, H. C.; Delany, J. M.; Nesbitt, H. W.; Bird, D. K. Am. J. Sci. 1978, 278-A, 1. Wood, B. J. In Advances in Physical Geochemistry, Vol. 1; Newton, R. C.; Navrotsky, A.; Wood, B. J., Eds.; Springer: New York, 1981; p 73. 9. All heat capacity values used here are from high-accuracy calorimetry and Cp is usually known to ±2% or better. Most are values recommended by NIST. See http://webbook.nist.gov/ (accessed May 2001). 10. Huntelaar, M. E; Booij, A. S.; Gordfunke, E. H. P; van der Laan, R. R. J. Chem. Thermodyn. 1998, 30, 497. 11. Barron, T. H. K.; Berg, W. T.; Morrison, J. A. Proc. R. Soc. London 1959, A250, 70. 12. Anderson, C. T. J. Am. Chem. Soc. 1935, 57, 429. 13. Anderson, C. T. J. Am. Chem. Soc. 1936, 58, 568. 14. Komada, N; Westrum, E. F. Jr.; Hemingway, B. S.; Zolotov, M. Y.; Semenov, Y. V.; Khodakovsky, I. L.; Anovitz, L. M. J. Chem. Thermodyn. 1995, 27, 1119. 15. Chase, M. W. NIST-JANAF Tables, J. Phys. Chem. Ref. Data; Monograph 9; 1998; p 1648. 16. Ditmars, D. A; Ishihara, S; Chang, S. S.; Berstein, G.; West, E. D. J. Res. Natl. Bur. Stand. (U.S.) 1982, 87, 159. 17. Barron, T. H. K.; Leadbetter, A. J.; Morrison, J. A. Proc. R. Soc. London 1964, A279, 62. 18. Dawson, R.; Brackett, E. B.; Brackett, T. E. J. Phys. Chem. 1963, 67, 1669; as reassessed by Chase, M. W., et al. J. Phys. Chem. Ref. Data 1985, 14 (Suppl. 1), 770. 19. Barrow, G. M. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 1988. 20. Gmelin, E. Z. Naturforsch. 1969, 24a, 1794. 21. White, M. A. Properties of Materials; Oxford University Press: New York, 1999. 22. Stølen, S.; Grønvold, F.; Brinks, H.; Atake, T.; Mori, H. J. Chem. Thermodyn. 1998, 30, 365. 23. Justice, B. H. Thermodynamic Properties and Electronic Energy Levels of Eight Rare Earth Sequioxides; Ph.D. Dissertation, University of Michigan, 1961. 24. Grønvold, F.; Westrum, E. F. Jr. J. Am. Chem. Soc. 1959, 81, 1780. 25. White, M. A.; Neshvad, G. J. Chem. Thermodyn. 1991, 23, 455. 26. Kelley, K. K. J. Am. Chem. Soc. 1940, 62, 818. 27. Chase, M. W. NIST-JANAF Tables, J. Phys. Chem. Ref. Data; Monograph 9; 1998; p 31. 28. Chase, M. W. NIST-JANAF Tables, J. Phys. Chem. Ref. Data; Monograph 9; 1998; p 1752. 29. Woodfield, B. E.; Boerio-Goates, J.; Shapiro, J. L.; Putnam, R. L; Navrotsky, A. J. Chem. Thermodyn. 1999, 31, 245. 30. NIST-JANAF; based on Kelley, K. K. Ind. Eng. Chem. 1944, 36, 377. 31. Naylor, B. F. J. Am. Chem. Soc. 1946, 68, 1077; as reassessed by Chase, M. W., et al. J. Phys. Chem. Ref. Data 1985, 14 (Suppl. 1), 1680. 32. Olafsen, A.; Fjellvag, H.; Stølen, S.; Atake, T.; Kawaji, H.; Matsuo, K. J. Chem. Thermodyn. 1999, 31, 433. 33. Goldstein, H. W.; Neilson, E. F.; Walsh, P. N.; White, D. J. Phys. Chem. 1959, 63, 1445. 34. Travis, C. T. J. Am. Chem. Soc. 1934, 56, 340. 35. Qiu, L.; Murashov, V.; White, M. A. Solid State Sci. 2000, 2, 841. 36. La Iglesia, A.; Aznar, A. J. Zeolites 1986, 6, 26.
JChemEd.chem.wisc.edu • Vol. 78 No. 8 August 2001 • Journal of Chemical Education
1079
