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Using Computer Simulations To Teach Salt Solubility The Role of Entropy in Solubility Equilibrium Victor M. S. Gil Department of Chemistry, University of Coimbra, Portugal João C. M. Paiva* Department of Chemistry, University of Porto, Portugal; *
[email protected] Teaching chemical equilibrium usually follows several steps that can be associated with questions of the type “how?” (e.g., how can we recognize chemical equilibrium?), “what?” (e.g., what is meant by a dynamic equilibrium?), “how much?” (e.g., how much do reactants transform into products?), “what if?” (e.g., what if a system at equilibrium is disturbed?), “why?” (e.g., why do some reactions go almost to completion, whereas other reactions hardly begin and still others fall in between?, i.e. why do equilibrium constants vary so much?). The answer to the “why?” questions above requires the application of the second law of thermodynamics (see, for example, ref 1 ). In this process, the central concept of entropy (total entropy of a system plus its surroundings) is involved. This is an important step towards a deeper understanding of the extent of reactions in terms of molecular structure and energy change. In any field of science, understanding occurs at different levels. Students ought to be conscious of this and avoid confusing a cursory understanding of a phenomenon with a deep, complex understanding of that phenomenon. For example, although Le Châtelier’s principle is important in learning chemical equilibrium changes, it is not desirable that it be taken by students as “the explanation” of such changes, nor that comparison of acidity constants is offered as the answer to questions like “why is acid A stronger than acid B?” For a discussion about the use of “because” by chemistry students, see ref 2.
of matter (particles), that is, ions of the solute mixed with solvent (usually, water) molecules. This reference to particles, in ever-present thermal motion (5), opens the way to the thermodynamic interpretation. The Role of Entropy and Equilibrium in Solubility As in any other chemical phenomenon, the extent to which dissolution occurs is indicated by an equilibrium constant and governed by the second law of thermodynamics: given an excess of solute, macroscopic changes cease to occur—namely, equilibrium is attained—when total entropy reaches a maximum. By total entropy change we mean either (a) or (b), below: (a) For a closed system, the entropy change of the system plus the entropy change of the surroundings, at a given temperature. (b) For an isolated system, the entropy of the system, due not only to configurational but also to thermal contributions. For our simulations, we chose to consider the dissolution as occurring in isolated systems, the interpretation being readily extended to closed systems.
In statistical terms, the entropy (S) of a given state for a given dynamic system is related to the probability of occurrence of that state, through Boltzmann’s logarithm relation S = k lnW
Comparing Salt Solubility Computer simulations can be very useful in teaching chemical equilibrium (3–4) and, in particular, in illustrating the answers at the thermodynamic level mentioned above. The solubility equilibrium of salts is especially appropriate for that. This paper compares the solubility of various salts and explores computer simulations in order to facilitate the thermodynamic interpretation of the differences. Specifically we address these questions about the solubility of various salts: 1. Why, for the same temperature, is calcium carbonate much less soluble in water than sodium chloride, two salts whose heats of dissolution are almost identical (nearly zero)? 2. Why, for the same temperature, is magnesium sulfate much more soluble in water than magnesium carbonate, two salts of similar composition and structure?
A first point to make is that the comparison of salt solubilities requires that solubility is quantitatively defined in terms of amount of solute and not mass of solute. Indeed, we are dealing with a property directly related with entities 170
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in which W is the number of the different ways of distributing the energy of particles among levels that lead to the same state, and k ⫽ 1.38 ⫻ 10᎑23 J/K (Boltzmann’s constant). This relation shows that a small change in S corresponds to an enormous change in probability as measured by W. It is such a large change in probability that justifies maximum total entropy to mean the final (equilibrium) state towards which every system spontaneously and inevitably evolves, rapidly in some cases, slowly in other. Dissolution is a rapid process. For very soluble salts, maximum total entropy (equilibrium in a saturated solution) only occurs after a large amount of salt has been dissolved; for very low solubility salts, maximum total entropy is reached with a slight amount of salt dissolved. For example, by mixing 10 mol of NaCl (s) with 1 dm3 of water, at 25 ⬚C, about 6 mol of salt dissolves, whereas by adding an equal amount of CaCO3(s) to 1 dm3 of water, at the same temperature, only 1.4 ⫻ 10᎑4 mol of carbonate dissolves (6). Similarly, starting with 1 mol of each salt and sufficient water to produce 1 dm3 of solution, at 25 ⬚C, total dissolution occurs with NaCl, while only 0.014% of CaCO3 enters solution. Total dissolution of 1 mol of NaCl
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(to produce 1 dm3 of solution) is accompanied by an increase of total entropy, whereas total dissolution of 1 mol of CaCO3 (to produce 1 dm3 of solution) would correspond to a decrease of total entropy and, as mentioned above, this is so improbable that it can be considered impossible. The increase of entropy in the former case and the decrease that would correspond to the latter is usually interpreted in a simpler language: increase of (molecular) disorder and increase of (molecular) order, respectively. Considering an isolated system, any disordered state is a more (much more) probable state, hence the tendency to less-ordered states. If we consider closed systems, it is the molecular disorder of the system plus surroundings that must increase (until equilibrium is reached). The concept of molecular order or disorder is linked to the way the energy of particles (atoms, ions, molecules) are distributed among vibrational, rotational, and translational energy levels. A particle distribution that uses only a
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small number of energy levels is a state of high order, hence low entropy. The more the dispersion of particle energy among levels and the more the variety of energy levels involved (greater energy dispersion), the larger the number W of equivalent ways (microstates) of having the same distribution (the same state) by exchanging identical particles, as shown in Figure 1. Should all particles be assigned to the same energy level, W ⫽ 1, order would be maximum, and entropy would be zero (ln 1 ⫽ 0). It is in this sense that expressions like “the tendency of an isolated system to evolve towards the maximum disorder possible” should be taken. There are two major contributions to such changes of molecular disorder for an isolated system: 1. Changes of temperature, an increase of T implying a greater particle energy dispersion among levels, hence an increase of entropy, ∆ST ⬎ 0 (Figure 2) 2. Changes of particle mobility, for example through changes of volume of the space used by some of the particles (from solid to liquid solution, for instance; note that, for isolated systems, there is no change of the volume of the entire system), an increase implying an increase of entropy, ∆SV ⬎ 0
Figure 1. Six ways of distributing three identical particles, one in each energy level, as compared to the only way of assigning them to the same level.
The latter increase corresponds to a greater particle energy dispersion that arises because, on increasing V used by some particles of the system, the spacing between the corresponding higher translational energy levels becomes smaller and the higher energy levels become more accessible (Figure 3). We chose to speak of global disorder of an isolated system as made up of thermal disorder plus configurational disorder, the changes in the latter including not only the volume factor, essentially related to translational motion, but also eventual changes in the rotational and vibrational energy levels. Selecting Salts for Simulation The following criteria should be considered in choosing examples of salts and simulations for your students to explore: A. Familiar salts of very different solubility B. Salts with similar packing structures
Figure 2. Higher temperature means greater dispersion of particle energy among levels (hence higher entropy).
C. Salts whose solubilities are either not significantly affected by ion reaction with water (such as acid–base and complex formation reactions), or which are affected in a way that opposes the observed differences D. Pairs of salts that essentially require discussion of only configurational disorder or thermal disorder
Figure 3. Higher volume means more accessible translational energy levels, hence greater dispersion of particle energy (higher entropy).
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We start by comparing NaCl and CaCO3. The hydrolysis of CO32᎑ as a base affects positively the solubility of calcium carbonate (see item C above) and yet, CaCO3 is still less soluble than sodium chloride. Both dissolution phenomena are almost athermic, as inferred from the small dissolution enthalpy changes: ∆H°diss ⫽ 3.9 kJ mol᎑1 for NaCl and ∆H°diss ⫽ ᎑12.3 kJ mol᎑1 for CaCO3. Therefore, the major discussion is centered in configurational disorder. The fact that the dissolution of CaCO3 is slightly exothermic would indeed contribute positively to the solubility of the carbonate, and this is still less soluble than sodium chloride.
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point that the total entropy would decrease for any significant dissolution of calcium carbonate. Ion solvation is less relevant both in the case of the monocharged ions Na⫹ and Cl᎑ and in the case of the comparatively large ion CO32᎑. We stress that the simulations are qualitative; no rigorous calculations of the contributions to entropy changes are performed. In a similar manner, Figure 5 compares the dissolutions of MgCO3 and MgSO4, the former being slightly less soluble than CaCO3 whereas the latter is very soluble (solubility over 2 mol dm᎑3). Little difference in the changes (decrease) of configurational disorder occurs due to the solvation of the magnesium ion; the important difference is that the dissolution of magnesium sulfate is a comparatively high exothermic phenomenon (∆H°diss ⫽ ᎑91 kJ mol᎑1). When we consider the dissolution to occur in a closed system, keeping the temperature constant (298 K), the ∆H°diss value above corresponds to an energy transfer, as heat, from the system to its surroundings, with a consequent increase of entropy of those surroundings.
Figure 4. Strong solvation of Ca2⫹ and decrease of solvent entropy as the main factor responsible for the lower solubility of CaCO3 as compared to NaCl.
Figure 5. Increase of temperature as the main contribution to the increase of global disorder, hence greater solubility of MgSO4 as compared to MgCO3.
Figure 4 is a screenshot of a salt solubility simulation we developed that is available as JCE WebWare (see page 173 of this issue) and also at http://www.molecularium.net/salts (accessed Nov 2005). The computer program simulates, in a qualitative manner, the “before” and the “after” states for the dissolution of equal amounts of NaCl and CaCO3 in identical amounts of water. Total entropy increases in the case of NaCl due to the increase of configurational disorder (∆SV) of the ions of the salt. In the case of CaCO3, however, an identical effect is offset by the decrease of mobility of the water molecules associated with the dipositive calcium ion, to the
Conclusion Note that similar simulations can be used to facilitate the understanding of strong and weak acids, such as HCl and HF, this time considering the effect of the halogen ion size on the entropy change of the solvent through ion solvation. We think that the qualitative illustrations and discussions provided above are valuable contributions towards a deep understanding of chemical equilibrium. In our instruction we offer this qualitative experience to students before we teach any quantitative interpretation of equilibrium constants (e.g., in terms of Gibbs free energy changes). Because of the qualitative nature of the simulations, they are meant to be graphical “artistic conceptions”, rather than rigorous models: instructors should view the simulations only in this regard. In addition, when using such graphical simulations, students should be aware that the small number of particles represented and the large spaces between them are unrealistic, and that the symbolic representation of ions and molecules and qualitative illustrations of only some molecular motions are meant to emphasize the aspects under study. Acknowledgments The authors thank one of the referees for important suggestions and corrections. Literature Cited 1. 2. 3. 4.
Hawkes, S. J. J. Chem. Educ. 1998, 75, 1179. Gil, Victor M. S. Int. Newsletter Chem. Educ. 1986, 25, 17. Paiva, João C.; Gil, V. M. S. J. Chem. Educ. 2001, 78, 222. Paiva, João C. M.; Gil, Victor M. S.; Correia, António Ferrer J. Chem. Educ. 2003, 80, 111. 5. Lambert, F. L. J. Chem. Educ. 1999, 76, 1385. 6. CRC Handbook of Chemistry and Physics, 74th ed.; Lide, David R., Ed.; CRC Press: London, 1993.
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An overview of this contribution to the peer-reviewed JCE WebWare collection is available on pages 173–174 of this issue.
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