Entropy Explained: The Origin of Some Simple Trends

1. Entropy Explained: The Origin of Some Simple Trends Lori A. Watsona, Odile Eisensteinb aDepartment of Chemistry, Indiana University, Bloomington, IN 47405 bLSDSMS, Université Montpellier 2, Montpellier, France

2. For the reaction: CaCO3 (s) CaO (s) + CO2 (g) ΔSº=38.0 cal/K ΔHº = 42.6 kcal ΔGº=31.3 kcal at 25 ºC ΔGº= -5.8 kcal at 1000 ºC Why calculate entropy? Δn=nproducts – nreactants (n=number of molecules) • For Δn=0 (isomerization): ΔGº  ΔHº as ΔSº is nearly 0 • For Δn0: ΔSº starts being important • Δn > 0 predicts ΔSº > 0, but it’s harder to know the magnitude of ΔSº • Many textbook examples exist where ΔSº opposes ΔHº and so ΔGº depends on the temperature.

3. Why use Density Functional Theory? • DFT is… • A relatively time-inexpensive computational method • Capable of calculating most elements in the periodic table • Used heavily by practicing chemists • Able to give highly accurate energies and structures of most molecules • Includes electron correlation—the fact that electrons in the molecule react to one another • Additionally… • Modern packages have easy to use graphical interfaces • Introduces the student to an important area of research—Computational Chemistry • “Breaks down” molecular properties (like entropy) into their components (like vibrational entropy)

4. How accurate is DFT in calculating entropies? • No significant dependence of error on molecular weight • No significant dependence of error on basis set

5. But be careful of molecular symmetry! • The symmetry number, σ, is different and incorrectly computed for molecules in different point groups, making the entropy incorrect by a factor of Rln(1/ σ). • There is confusion as to which frequency to remove when going from a non-linear molecule to a linear molecule. • Commercial programs will optimize the geometry of your molecule in the point group you submit it in (even if it’s not the “right” one!). • An incorrect point group, while giving you nearly identical geometric parameters, will result in very incorrect entropies.

6. Entropy in 12 particle systems (298 K) • Average TΔS for all reactions: 9.38 kcal/mol Range: 7.38-12.66 kcal/mol

7. TΔS (kcal/mol) For an ideal gas, the translational contribution of entropy for independent particles as a function of pressure can be written as: TΔS (kcal/mol) Graph of translational entropy contributions (at 298.15 K) to a reaction system with daughter particles of mass x and y (amu) [slice at x=2] 12 particle reactions that produce H2 have TΔS = 81 kcal/mol at 298 K • Reactions that produce H2 as one of the two particles have an average entropy change of 8.4 kcal/mol, largely determined by the translational entropy.

8. Why is there more entropy in reactions without linear molecules? • When the mass of one of the daughter particles is not 2, the translational entropy will be slightly higher than the 8.31 kcal/mol observed with H2. • The rotational entropy, near zero when H2 was liberated, is now increasing. • Look at the shape of the molecules—none are linear. • Linear molecules with smaller moments of inertia have small rotational partition functions and small contributions to Sº compared withnonlinear moleculeswith larger, multiple moments of inertia and correspondinglylarger contributions to Sº. • Average of TΔSº for: • 2 linear molecules produced: 7.73 kcal/mol • 1 linear molecules produced: 8.40 kcal/mol • 0 linear molecules produced: 11.70 kcal/mol

9. The role of vibrational and electronic entropy • Vibrational entropy only plays a significant role in the overall reaction entropy if the number of low frequency vibrations changes significantly from reactant to product. • The vibrations that play the largest role in the calculated Svib values must be low-energy (low frequency) vibrations, such as rotation of a CH3 group. • Usually, the change in vibrational entropy is near zero, reflecting the small change in rigidity of the reactant and product molecules. In some cases, larger Svib contributions are observed. • In other words, molecules lose their unique differences and become, nearly, billiard balls. • All molecules have an Selec contribution of Rln(g) (where g is the degeneracy of the spin multiplicity (g=2S+1)—zero for a singlet!). So for molecules which are ground state triplets, there is an added Selec of 0.65 kcal/mol at 298.15 K.

10. Application to 13 particle systems • Similar trends can be observed for 13 particle systems. • Largest contributor is the translational entropy—for 2 molecules of H2,it is (8.312)=16.62 kcal/mol • Translational contribution increases with heavier products; rotational contribution increases with non-linear products. • Somewhat larger negative vibrational entropies are observed, consistent with loss of easy rotation around C-C single bonds.

11. Extension to heavier main group compounds • Hypothesis: Vibrational contributions of entropy should be more important because heavier analogues of 1st row compounds have lower vibrational modes associated with them. • Conclusion: Vibrational contributions make no significant difference in the 81 kcal/mol TΔS observed for 1st row compounds. • Rotational entropy is more important (especially for Si2H6), as the molecules are not planar.

12. Entropy calculations for transition metal systems • Entropic contributions can make a large difference in the spontaneity of organometallic reactions. • For reactions that produce a linear molecule of low molecular weight, TΔS remains near 8 kcal/mol. • For non-linear molecule producing reactions, or when the product molecule has a particularly low energy vibration, a value of 10 kcal/mol is a good “back of the napkin” number. • Increase in vibrational entropy reflects the “softer” nature of metal–to-ligand bonds.

13. Why would you use this in your classroom? • “Doing science” means observing and then explaining trends in recorded measurements • Here, students must “observe” reaction entropies and “explain trends” based on their knowledge of molecular structure and vibrational frequencies. • A student project based on exploring entropy complements existing discussions of… • Thermodynamics (when is a reaction favored?) • Statistical mechanics (what molecular properties influence the observed value?) • Quantum mechanics (can an “approximate” wave function generate useful and relevant predictions of molecular properties?)

14. What will this teach my students? • Experimental design • What reactions will be calculated? Why? • Modern computational methods • What factors—method, basis set, input symmetry, etc.—will influence the result? • Writing about chemistry • What trends are expected? Observed? Why? A good example of Discovery Based Learning in the curriculum

15. What will I need to do this? • A computational package that can perform DFT calculations and some mathematical software for plotting. • For example: Gaussian 98 (has the option of a graphical user interface) and Maple • Access to at least one desktop PC or UNIX system (for organic molecules) or a larger computing system (for larger inorganic molecules) • One or two lecture periods to explain the basics of computational chemistry and DFT • A recitation or lab period to give a short demonstration of the software • This project could be carried out as a class (assigning different molecules to each student), as a lab team, or as an individual assignment/project.

16. Conclusions and Acknowledgements • Conclusions: • For 12 particle organic reactions that produce a linear molecule, TΔS is 81 kcal/mol. • Rotational entropy increases TΔS for non-linear products. • Molecular identity is less important. • Trends are mirrored for main group and transition metal species. • The use of modern computational methods to explore trends in chemical systems introduces students to discovery based learning and a new area of research. • Acknowledgements: • Kenneth G. Caulton, Ernest R. Davidson, and Odile Eisenstein • National Science Foundation and Indiana University Chemistry Department