1 Entropy Explained: The Origin of Some Simple Trends Lori A. Watson a, Odile Eisenstein b a Department of Chemistry, Indiana University, Bloomington, IN b LSDSMS, Université Montpellier 2, Montpellier, France

2 Why calculate entropy? Δn=n products – n reactants (n=number of molecules)  For Δn=0 (isomerization): ΔGº  ΔHº as ΔSº is nearly 0  For Δn  0: ΔSº starts being important For the reaction: CaCO 3 (s) CaO (s) + CO 2 (g) ΔSº=38.0 cal/K ΔHº = 42.6 kcal ΔGº=31.3 kcal at 25 ºC ΔGº= -5.8 kcal at 1000 ºC  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 p article systems (298 K)  Average TΔS for all reactions: 9.38 kcal/mol Range: kcal/mol

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

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 H 2.  The rotational entropy, near zero when H 2 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 with nonlinear molecules with larger, multiple moments of inertia and correspondingly larger 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: 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 S vib values must be low-energy (low frequency) vibrations, such as rotation of a CH 3 group.  All molecules have an S elec 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 S elec of 0.65 kcal/mol at K.  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 S vib contributions are observed.  In other words, molecules lose their unique differences and become, nearly, billiard balls.

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 H 2, 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 1 st 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 1 st row compounds.  Rotational entropy is more important (especially for Si 2 H 6 ), 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