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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

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Why calculate entropy?

Δn=nproducts – nreactants

(n=number of molecules)

For Δn=0 (isomerization): ΔGº ΔHº as ΔSº is nearly 0 For Δn0: ΔSº starts being important

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

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.

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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)

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How accurate is DFT in calculating entropies?

Molecule Label H2 a HF b CH4 c HCl d N2 e NH3 f :CH2 (triplet) g CO h HCN i HCCH j F2 k PH3 l CO2 m O=CH2 n H2C=CH2 o H3C-CH3 p cyclopropane q H2N-NH2 r HOCH3 s CH2=C=CH2 t ethylene oxide u O=CH(CH3) v H2C=CH(CH3) w

0 10 20 30 40 50-0.5

0.0

0.5

1.0

abc de

f

ghi

j

k

l m

p

q

r

s

t

u

v

w

Sob

s-S

calc (

cal m

ol-1

K-1)

Molecular Weight

No significant dependence of error on molecular weight No significant dependence of error on basis set

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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.

Molecule Calculated So (cal mol-1 K-1)

Point Group

CH4 44.48 Td 47.23 C3v CH2=C=CH2 57.84 D2h 59.26 C2 cyclopropane 56.57 D3h 58.76 C2v 60.14 C1

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.

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Entropy in 12 particle systems (298 K)

Average TΔS for all reactions: 9.38 kcal/mol Range: 7.38-12.66 kcal/mol

Reaction

Calculated TΔS

(kcal/mol)

TΔS (elec)

TΔS (trans)

TΔS (rot)

TΔS (vib)

N2H2 N2 + H2 7.38 0.00 8.31 -0.91 -0.02 CH2=C(H)CH3 H2 + CH2CCH2 7.68 0.00 8.33 -0.25 -0.40 OC(H)CH3 H2 + OCCH2 7.78 0.00 8.33 0.00 -0.55 OCH2 CO + H2 7.79 0.00 8.31 -0.49 -0.02 CH3OH H2 + OCH2 7.95 0.00 8.31 0.03 -0.39 N2H4 H2 + N2H2 7.96 0.00 8.31 0.43 -0.78 C2H4 H2 + HCCH 8.02 0.00 8.31 -0.57 0.29 OC(H)CH3 CO + CH4 8.64 0.00 9.81 -0.06 -1.12 C2H6 H2 + C2H4 8.68 0.00 8.31 0.79 -0.42 CH4 H2 + :CH2 (T) 9.94 0.65 8.25 1.04 0.00 C2H4 H2 + :CCH2 (S) 9.54 0.00 8.31 0.95 0.27 cyclopropane H2 + cyclopropene 9.70 0.00 8.33 1.39 -0.02 H2C=C(H)CH3 CH4 + :CCH2 (S) 10.26 0.00 9.79 1.22 -0.75 OCCH2 CO + :CH2 (T) 10.80 0.65 9.73 0.99 -0.57 H2CCH2 2 :CH2 (T) 12.25 1.30 9.48 1.59 -0.12 CH2CCH2 :CH2 (T) + :CCH2 (S) 12.52 0.65 9.71 2.51 -0.35 cyclopropane C2H4 + :CH2 (T) 12.66 0.65 9.73 2.49 -0.21

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12 particle reactions that produce H2 have TΔS = 81 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.

For an ideal gas, the translational contribution of entropy for independent particles as a function of pressure can be written as:

2

52ln

2/3

2 o

o

t P

kT

h

mkTNkS

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]

TΔS (kcal/mol)

TΔS (kcal/mol)

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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 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: 11.70 kcal/mol

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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.

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.

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.

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Application to 13 particle systems

Similar trends can be observed for 13 particle systems.

Largest contributor is the translational entropy—for 2 molecules of H2, it is (8.312)=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.

Reaction TΔS TΔS (elec)

TΔS (trans)

TΔS (rot)

TΔS (vib)

N2H4 2H2 + N2 15.34 0.00 16.62 -0.48 -0.80 CH3OH 2H2 + CO 15.74 0.00 16.62 -0.47 -0.42 CH2=C(H)CH3 H2 + :CH2 (T) + :CCH2 (S) 20.21 0.65 18.04 2.27 -0.75 CH2=C(H)CH3 3 :CH2 (S) 22.93 1.95 19.21 2.90 -1.14

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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 81 kcal/mol TΔS observed for 1st row compounds.

Rotational entropy is more important (especially for Si2H6), as the molecules are not planar.

Reaction TΔS TΔS

(trans) TΔS (rot)

TΔS (vib)

C2H6 H2 + H2C=CH2 8.68 8.31 0.79 -0.42 Si2H6 H2 + H2Si=SiH2 9.34 8.35 1.26 -0.26 Ge2H6 H2 + H2Ge=GeH2 8.84 8.36 1.34 -0.86 Sn2H6 H2 + H2Sn=SnH2 8.75 8.37 1.38 -1.00

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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.

Reaction TΔS TΔS

(trans) TΔS (rot)

TΔS (vib)

(PH3)2Ir(H)2(H2)Cl H2 + (PH3)2Ir(H)2Cl 10.28 8.37 0.87 1.05 (PH3)2Ir(H)2Cl H2 + (PH3)2IrCl 7.16 8.37 0.85 -2.06 (PH3)2Ir(H)2Cl HCl + (PH3)2IrH 8.51 10.82 1.40 -3.72 (PH3)3IrH (PH3)2IrH + PH3 10.42 10.77 3.10 -3.45 (PH3)3IrCl (PH3)2IrCl + PH3 10.62 10.78 3.70 -3.86

Ir

PH3

PH3

H

Cl

H

H2

-PH3

-H2

-HCl

-PH3

Ir

PH3

PH3

Cl

Ir

PH3

PH3

H

Ir

PH3

PH3

ClH

H

Ir

PH3

PH3

ClH3P

Ir

PH3

PH3

HH3P

-H2

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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?)

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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

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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.

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Conclusions and Acknowledgements

Conclusions: For 12 particle organic reactions that produce a linear

molecule, TΔS is 81 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