Chemistry 125: Lecture 37 December 8, 2010 Statistics / Equilibrium / Rate: Boltzmann Factor, Entropy, Mass Action, Transition State Theory The values of bond dissociation energies and average bond energies, when corrected for certain “effects” (i.e. predictable errors) can lead to understanding equilibrium and rate processes through statistical mechanics. The Boltzmann factor favors minimal energy in order to provide the largest number of different arrangements of “bits’ of energy. The slippery concept of disorder is illustrated using Couette flow. Entropy favors “disordered arrangements” because there are more of them than there are of recognizable ordered arrangements. The Law of Mass Action explains the exponentiation of concentration in equilibrium equations. Equilibrium ideas allow understanding reaction rates by using transition state theory instead of considering all trajectories on the potential energy surface. This semester’s study of structure and energy leads to next semester’s study of reaction mechanisms and synthesis. For copyright notice see final page of this file

Exam Help Sessions Friday, Dec. 10, 10:30-11:30 am (here) Sunday, Dec. 12, 3-5 pm Monday, Dec. 13, 7-9 pm Wednesday, Dec. 15, 8-10pm Thursday, Dec. 16, 7-9 pm

from Barney Ellison & his friends (good experiments!)  Average Bond Energy = / 4 = 99.4 kcal mol -1 No individual bond actually equals the “average” C-H bond. (because of changes in hybridization, etc.) Sum ± 0.6 Bond Strengths in CH 4 Heat of Atomization of CH 4 = kcal mol -1 (from heat of combustion, Chupka, etc.) Individual Bond Dissociation Energies (from spectroscopy etc.) CH 3 -H ± 0.03 CH 2 -H110.4 ± 0.2 CH-H101.3 ± 0.3 C-H80.9 ± 0.2

 ve Bond Energies How good are these average bond energies for calculating "Heats of Atomization” for other molecules? “2 nd C-C bond” 63 kcal/mole “3 rd bond” 54 kcal/mole From Streitwieser, Heathcock, & Kosower “2 nd C-O bond” kcal/mole! (Carbonyl group very “stable”) (  bond even weaker, since sp 2 -sp 2  bond is stronger)

 H Atomization by Additivity of Average Bond Energies? Ethene Ave. Bond Energy (kcal/mole) C-C C-H C=C C-O O-H c-Hexane c-Hexanol  -Glucose Seems Pretty Impressive! How accurate must you be to be useful? K calc = 10 -(3/4)(  H true +  H error ) K calc = 10 -(3/4)(  H calc ) K calc = K true  10 -(3/4)(  H error ) kcal error not % error determines K error factor To keep error less than  10 need <1.3 kcal/mole error!  Bond Energies 542  H atomization Error kcal/mole -4.3 Error % ~10× better than composition

 ve Bond Energies Can one sum bond energies to get accurate"Heats of Atomization"? H C O H C C H H H H H C O H C C H H H H Ketone "Enol" C O C H C O C H C=O179 C-C83 C-H99 sum361 C-O86 C=C146 O-H111 sum343 K calc = 10 -(3/4) 18 = K obs = = 10 -(3/4) 9.3 Bonds that change (the others should cancel in taking the difference)

H C O H C C H H H H H C O H C C H H H H Ketone "Enol" H Why is Enol 9 kcal/mole "Too" Stable? O C=O179 C-C83 C-H99 sum361 C-O86 C=C146 O-H111 sum343 K calc = 10 -(3/4) 18 = K obs = = 10 -(3/4) 9.3 C(sp 2 )-H stronger than C(sp 3 )-H (they shouldn’t actually cancel) Intramolecular HOMO-LUMO Mixing H C O H C C H H H H + "Resonance Stabilization” from

“Constitutional Energy” from bond additivity needs correction for effects such as: Resonance (HOMO/LUMO) Hybridization Strain C H H C H H H vs. HO C CH 2 H sp 2 sp 3 * * Polite name for error in simplistic scheme

Energy determines what can happen (equilibrium) K = e -  E/kT and how fast (kinetics) = 10 -(3/4)  E room Temp k (/sec) = e -  E /kT ‡ ‡ = (3/4)  E room Temp

What's so great about low energy? Statistics

Gibbs

Exponents & Three Flavors of Statistics 1) The Boltzmann Factor 2) The Entropy Factor 3) The Law of Mass Action

On the Relationship between the Second Law of Thermodynamics and Probability Calculation regarding the laws of Thermal Equilibrium (1877) S = k ln W Ludwig Boltzmann Considered the implications of random distribution of energy among real atoms. “I am conscious of being only an individual struggling weakly against the stream of time. But it still remains in my power to contribute in such a way that when the theory of gases is again revived, not too much will have to be rediscovered.”

Random Distribution of 3 “Bits” of Energy among 4 “Containers” How many “complexions” have N bits in the first container? 3 N#N# 3 1 2

Random Distribution of 3 “Bits” of Energy among 4 “Containers” How many “complexions” have N bits in the first container? 6 N#N#

How many “complexions” have N bits in the first container? 0 6 N#N# Random Distribution of 3 “Bits” of Energy among 4 “Containers” 10

0 6 N#N# bits of energy in 20 molecules 3 bits of energy in 4 “molecules” 30 in 20

(N)(N) E  (E) e -E/kT Boltzmann showed Exponential limit for lots of infinitesimal energy bits E ave = 1/2 kT If all “complexions” for a given E total are equally likely, shifting energy to any one degree of freedom of any one molecule is disfavored. By reducing the energy available elsewhere, this reduces the number of relevant complexions. Boltzmann Constant cal/moleK (Note: temperature is average energy)

Exponents & Three Flavors of Statistics 1) The Boltzmann Factor 2) The Entropy Factor 3) The Law of Mass Action

Disorder and Entropy "It is the change from an ordered arrangement to a disordered arrangement which is the source of the irreversibility.” The Feynman Lectures on Physics, Vol. I, 46-7

Disorder and Entropy Which is more ordered?

Disorder, Reversibility, & Couette Flow Click for webpage and "Magic" movie

Couette Flow If disorder is in the eye of the beholder, how can it measure a fundamental property? The rotated state only seemed to be disordered. Top View Ink line Syrup

Entropy is Counting in Disguise. “A disordered arrangement” seems to be an oxymoron. “A disordered arrangement” is code for a collection of random distributions whose individual structures are not obvious. It is favored at equilibrium, because it includes so many individual distributions. The situation favored at equilibrium has particles that have diffused every whichaway. every

Free Energy & entropy units K = e -  G/RT e -(  H - T  S)/RT e -  H /RT e T  S/RT e -  H /RT e  S/R e -  H /RT e R ln 2/R e -  H /RT e ln 2 e -  H /RT x e.u. (R ln 2) is a common  S. Conclusions: e.u. just means a factor of two. K depends on T because of  H, not  S. G (and S) sometimes obscure what is fundamentally simple. e.g. difference in entropy between gauche and anti butane Gauche Anti Gauche

3) The Law of Mass Action from counting random arrangements of a fixed number of energy bits 2) The Entropy Factor e TS/kT random Exponents & Three Flavors of Statistics 1) The Boltzmann Factor e -H/RT = W from counting W, the number of molecular structures being grouped R k Same thing: k is per individual molecule R is per mole (= k  N A ) “there’s a divinity that shapes our ends” Hamlet V:2

Cyclohexane Conformers few "structures" few "structures" many "structures" many quantum states few quantum states few quantum states Chair (stiff) Chair (stiff) Twist-Boat (flexible) kcal/mole Both classical and quantum views suggest a statistical "entropy" factor (of ~  7) favoring twist-boat. This reduces the room-temperature Boltzmann "enthalpy" bias of 10 - (3/4) 5.5 (= 14,000) in favor of chair to about 2,000.

Experimental Entropy Although we discuss entropy theoretically (in statistical terms), physical chemists can measure it experimentally. The entropy of a perfectly ordered crystalline material at zero Kelvin is zero ( ln 1 ). As the material is warmed it gains entropy in increments of (Heat Absorbed)/Temperature.  S =  H/T “Floppy” molecules with closely spaced energy levels absorb more energy, and at lower temperatures, and thus gain more S on warming. Cf. Ethane rotation - Lecture 33

3) The Law of Mass Action from counting random arrangements of a fixed number of energy bits 2) The Entropy Factor e TS/kT Exponents & Three Flavors of Statistics 1) The Boltzmann Factor e -H/RT = W from counting W, the number of quantum states being grouped K = e -  G/RT from counting molecules per volume weighted

Law of Mass Action Late 1700s : Attempts to assemble. hierarchy of “Affinities” Mid 1800s : Equilibrium “K” as balance of forward and reverse rates... Early 1800s : Amounts [concentration] can shift reaction direction away. from “affinity” prediction. …

[concentration] [A 2 ] [A] 2 = K 2 A A 2 [A 2 ][A] 2 = K Where does the exponent come from? Law of Mass Action

Randomly Distributed “Particles” # Particles# Dimers

[D] = K [P] 2 Randomly Distributed “Particles” # Particles# Dimers # of Particles # of Dimers Increasing concentration increases both the number of units and the fraction of units that have near neighbors. number fraction

Equilibrium, Statistics & Exponents Particle Distribution : Law of Mass Action [A 2 ] [A] 2 = K Energy Distribution :  H, Boltzmann Factor K  e -  H/RT Counting Quantum States :  S K  e  S/R

Free energy determines what can happen (equilibrium) K = e -  G/RT = 10 -(3/4)  G room Temp But how quickly will it happen? (kinetics) Energy & Entropy

Classical Trajectories & The Potential Energy Surface Visualizing Reaction

Time-Lapse “Classical” (Molecular Mechanics) Trajectory for non-reactive collision of 13 atoms 6 molecules 40 Dimensions (3n + time) by E. Heller faster slower heavier lighter rotating slowly rotating rapidly & vibrating Too Complicated (for our purposes)

Potential Energy “Surface” for Stretching Diatomic Molecule A-B A-B Distance Potential Energy Rolling Ball Maps A-B Vibration

Potential Energy Surface for Linear Triatomic A-B-C Cliff Pass (Transition State or Transition Structure) Plateau Valley ridge + maximum minimum * * So 2-D specifies structure

Vibration of A-B with distant C spectator Slice and fold back Potential Energy Surface for Linear Triatomic A-B-C Vibration of B-C with distant A spectator

Unreactive Trajectory: (A bounces off vibrating B-C) Potential Energy Surface for Linear Triatomic A-B-C

C flies away from vibrating A-B Reactive Trajectory A approaches non-vibrating B-C Potential Energy Surface for Linear Triatomic A-B-C “classical” trajectory (not quantum)

H 3 Surface Henry Eyring (1935) Crazy angle of axes means that classical trajectories can be modeled by rolling marble. Transition State (“Lake Eyring”)

H + H-Br

John McBride (1973) “I wanted to catch a little one”

Studying Lots of Random Trajectories Provides Too Much Detail Summarize Statistically with Collective Enthalpy (H) & Entropy (S)

“steepest descent” path Slice along this path, then flatten and tip up to create… (not a trajectory)

“Reaction Coordinate” Diagram (for one-step transfer of atom B) Not a realistic trajectory, but rather a sequence of three species Starting Materials Products Transition “State” G each with H and S, i.e. Free Energy (G)

Free Energy determines what can happen (equilibrium) K = e -  G/RT = 10 -(3/4)  G room Temp and how rapidly (kinetics) k (/sec) = e -  G /RT ‡ ‡ = (3/4)  G room Temp Amount of ts (universal) Velocity of ts theory Since the transition state is not truly in equilibrium with starting materials, and the velocity is not universal, the theory is approximate.

(1959) (1953) (2007)

Organic Chemistry Paul D. Bartlett Physical

Jack Dunitz: At the time when I was reading that book I was wondering whether chemistry was really as interesting as I had hoped it was going to be. And I think I was almost ready to give it up and do something else. I didn't care very much for this chemistry which was full of facts and recipes and very little thought, very little intellectual structure. And Pauling's book gave me a glimpse of what the future of chemistry was going to be and particularly, perhaps, my future. 1939

The Chemical Bond Is there an Atomic Force Law? Feeling & Seeing Molecules and Bonds Understanding Bonding & Reactivity through H  = E  How chemists learned to treasure Composition, Constitution, Configuration, Conformation and Energy

Is there an Atomic Force Law? Feeling & Seeing Molecules and Bonds Understanding Bonding & Reactivity through H  = E  How chemists learned to treasure Composition, Constitution, Configuration, Conformation and Energy The Chemical Bond How does science know? Compared to what? Were chemical bonds discovered or invented? Some Big Questions: Would we even have chemical bonds without our particular chemical history?

End of Lecture 37 Dec. 8, 2010 Copyright © J. M. McBride 2009, Some rights reserved. Except for cited third-party materials, and those used by visiting speakers, all content is licensed under a Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0).Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0) Use of this content constitutes your acceptance of the noted license and the terms and conditions of use. Materials from Wikimedia Commons are denoted by the symbol. Third party materials may be subject to additional intellectual property notices, information, or restrictions. The following attribution may be used when reusing material that is not identified as third-party content: J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0