Chem 125 Lecture 37 12/10/08 This material is for the exclusive use of Chem 125 students at Yale and may not be copied or distributed further. It is not readily understood without reference to notes or the wiki from the lecture.

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 31

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

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) 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

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

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

“Reaction Coordinate” Diagram (for a one-step atom transfer) Not a trajectory, but 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.

Using Energies to Predict Equilibria and Rates for One-Step Reactions: Free-Radical Halogenation H CH 3 Cl H Cl CH 3 Cl CH 3 Cl Cl "free-radical chain"

Are Average Bond Energies “Real” or just a trick for reckoning molecular enthalpy ? Bond Dissociation Energies are real.

BondDissn Energies best values as of 2003

Ellison I Larger halogen  Poorer overlap with H (at normal bond distance) & less e-transfer to halogen H I H F less e-stabilization  weaker bond Diagram qualitative; not to scale.

Ellison II No special stabilization  SOMO orthogonal to  *) C - H bond unusually strong (good overlap from sp 2 C ) Vinyl C - H bond normal (sp 3 C, as in alkane) Allyl Special stabilization  SOMO overlaps  *) hard 111 Phenyl Ditto hard 113 easy 89 Ditto Benzyl easy 90 All H-Alkyl 100 ± 5 Same trend as H-Halogen Special Cases SOMO C   Are unusual BDE values due to unusual bonds or unusual radicals? or actually

H 3 C H + X X  H 3 C X + H X F Cl Br I ” Possibility of Halogenation (Equilibrium) CostReturnProfit

H 3 C H + X X  H 3 C X + H X Possibility of Halogenation (Equilibrium) F Cl Br I ” CostReturnProfit Is break-two-bonds-then-make-two a plausible Mechanism? at RT (~300K)? at ~3000K?  = /sec  = 250/sec How about rate (which depends on Mechanism)? No Way! Yes (unless there is a faster one)

H H 2 H 2 H HHHHHH H H H Henry Eyring (1935) Dissociation followed by association requires high activation energy. SLOW Make-as-you-break “displacement” is much easier. FAST

Free-Radical Chain Substitution X-HR-H X-X R-X X R cyclic machinery

H 3 C-H + X 2  HX + H 3 CX F Cl Br I ” Possibility of Halogenation (Equilibrium) CostReturnProfit H 3 C-H  HX X X2  H3CXX2  H3CX H 3 C Step Step (Mechanism for Reasonable Rate) How can we predict activation energy?

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 in it, 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 own chemical forbearers?

End of Lecture 37 Dec. 10, 2008 Good Luck on the Final!